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Topology of arrangements and representation stability. Abstracts from the workshop held January 14–20, 2018. (English) Zbl 1409.00084
Summary: The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research.
00B05 Collections of abstracts of lectures
00B25 Proceedings of conferences of miscellaneous specific interest
14N20 Configurations and arrangements of linear subspaces
20F55 Reflection and Coxeter groups (group-theoretic aspects)
32S22 Relations with arrangements of hyperplanes
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55P15 Classification of homotopy type
55P62 Rational homotopy theory
55R80 Discriminantal varieties and configuration spaces in algebraic topology
57M07 Topological methods in group theory
52-06 Proceedings, conferences, collections, etc. pertaining to convex and discrete geometry
32-06 Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces
55-06 Proceedings, conferences, collections, etc. pertaining to algebraic topology
14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry
20F65 Geometric group theory
Full Text: DOI
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[21] W. van der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980), no. 3, 269–295. 52Oberwolfach Report 2/2018 Configuration spaces, KZ connections and conformal blocks Toshitake Kohno The purpose of this report is to clarify a relation among the following three approaches for linear representations of braid groups. (1) Homological representations (Burau and Lawrence–Krammer–Bigelow (LKB) representations). (2) Monodromy representations of Kniznhnik–Zamolodchikov (KZ) connections. (3) Generalized Jones representations arising from R matrices in the theory of quantum groups. We first recall basic notions concerning hyperplane arrangements. LetA = H1, . . . , Hℓ be an arrangement of affine hyperplanes in the complex vector spaceS Cn. We consider the complement M (A) = Cn\H∈AH. LetL be a complex rank one local system over M (A) associated with a representation of the fundamental group r : π1(M (A), x0)−→ C∗. We denote by fjbe a linear form defining the hyperplane Hj, 1≤ j ≤ ℓ. We associate a complex number aj= a(Hj) called an exponent to each hyperplane and consider a multivalued function Φ = f1a1· · · fℓaℓ. The associated local system is denoted byLΦ. We choose a smooth compactification i : M (A) −→ X. We shall say that the local system L is generic if and only if there is an isomorphism i∗L ∼= i!L where i∗is the direct image and i!is the extension by 0. If the local systemL is generic in the above sense, then there is an isomorphism H∗(M (A), L) ∼= H∗lf(M (A), L) and we have Hk(M (A), L) = 0 for any k6= n. Here H∗lfstands for the homology with locally finite chains. Let Dnbe a disk with n-punctured points and consider the configuration space of unordered distinct m points in Dn, which is denoted by Conf(m, Dn). We have H1(Conf(m, Dn); Z) ∼= Z⊕n⊕ Z. Consider the homomorphism α : H1(Conf(m, Dn); Z)−→ Z ⊕ Z defined by α(x1, . . . , xn, y) = (x1+· · ·+xn, y). Composing with the abelianization map, we obtain the homomorphism β : π1(Conf(m, Dn), x0)−→ Z⊕ Z. We denote by eCn,mthe abelian covering of Conf(m, Dn) corresponding to Ker β. The homology group H∗( eCn,m; Z) is considered to be a Z[Z⊕ Z]-module by deck transformations. We express Z[Z⊕Z] as the ring of Laurent polynomials R = Z[q±1, t±1]. We put Hn,m= Hm( eCn,m; Z), which is a free R-module. There is a homomorphism ρ : Bn−→ AutRHn,m called the homological (LKB) representation of the braid group. The case m = 1 corresponds to the Burau representation. Let g be a complex semi-simple Lie algebra andIµ be an orthonormal basis of g with respect to the Cartan–Killing form. Let ri: g→ End(VPi), 1≤ i ≤ n, be representations of g. We consider the Casimir element Ω =µIµ⊗ Iµand denote Topology of Arrangements and Representation Stability53 by Ωijthe action of Ω on the i-th and j-th components of V1⊗ · · · ⊗ Vn. We set 1X κΩijd log(zi− zj),κ∈ C 0. i,j The 1-form ω defines a flat connection for a trivial vector bundle over Xn, the configuration space of ordered distinct n points in C with fiber V1⊗ · · · ⊗ Vn. As the holonomy we have representations of pure braid groups θκ: Pn−→ Aut(V1⊗ · · · ⊗ Vn), which are called the monodromy representations of KZ connections. In the following, we consider the case g = sl2(C) with the standard basis H, E, F . For a complex number λ we denote by Mλthe Verma module of g with highest weight vector v such that Hv = λv and Ev = 0. For an n-tuple Λ = (λ1,· · · , λn)∈ Cnwe set|Λ| = λ1+· · · + λn. We consider the tensor product Mλ1⊗ · · · ⊗ Mλn. For a non-negative integer m we set W [|Λ| − 2m] = x ∈ Mλ1⊗ · · · ⊗ Mλn; Hx = (|Λ| − 2m)x and define the space of null vectors by N [|Λ| − 2m] = x ∈ W [|Λ| − 2m] ; Ex = 0. The KZ connection ω commutes with the diagonal action of g on Mλ1⊗ · · · ⊗ Mλn and acts on the space of null vectors N [|Λ| − 2m]. For parameters κ and λ we consider the multi-valued function Φn,m=(zi− zj)2κ(ti− zℓ)−λℓκY(ti− tj)2κ 1≤i<j≤n1≤i≤m,1≤ℓ≤n1≤i<j≤m defined over Xn+m. LetL be the local system over Xn+massociated to the multivalued function Φn,m. We denote by π : Xm+n→ Xnthe projection defined by π(x1,· · · , xn, t1,· · · , tm) = (x1,· · · , xn). Let Xn,mdenote a fiber of π and put Yn,m= Xn,m/Sm, where Smacts as the permutation of coordinates. Let us notice that Yn,mis homotopy equivalent to Conf(m, Dn). We denote byL the induced local system on Yn,m. The symbolL∗stands for the dual local system of L. We have the monodromy of the KZ connection θκ,λ: Pn→ Aut N[|Λ| − 2m]. On the other hand, we have a homological representation ρn,m: Pn→ Aut Hm(Yn,m,L∗). By using a construction of horizontal sections of KZ connections by hypergeometric integrals using Φn,mdue to Schechtman and Varchenko [6], we can construct a period map φ : Hm(Yn,m,L∗)−→ N[|Λ| − 2m]∗. It turns out that φ is an isomorphism for generic parameters λ, κ and is equivariant with respect to the action of the pure braid group Pn. We fix a complex number λ and consider the case λ1=· · · = λn= λ. Then the above representation is the 54Oberwolfach Report 2/2018 √√ specialization of the LKB representation with q = e−2π−1λ/κ,t = e2π−1/κ (see [4]). A relation between the monodromy representations of KZ connections and R matrices in the theory of quantum groups Uh(g) was originally found in [2] and [1]. By defining the action of Uh(g) on chains with local system coefficients and identifying the action of E∈ Uh(g) with the twisted boundary operator, we can recover the quantum group symmetry in homological representations. Finally, we briefly discuss a relation to conformal field theory (see [3]). We consider the affine Lie algebrabg = g ⊗ C((ξ)) ⊕ Cc with the commutation relation [X⊗ f, Y ⊗ g] = [X, Y ] ⊗ fg + Resξ=0df ghX, Y ic. We fix a positive integer K called a level. For an integer λ with 0≤ λ ≤ K we can associate the integrable highest weight moduleMλ, which is an irreduciblebg module containing Vλand c acts as K· id. We call such λ a level K highest weight. We consider the Riemann sphere CP1with n+1 marked points p1,· · · , pn, pn+1, where pn+1=∞. We assign level K highest weights λ1,· · · , λn, λn+1to p1,· · · , pn, pn+1. We denote byMpthe set of meromorphic functions on CP1with poles at most at p1,· · · , pn+1. The space of conformal blocks is defined as the space of coinvariants HΣ(p, λ) =Hλ1⊗ · · · ⊗ Hλn+1/(g⊗ Mp) where g⊗ Mpacts diagonally via Laurent expansions at p1, . . . , pn+1. The space of conformal blocks forms a vector bundle over Xnwith the KZ connection such that κ = K + 2. By means of horizontal sections of the KZ connection using hypergeometric integrals we can construct a period map φ : Hm(Yn,m,L∗)→ H(p, λ)∗. with m =12(λ1+· · · + λn− λn+1) . This might not be a generic case and the period map is not an isomorphism in general. There is a subtle point concerning fusion rules and resonance at infinity. We refer the reader to [5] for recent progress on this aspect. References
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[128] J. D. Wiltshire-Gordon. On computing the eventual behavior of a finitely presented FImodule. 2016. Arrangements and Artin groups Michael J. Falk We give an introduction to the general theory of complex hyperplane arrangements and its origins in the study of finite reflection groups and their discriminants. We state some solved and still open problems, in particular involving the Milnor fibration. We present recent constructions of the author with E. Delucchi [4], and from work in progress with D. Ernst and S. Riedel, of finite combinatorial models for the complements of general complexified real arrangements, and of complements of discriminants of finite real reflection groups, respectively. Natural generalizations to Artin groups and pure Artin groups on one hand, and to finite complex reflection groups and complex braid groups on the other, should be of interest in the study of representation and homological stability. Complex hyperplane arrangements. A complex hyperplane arrangement is a finite setA of linear hyperplanes in V = Cℓ. For H∈ A choose αQH: V→ C, a nonzero linear form satisfying H = ker(αH). The product Q :=H∈ASαHis called the defining polynomial ofA. The union of A, D := Q = 0 =H∈AH is an affine algebraic variety with a (homogeneous) singularity at 0 for ℓ≥ 2, isolated only if ℓ = 2. The complement M := V D of A is a non-compact 2ℓ-manifold, connected but not simply-connected. In case all αHcan be chosen to have real coefficients, we sayA is a complexified real arrangement. Then one has real hyperplanes HR= H∩ Rℓfor H∈ A comprising the associated real arrangementARin Rℓ. As an example, consider the arrangementA in V = C3with defining polynomial Q = (x− y)(x − z)(y − z). The complement of A consists of ordered 60Oberwolfach Report 2/2018 triples of distinct points in the plane; that is, M is the ordered configuration space Conf(R2, 3). All three hyperplanes contain the line L given by x = y = z, soA determines an arrangement of hyperplanes in the quotient vector space V /L. This is a complexified real arrangement, whose real part is pictured in Figure 1. 0 C Figure 1.The Coxeter arrangement of type A2. A quick sketch of the general theory. Since the αHare homogeneous, they define hyperplanes H in complex projective space Pℓ−1, comprising the associated projective arrangementA, with complement M. The projectivization map M→ M is a trivial C×-bundle, and M is diffeomorphic to the complement of an arrangement of|A| − 1 affine hyperplanes in Cℓ−1. This “deconing” process is useful for induction arguments [2]. One motivation for research in the field is the theorem of Orlik and Solomon
[129] : the cohomology ring H∗(M, C) has a presentation that depends only on the combinatorics ofA. Here the “combinatorics of A” means the function dTAgiven by dA(S) = dimC(H∈SH), for S⊆ A. There are arrangements with different combinatorics but isomorphic cohomology rings; a complete classification of these rings has not yet been accomplished. There are known presentations for the arrangement group π1(M ), all depending on a choice of coordinates. Rybnikov [6] showed that π1(M ) is not determined by dA; the counterexample is a pair of thirteen-line (projective) arrangements with no real form. For complexified real arrangements one has, for instance, the Salvetti complexS of A, a finite regular cell complex of dimension ℓ with the homotopy type of M , determined by the stratification of Rℓcoming fromAR.S is the nerve of a partially-ordered set on the set of pairs (C, F ), where C is a chamber ofARand F is a face of C. The boundary of a typical top-dimensional cell, corresponding to the pair (C, 0), is illustrated in Figure 1. Recently a pair of complexified real arrangements with the same combinatorics but different arrangement groups has been found [1], resolving a long-standing open problem. These arrangements also have 13 lines. Milnor fibers. The restriction Q : M→ C×of the defining polynomial Q is a locally trivial fibration, known as the Milnor fibration; the fiber F =Q = 1 is the Milnor fiber ofA. It is an open question whether the betti numbers bi(F ) Topology of Arrangements and Representation Stability61 are determined by dA, although the author has a conjectural solution for the case i = 1. Since C×is aspherical, F has the homotopy type of the Z-cover of M classified by Q∗: π1(M )→ π1(C×) = Z. The cyclic group of order n acts freely on F , by scalar multiplication, with orbit space M . It is also an open problem whether the monodromy homomorphism H1(F )→ H1(F ) is determined by dA. Coxeter arrangements and discriminants. SupposeA is a complexified real arrangement. For H∈ A, let sH: Rℓ→ Rℓdenote the orthogonal reflection across the hyperplane HR. If sH(KR)∈ ARfor every H, K∈ A, A is a Coxeter arrangement. This is the case for the example in Figure 1. The finite real reflection group W generated bysH| H ∈ A is a Coxeter group. It acts on V and the space of orbits is isomorphic to Cℓ; coordinates on the orbit space are given by a set of homogeneous invariant polynomials for W . The image ∆Wof D under the orbit map Cℓ→ Cℓis the discriminant associated with W . The fundamental group of Cℓ ∆Wis the Artin group associated with the Coxeter presentation of W —see [2]. These spaces are aspherical [3]. In the example, W is the symmetric group S3and the associated Artin group is the full braid group on three strands. The Salvetti complexS can be constructed equivariantly, so as to yield a finite cell complex with the homotopy type of Cℓ ∆W, and hence a finite model for the associated Artin group of finite type. Some new models. In [4] we defined a variation on the Salvetti complex for complexified real arrangements. One defines a partial ordering on the set Q of ordered pairs of chambers ofAR: (R, S)≤ (U, V ) if there is a minimal gallery from R to S that can be extended to a minimal gallery from U to V . The nerve of this poset is homotopy equivalent to M . IfA is a Coxeter arrangement, the poset Qcarries an action of W , and yields a model for the complement Cℓ ∆Wof the discriminant, and hence for the associated finite-type Artin group. This model is the nerve of the acyclic category with set of objects W and morphisms u→ v labelled by pairs (x, y) of group elements giving a reduced factorization v = xuy relative to the Coxeter generators. The composite of morphisms (x, y) : u→ v and (s, t) : w→ u is (xs, ty): w → y. In Figure 2 we illustrate the category for W of sts = tst s66BB
hht tsst OOjj❚❚❚❚❚❥❥❥❥44OO ❚❚❥❥ ❚❚❥❥ t❚❚❥❥❥ ❥❥❚❚s ❥❚❚❚❚ ❥❥ ❥❥❚❚ s❥❥❚❚t
jjBB44 st e Figure 2.A model for the braid group on three strands. 62Oberwolfach Report 2/2018 type A2, the example in Figure 1. Only non-identity indecomposable morphisms are pictured; the morphism (x, y) is labeled x or is unlabelled if x = e. References
[130] E. Artal Bartolo, B. Guerville-Balle, and J. Viu-Sos. Fundamental groups of real arrangements and torsion in the lower central series quotients. Available at arXiv:1704.04152, (2017).
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[134] P. Orlik and L. Solomon. Topology and combinatorics of complements of hyperplanes. Inventiones mathematicae, 56:167–189, 1980. · Zbl 0432.14016
[135] G. Rybnikov. On the fundamental group of a complex hyperplane arrangement. DIMACS Technical Reports, 13, 1994. Polynomial functors and homological stability Christine Vespa (joint work with Aur´elien Djament) The definition of polynomial functors on a category of modules over a ring R has been introduced by Eilenberg and Mac Lane [5] using the notion of cross-effects. A typical example of a polynomial functor of degree n is the n-th tensor power Tn: R-M od→ R-Mod defined by Tn(V ) = V⊗n. The definition of Eilenberg and Mac Lane can easily be extended to functors on a monoidal category whose unit is a null object. Several natural functors having polynomial properties are defined only on monoidal categories (M, ⊕, 0) whose unit 0 is an initial object but is not a terminal object. Examples of such categories are • the category (F I, ∐, ∅) of finite sets and injections; • the category (S(Z), ⊕, 0) having as objects the finitely generated free abelian groups and as morphisms S(Z)(Z⊕n, Z⊕m) =(u, v)∈ Ab(Z⊕n, Z⊕m)× Ab(Z⊕m, Z⊕n)/v◦ u = Id where Ab is the category of abelian groups, • the homogeneous category associated to braid groups (Uβ, ∐, ∅). (See [7] for the definition of homogeneous category. For examples of polynomial functors on this category see [9] or the extended abstract of Souli´e in this report). In [4] we introduce two notions of polynomial functors on a symmetric monoidal category whose unit is an initial object, extending the original definition of Eilenberg and Mac Lane: the strong polynomial and the weak polynomial functors. This talk is an overview of [4]. The strong polynomial functors are related to representation stability by the following proposition. Topology of Arrangements and Representation Stability63 Proposition 1. [4] Let F be a functor from F I to Ab. The functor F is strong polynomial with finitely generated values iff it is finitely generated. To describe stable phenomena the weak polynomial degree is more suitable than the strong polynomial degree. For example, the functor T≥in: F I→ Ab defined by T≥in(k) = Tn(Zk) for k≥ i and 0 otherwise, is strong polynomial of degree n + i and weak polynomial of degree n. Stably this functor behaves as Tn. Let (M, ⊕, 0) be a small symmetric monoidal category where 0 is an initial object and generated by an object x (i.e. for each object m∈ M there exists k∈ N such that m ≃ x⊕k). For example F I is generated by 1 the set having one element. 1. Strong polynomial functors 1.1. Definition. Let F unc(M, Ab) be the category of functors from M to Ab. The shift functor τx: F unc(M, Ab) → F unc(M, Ab) is defined by τx(F ) = F (x⊕ −). Since 0 is initial, there is a unique map 0 → x inducing a natural transformation ix: Id→ τx. The cokernel of this transformation is the difference functor denoted by δxand the kernel is the evanescence functor denoted by κx. Definition 2. A functor F :M → Ab is strong polynomial of degree ≤ d if δxd+1F = 0. If the unit 0 is also terminal, ixsplits so κx= 0. We recover the definition of usual polynomial functors using the difference functor (see for example [6]). This definition is equivalent to the original definition of Eilenberg and Mac Lane. 1.2. Examples. • The constant functor Z : F I → Ab defined by Z(k) = Z ∀k ∈ N is strong polynomial of degree 0. • The atomic functor Zi: F I→ Ab defined by Zi(k) = Z for k = i and 0 otherwise, is strong polynomial of degree i. • The functor Z≥i: F I→ Ab defined by Z≥i(k) = Z for k≥ i and 0 otherwise, is strong polynomial of degree i. Since Z≥i(k) is a subfunctor of Z we deduce that the category of strong polynomial functors of degree≤ d is not closed under subobjects. By Proposition 1, the examples of finitely generated F I-modules given in [1] are examples of strong polynomial functors. 2. Weak polynomial functors Stably the functors Z and Z≥iare equal. We will introduce a quotient of the category F unc(M, Ab), named the stable category, in which these two functors are equal and we will define polynomial functors in this quotient category. 64Oberwolfach Report 2/2018 2.1. The stable category St(M, Ab). A functor F : M → Ab is stably zero if colimF (x⊕n) = 0. For example, Ziis stably zero. We denote bySN (M, Ab) the n∈N full subcategory of F unc(M, Ab) of stably zero functors. The category SN (M, Ab) is a thick subcategory of F unc(M, Ab) so we can give the following definition. Definition 3. The stable category St(M, Ab) is the quotient category F unc(M, Ab)/SN (M, Ab). We denote by πMthe functor F unc(M, Ab) → F unc(M, Ab)/SN (M, Ab). The functor κxtakes its values inSN (M, Ab). Definition 4.(1) A functor F∈ St(M, Ab) is polynomial of degree ≤ d if δxd+1F = 0. (2) A functor F∈ F unc(M, Ab) is weak polynomial of degree ≤ d if πM(F ) is polynomial of degree≤ d. A strong polynomial functor of degree d is weak polynomial of degree≤ d. For example, the functor Z≥iis strong polynomial of degree i and weak polynomial of degree 0. The converse of the previous statement is not true. For example, theL functorZ≥iis weak polynomial of degree 0 but is not strong polynomial. i∈N If the unit 0 is also terminal St(M, Ab) = F unc(M, Ab). In this case the notions of strong polynomiality, weak polynomiality, polynomiality in St(M, Ab) and polynomiality in the sense of Eilenberg and Mac Lane are equivalent. The category of polynomial functors of degree≤ d in St(M, Ab) (denoted by P old(M, Ab)) is thick. In [4] we study the quotient categories P old(M, Ab)/P old−1(M, Ab). Note that in [2] the authors call “stable degree” the weak polynomial degree. 2.2. Examples. (1) In [3] (see also the extended abstract of Djament in this report) Djament computes the weak polynomial degree of the homology of congruence subgroups. (2) Let φ : Aut(Fn)→ GLn(Z) be the map induced by the abelianisation and IAn= ker(φ), Djament gives the following conjecture. Conjecture 5. The functor Hk(IA•) : S(Z)→ Ab is weak polynomial of degree 3k. (3) Let γk+1be the lower central series, ψ : Aut(Fn)→ Aut(Fn/γk+1(Fn)) andAk(Fn) = ker(ψ). We have a functorAk/Ak+1: S(Z)→ Ab. Proposition 6. [4, Proposition 6.3] The functorAk/Ak+1: S(Z)→ Ab is weak polynomial of degree k + 2. The keystone of the proof of this proposition is the description of the cokernel of the Johnson homomorphism for Aut(Fn) given by Satoh in [8]. Topology of Arrangements and Representation Stability65 References
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[144] A. Souli´e, The Long–Moody construction and polynomial functors, ArXiv: 1702.08279 Homology of braid group with coefficients in symplectic representations Filippo Callegaro (joint work with Mario Salvetti) We consider the family of hyperelliptic curves Edn:=(P, z, y) ∈ Cn×D × C|yd= (z− x1)· · · (z − xn). where D is the unit open disk in C, Cnis the configuration space of n distinct unordered points in D and P =x1, . . . , xn ∈ Cn. For each configuration P∈ Cn the equation yd= (z− x1)· · · (z − xn) defines a curve that we call Σdn. Each curve Σdnin the family is a d-fold covering of the disk D ramified along the set P and there is a fibration π : Edn→ Cnwhich takes Σdnonto its set of ramification points. The bundle π : Edn→ Cnhas a global section, so H∗(Edn) splits as a direct sum H∗(Cn)⊕ H∗(Edn, Cn) and by the Serre spectral sequence H∗(Edn, Cn) = H∗−1(Brn; H1(Σdn)), where Brnis the classical Artin braid group on n strands. The surface Σdnhas Euler characteristic χ = d− n(d − 1) and the number of connected component of the boundary is gcd(n, d). In particular when d = 2 the surface Σdnhas genusn−12if n is odd andn−22if n is even. The representation of the group Brnon the fundamental group of the surface Σ2nis described in [5] (see also [4]). Following some ideas in [1] we can project the space Ednto the product Cn×D, that decomposes as a union of two open sets: the first one is homotopy equivalent to the configuration space C1,nof n distinct marked points and one additional distinguished point in D, the second one is homotopy equivalent to the configuration space C1,n-1and their intersection is homotopy equivalent to the product 66Oberwolfach Report 2/2018 C1,n-1×S1. This induces a decomposition of Edn. The associated Mayer-Vietoris long exact sequence can be used to compute the homology of Edn. The rational homology of the space Ednhas been computed in [3]. Our main results give a complete description of the homology of E2nfor n odd: Theorem 1. For odd n : (1) the integral homology Hi(Brn; H1(Σ2n; Z)) has only 2-torsion. (2) the rank of Hi(Brn; H1(Σ2n; Z)) as a Z2-module is the coefficient of qitnin the series Pe2(q, t) =qt3Y1. (1− t2q2)1− q2i−1t2i i≥0 In particular the series eP2(q, t) is the Poincar´e series of the homology groupM H∗(Brn; H1(Σ2n; Z)) nodd as a Z2-module. The homology group H1(Σdn) can be seen as a polynomial coefficient system for Brn(see [6] for a definition of polynomial coefficient system). Hence the homology computed in the previous theorem stabilizes. For d = 2 the stable homology is described in the following result. Theorem 2. Let us consider homology with integer coefficients. (1) The homomorphism Hi(Brn; H1(Σ2n))→ Hi(Brn+1; H1(Σ2n+1)) is an epimorphism for i≤n2− 1 and an isomorphism for i <n2− 1. (2) For n even Hi(Brn; H1(Σ2n)) has no p torsion (for p > 2) whenppi+3≤ n −1 and no free part for i + 3≤ n. In particular for n even, when3i2+ 3≤ n the group Hi(Brn; H1(Σ2n)) has only 2-torsion. (3) The Poincar´e polynomial of the stable homology Hi(Brn; H1(Σ2n; Z)) as a Z2-module is the following: P2(Br; H1(Σ2))(q) =qY1. 1− q21− q2j−1 j≥1 When d is greater than 2 the same argument gives a partial description of the homology of Ednand of its stabilization. References
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[150] N. Wahl and O. Randal-Williams. Homological stability for automorphism groups, 2017. ArXiv:1409.3541v4. Homology of surface and graph braid groups Ben Knudsen (joint work with Byung Hee An and Gabriel C. Drummond-Cole) For a topological space X, we consider the unordered configuration space of k points in X, which is the quotient  Bk(X) =(x1, . . . , xk)∈ Xk: xi6= xjif i6= j/Σ. k ‘ It is convenient to consider the graded space B(X) =k≥0Bk(X). Example. If X is an aspherical surface or a graph, then Bk(X) is a classifying space for its fundamental group, the kth surface or graph braid group, respectively. When the background space X is a manifold, we have the following calculation
[151] , which unifies and extends partial results of [3, 6]. Theorem (K). Let M be an n-manifold. There is an isomorphism of bigraded Abelian groups H∗(B(M ); Q) ∼= HL(gM), where gMis the graded Lie algebra defined by ( Hc−∗(M ; Qw)⊗ vn odd gM= Hc−∗(M ; Qw)⊗ v ⊕ Hc−∗(M ; Q)⊗ [v, v]n even. Here, • HLdenotes Lie algebra homology, • Hcdenotes compactly supported cohomology, • Qwdenotes the orientation sheaf of M , and • v and [v, v] are formal parameters in bigrading (n − 1, 1) and (2n − 2, 2), respectively. The Lie algebra homology in question may be computed by means of the classical Chevalley–Eilenberg complex. This complex is very amenable to computation; for example, we are able to determine explicit formulas for dim Hi(Bk(Σ); Q) for every i, k≥ 0 and surface Σ [5]. Remark. The dual of the Chevalley–Eilenberg complex for gMcoincides with the direct sum over k of the Σk-invariant part of the E2page of the spectral sequence considered in [11]. Thus, our result may be interpreted as asserting the vanishing of all higher differentials in these spectral sequences of Σk-invariants. In contrast, the full spectral sequence is known not to collapse in general. 68Oberwolfach Report 2/2018 The Lie algebra homology of any Lie algebra is naturally a cocommutative coalgebra. Consideration of this structure leads to an alternate proof of homological stability for configuration spaces [4]. Corollary. Suppose that M is connected and n > 1. The cap product with 1∈ H0(M ; Q)⊆ H∗(B(M ); Q)∨induces an isomorphism Hi(Bk+1(M ); Q)−→ H≃i(Bk(M ); Q) for i≤ k and a surjection in the next degree. The proof is based on elementary combinatorial facts about the Chevalley– Eilenberg complex. If M is not an orientable surface, a slightly better stable range obtains. We turn now to the case of a graph Γ with set of vertices V , set of edges E, and set of half-edges H. For v∈ V , we write S(v) = Zh∅, v, h ∈ H(v)i, where H(v) denotes the set of half-edges incident on v. The Swiatkowski complex of Γ is the Abelian group O S(Γ) = Z[E]⊗S(v), v∈V endowed with the differential determined by the equation ∂(h) = e(v)− v(h) and the bigrading determined by declaring that|∅| = (0, 0), |v| = |e| = (0, 1), and |h| = (1, 1). We prove the following [1]. Theorem (A–D-C–K). There is a natural isomorphism of bigraded Z[E]-modules H∗(B(Γ); Z) ∼= H∗(S(Γ)). The complex S(Γ) is a functorial and algebraic enhancement of the cellular chains on the cubical model considered in [10] (see also [8]). The Z[E]-action arises geometrically from the process of edge stabilization, which replaces a subconfiguration on the edge e with the collection of pairwise averages of the points in this subconfiguration and the endpoints (a similar stabilization mechanism for trees is considered in [9]). This structure provides a natural setting in which to study analogues of classical homological stability phenomena. Corollary. The bigraded Z[E]-module H∗(B(Γ); Z) is finitely presented. This algebraic structure is usually rather complicated; indeed, S(Γ) is formal as a Z[E]-module if and only if each component of Γ is homeomorphic to a graph in which each vertex has at most two edges [2]. References
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[162] B. Totaro, Configuration spaces of algebraic varieties, Topology 35, 1996. Topology of Enumerative Problems: Inflection Points on Cubic Curves Weiyan Chen A plane cubic curve is given by the vanishing locus of a complex homogeneous polynomial F (x, y, z) of degree 3. It is a classical result that every smooth cubic plane curve has exactly 9 inflection points (also called “flexes”), i.e. points where the Hessian vanishes. In other words, every smooth cubic plane curve naturally comes with 9 marked points. Motivated by this classical result, Benson Farb asked the following question: Question 1 (Farb). What are all the possible ways to continuously choose n distinct unlabeled points on any smooth cubic plane curve, as the curve varies in family? To make this question precise, we define the following space parameterizing smooth cubic curves: X := F (x, y, z) : F is a homogeneous polynomial of degree 3 and is smooth/ ∼ where F∼ cF for any c ∈ C×. Each F∈ X gives a well-defined smooth cubic curve CFin CP2. This construction gives the following fiber bundle overX : CFE ξ X  where the total space E :=(F, p)∈ X × CP2: p∈ CFcan be viewed as the universal cubic curve. Each fiber CFis a Riemann surface of genus 1. 70Oberwolfach Report 2/2018 Definition 1. A multisection for ξ of degree n is a triple ( eX , p, i) where p : eX → X is a cover of degree n, and i : eX → E is a continuous injection such that p = ξ ◦ i, making the following diagram commute: n pointsXepX (1)i= CFEX ξ Thus, a multisection is a continuous choice of n distinct points on the cubic curve CFas F varies in the familyX . Sometimes we will just call eX a multisection when p and i are clear from the context. Example 1 (Inflection points). Define eXflexto be Xeflex:=(F, q) ∈ X × CP2: q is an inflection point on CF. Let pflex: eXflex→ X be the projection onto the first factor. Then eXflexdefines a multisection for ξ of degree 9. Question 1 asks for a classification of multisections of ξ. To aim for a partial answer, Farb made the following conjecture: Conjecture 1 (Farb). There is no multisection for ξ of degree n < 9. It turns out that ξ does admit multisections of degree n > 9: Example 2 (A multisection of degree 36). Define eX36to be n Xe36:=(F, q)∈ X × CP2: o q is a point on CFwhose tangent line passes through a flex Let p : eX36→ X be the projection onto the first factor. Then eX36defines a multisection for ξ of degree 36. Let us make the following two observations from Example 2. First, eX36has an intermediate cover which is exactly eXflex. Thus, the multisection eX36“factors through” the multisection of 9 flexes. Second, if we choose a flex p on CFto be the identity for the elliptic curve, then the multisection eXflexpicks the 3-torsions of (CF, p), while eX36picks the 6-torsions of (CF, p). Thus, eX36comes from an algebraic construction. Therefore, what is behind Conjecture 1 is the following metaconjecture: Metaconjecture 2 (Farb). There is no multisection for ξ unless there is an algebraic reason for it to exist. Topology of Arrangements and Representation Stability71 To state the main theorem, we need to first introduce a cover eXncfofX :  Xencf:=F,p, q, r∈ X × Sym3(CP2) :p, q, r is a triple of  three non-collinear infection points on CF.  3= 84 unordered triples of flexes, 12 of which are collinear. We will say a cover eX1/X factors through another cover eX2/X if the later is an intermediate cover of the former. Theorem 1. If eX /X gives a multisection of ξ, then each connected component of X must factor through either eeXflex/X or eXncf/X . We already knew that eXflexis a multisection of degree 9 (Example 1). However, currently it is not known whether eXncfcan be made into a multisection or not. What is missing is the injection i in the commutative diagram (1). Question 2. Does there exist a continuous injective map i : eXncf→ E making the diagram (1) commute? Equivalently, is it possible to associate 72 distinct points xp,q,rto the 72 triplesp, q, r such that the choice varies continuously with (F,p, q, r) ∈ eXncf? Either a positive or a negative answer to Question 2 will be very interesting because: • If eXncf/X is not a multisection, then Theorem 1 implies that every multisection must factor through eXflex, and therefore eXflexis the universal multisection. • If eXncf/X is a multisection given by certain algebraic construction, then every smooth cubic curve naturally has 72 special points on it. These 72 special points are perhaps as interesting as the 9 flexes. • If eXncf/X is a multisection that is continuous but is not from any algebraic construction, then it is a counter-example to Farb’s Metaconjecture 2. Theorem 1 implies that Farb’s Conjecture 1 is true. In fact, it implies the following corollary which is stronger than Conjecture 1: Corollary 2. The bundle ξ admits no multisection of degree n if n is not a multiple of 9. Proof. The degree of a cover is multiplicative when two covers are composed. Either eXflex/X or eXncf/X is of degree a multiple of 9. Thus, Theorem 1 implies that degree of any multisection eX /X must be a multiple of 9. Every smooth cubic plane curve is a Riemann surface of genus 1, and thus is an elliptic curve. However, Corollary 2 implies that the bundle ξ does not admit a multisection of degree 1, or equivalently, that it is not possible to continuously 72Oberwolfach Report 2/2018 choose one point on every smooth cubic curve to serve as the identity. Therefore, we conclude: Corollary 3. It is not possible to continuously choose an elliptic curve structure for all smooth cubic plane curves. References
[163] W. Chen, Topology of Enumerative Problems: Inflection Points on Cubic Curves, work in progress. Milnor monodromy of plane curves, space surfaces and hyperplane arrangements Alexandru Dimca (joint work with Gabriel Sticlaru) Let C : f (x, y, z) = 0 be a reduced plane curve in the complex projective plane P2, defined by a degree d homogeneous polynomial f in the graded polynomial ring S = C[x, y, z]. The smooth affine surface F : f (x, y, z) = 1 in C3is called the Milnor fiber of f . The mapping h : F→ F given by (x, y, z) 7→ (θx, θy, θz) for θ = exp(2πi/d) is called the monodromy of f . There are induced monodromy operators hj: Hj(F, C)→ Hj(F, C), hj(ω) = (h−1)∗(ω), for j = 0, 1, 2, and we can look at the corresponding characteristic polynomials ∆jC(t) = det(t· Id − hj|Hj(F, C)). Given the degree d reduced plane curve C : f = 0 and a d-th root of unity λ6= 1, how to determine the multiplicity m(λ) of λ as a root of the Alexander polynomial ∆1C(t)? This question, or higher dimensional versions of it, has a very long tradition, see for instance [1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 16]. A general answer is given by the following result, see [5, 6, 8, 10, 15]. Let Kf∗= (Ω∗, df∧) be the Koszul complex of the partial derivatives fx, fy, fzof f in S. Theorem 1. For any integer k with 1≤ k ≤ d, there is an E1- spectral sequence E∗(f )ksuch that E1s,t(f )k= Hs+t+1(Kf∗)td+k and converging to E∞s,t(f )k= GrPsHs+t(F, C)λ, where P∗is a decreasing filtration on the Milnor fiber cohomology, called the pole order filtration. Moreover, E2(f )k= E∞(f )kif and only if C has only weighted homogeneous singularities. In fact this result is valid for projective hypersurfaces of any dimension. In the case of plane curves this gives the following, see [8, 10]. This results says that for a plane curve the computations of a limited number of the terms in the second page of the spectral sequence is enough to determine all the Alexander polynomials ∆jC(t). Topology of Arrangements and Representation Stability73 Theorem 2. Let C : f = 0 be a reduced degree d curve , and let λ = exp(−2πik/d), with k∈ (0, d) an integer. Then λ is a root of the Alexander polynomial ∆1C(t) of multiplicity m(λ) given by m(λ) = dim E21,0(f )k+ dim E1,02(f )k′. In the case of hyperplane arrangements in Pn, one has the following Conjecture. For any hyperplane arrangement V : f = 0 in Pnand any integer k with 1≤ k ≤ d, d being the number of hyperplanes in V , the E1- spectral sequence E∗(f )kdegenerates at the second page, i.e. E2(f )k= E∞(f )k. Moreover, one has Es,t2(f )k= 0 for t > 1, see [10]. A lot of examples suggest that this conjecture holds, and this is very important for doing computations in terms of computing time. The corresponding SINGULAR codes are available at http://math1.unice.fr/ dimca/singular.html. These codes are very effective for plane curves, especially for the free and nearly free curves, as explained in [8, 10]. Indeed, for plane curves, the graded cohomology group H2(Kf∗) is determine by the graded S-module of Jacobian syzygies AR(f ) =(a, b, c) ∈ S3| afx+ fy+ cfz= 0, which is free for a free curve, and has a very precise resolution in the case of a nearly free curve, see [9]. There are many interesting relations of the above results with the roots of the Bernstein–Sato polynomials for projective hypersurfaces, which can be found in [10, 15]. References
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[168] A. Dimca, On the Milnor fibrations of weighted homogeneous polynomials, Compositio Math. 76 (1990), 19–47. · Zbl 0726.14002
[169] A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. · Zbl 0753.57001
[170] A. Dimca, Hyperplane Arrangements: An Introduction, Universitext, Springer-Verlag, 2017 · Zbl 1362.14001
[171] A. Dimca, G. Sticlaru, A computational approach to Milnor fiber cohomology, Forum Math. 29(2017), 831– 846. · Zbl 1384.32024
[172] A. Dimca, G. Sticlaru, Free and nearly free curves vs. rational cuspidal plane curves, Publ. RIMS Kyoto Univ. 54 (2018), 163–179. · Zbl 1391.14057
[173] A. Dimca, G. Sticlaru, Computing Milnor fiber monodromy for some projective hypersurfaces, arXiv:1703.07146. · Zbl 1416.32014
[174] H. Esnault, Fibre de Milnor d’un cˆone sur une courbe plane singuli‘ere, Invent. Math. 68 (1982), 477–496. 74Oberwolfach Report 2/2018 · Zbl 0475.14018
[175] A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes, Duke Math. J. 49 (1982), 833–851. · Zbl 0524.14026
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[179] A. Suciu, Hyperplane arrangements and Milnor fibrations, Annales de la Facult´e des Sciences de Toulouse 23 (2014), 417–481. Long-Moody constructions and generalizations Arthur Souli´e In [4], Long and Moody gave a construction on representations of braid groups which associates a representation of Bnwith a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. This construction inspires endofunctors, called Long–Moody functors, on a suitable category of functors (see [7]). This construction also generalizes to other families of groups such as automorphism groups of free groups, mapping class groups of orientable and non-orientable surfaces or mapping class groups of 3-manifolds (see [8]). Moreover adapting notions of strong polynomial functors in this context, the Long–Moody functors increase by one the degree of polynomiality (see [7, 8]). 1. Case of braid groups 1.1. Categories. The braid groupoid β has the natural numbers n∈ N as objects and the braid groups Bnas automorphisms. A monoidal product ♮ : β× β → β is defined assigning the usual addition for the objects and connecting two braids side by side for the morphisms (see [5]). The object 0 is the unit of this monoidal product. The strict monoidal groupoid (β, ♮, 0) is braided, its braiding is denoted by b−,−. Remark that a family of representations of braid groups is this way equivalent to define a functor β→ K-Mod where K-Mod is the category of K-modules for Ka commutative ring. Note that all the classical representations of braid groups, such as Burau representations (see [2]), Tong–Yang–Ma representations (see [9]) or Lawrence–Krammer representations (see [3]) satisfy compatibility relations when one passes from Bnto Bn+1. This motivates the use of Quillen’s bracket construction over β. Definition 1.1. [6] The category Uβ is defined by: • Obj (Uβ) = Obj (β) = N; • HomUβ(n, n′) = colim[Homβ(−♮n, n′)] . β Topology of Arrangements and Representation Stability75 Proposition 1.2. [6] The category Uβ satisfies the following properties. • The unit 0 is initial object. We denote by ιn: 0→ n the unique morphism from 0 to N . • ♮ extends to give a monoidal structure (Uβ, ♮, 0). This category is not braided but pre-braided (see [6]). In [6, 7], it is proven that the families of Burau, Tong-Yang-Ma and LawrenceKrammer representations define functors over the category Uβ. 1.2. Definition. Consider an: Bn→ Aut (Fn) a Wada representation (see [10]), where Fnis the free group on n generators. Let ςn: Fn→ Fn⋊Bn→ Bn+1be a an group morphism. Denote byIK[Fn]the augmentation ideal of the free group Fn. Long–Moody functors are defined by: Theorem 1.3. [4, 7] Let G∈ Obj (Fct (Uβ, K-Mod)). Assign: • ∀n ∈ N, LM (G) (n) = IK[Fn]⊗G (n + 1). K[Fn] • For [n′− n, σ] ∈ HomUβ(n, n′): LM (G) ([n′− n, σ]) = an′(σ)⊗G (id1♮ [n′− n, σ]) . K[Fn′] Then we define this way an object of Fct (Uβ, K-Mod) and this naturally extends to give an exact functor: LM : Fct (Uβ, K-Mod)→ Fct (Uβ, K-Mod) . Using the Artin representation as an, the induced Long–Moody functor recovers the unreduced Burau functor, and its iteration allows to obtain Lawrence– Krammer as subfunctor. Another choice of andefines a Long–Moody functor which recovers the Tong–Yang–Ma functor. 2. Generalizations We can generalize the principle of Long–Moody functors to other families of groups. Consider (G, ♮, 0) a braided monoidal groupoid, such that Obj (G) = N. We denote the automorphism groups ofG by Gn. Quillen’s construction U applies in the same way as before and defines a pre-braided homogenous category (UG, ♮, 0). Let Hmm∈Nbe a family of free groups with injections Hm֒→ Hm+1. Let us make the following analogy: • Bn←→ Gn • Fn←→ Hn • (Bn→ Aut (Fn))←→ (Gn→ Aut (Hn)) • (Fn→ Fn⋊Bn→ Bn+1)←→ (Gn→ Hn⋊Gn→ Hn+1) Theorem 2.1. [8] Repeating mutatis mutandis the assignments of Theorem 1.3, we define an exact functor: LM : Fct (UG, K-Mod) → Fct (UG, K-Mod) . Applications 2.2. [8] The following families of groups fit into this framework. 76Oberwolfach Report 2/2018 • The automorphism groups of free groups Aut (Fn). We can obtain the abelianization functor thanks to a Long–Moody functor. • The mapping class groups of compact orientable connected surfaces with genus g and one boundary componentΓg,1g∈N. For example, a Long– Moody functor recovers the family of symplectic representations. • The mapping class groups of compact, connected, oriented 3-manifold with boundary. This includes handlebody mapping class groupsHn,1or symmetric automorphisms of free groups ΣAut (Fn). 3. Polynomial behaviour 3.1. Polynomial functors. The notions of strong and weak polynomial functors for symmetric monoidal categories are introduced by Djament and Vespa in [1]. This is extended to the case of a pre-braided monoidal category in [7]. We take up the framework, definitions and terminology of the extended abstract of Vespa in this report. Recall that F∈ Obj (Fct (UG, K-Mod)), the shift, difference and evanescence functors define a short exact sequence: 0→ κ1F→ F → τ1F→ δ1F→ 0. Definition 3.1. F is very strong polynomial of degree 0 if it is constant. For d≥ 1, F is very strong polynomial of degree ≤ d if it is strong polynomial of degree≤ d, κ1F = 0 and δ1F is very strong polynomial of degree≤ d − 1. The concept of very strong polynomial functor corresponds to the one of coefficient system of finite degree at 0 in the terminology of [6]. 3.2. Effect of Long–Moody functors. Theorem 3.2. [8] Let LM be any generalized Long–Moody functor. It induces a functor: LM :Polstrongd(UG) → Polstrongd+1(UG) . If F∈ Obj (Fct (UG, K-Mod)) is very strong (resp. weak) polynomial of degree ≤ d, then LM (F ) is very strong (resp. weak) polynomial of degree ≤ d + 1. Thus, the Long–Moody constructions will provide new examples of twisted coefficients fitting into the framework developed by Randal-Williams and Wahl in
[180] where prove homological stability for different families of groups, in particular for braid groups, mapping class groups of surfaces and 3-manifolds. References
[181] A. Djament and C. Vespa, Foncteurs faiblement polynomiaux, to appear in IMRN, ArXiv: 1308.4106
[182] C. Kassel and V. Turaev, Braid groups, Graduate Texts in Mathematics, Springer, New York, 2008 · Zbl 1208.20041
[183] T. Kohno, Quantum and homological representations of braid groups, Communications in Analysis and Geometry, 217–238, 1994
[184] D. Long, Constructing representations of braid groups, Configuration spaces, Geometry, Combinatorics and Topology, 355–372, 2012 Topology of Arrangements and Representation Stability77
[185] S. Mac Lane, Categories for the working mathematician, Springer Science & Business Media, Volume 5, 2013 · Zbl 0232.18001
[186] O. Randal-Williams and N. Wahl Homological stability for automorphism groups, Adv. Math. 318 (2017), 534–626 · Zbl 1393.18006
[187] A. Souli´e, The Long–Moody construction and polynomial functors, ArXiv: 1702.08279
[188] A. Souli´e, The generalized Long–Moody functors, ArXiv: 1709.04278
[189] D. Tong and S. Yang and Z. Ma, A New Class of Representations of Braid Groups, Communications in Theoretical Physics, 483-486, 1996 · Zbl 1002.20500
[190] M. Wada, Group invariants of links, Topology, 399–406, 1992 Introduction to twisted commutative algebras Andrew Snowden 1. Overview In representation stability, one often has a sequence of representations connected by some kind of transition maps. Such a structure can often be conveniently encoded as a representation of a category. Recall that a representation of a category C is a functor from C to the category of vector spaces. Here are some examples of the kinds of categories that come up, and some sample applications of them. NameDefinitionApplication FIfinite sets / injectionsCohomology of configuration spaces [1] FIdfinite sets / injections withConfiguration spaces of graphs [6] a d-coloring on the com-Syzygies of Segre embeddings [10] plement of the image FIMfinite sets / injections withSecondary stability [4] a perfect matching on theRepresentations of O∞
[191] complement of the image OIfinite totally ordered sets /Homology of unipotent groups [5] order preserving injections FSopfinite sets / surjectionsHomology ofMg,n
[192] (opposite category) VA(F)finite dimensional vectorSteenrod algebra [3] spaces over F / linear maps Representations of the first three categories are equivalent to modules over three specific twisted commutative algebras (tca’s). The second three categories are not directly related to tca’s. We hope this gives the reader some insight into the place that tca’s occupy within representation stability. 78Oberwolfach Report 2/2018 2. Twisted commutative algebras: three definitions We now give three equivalent ways to define tca’s. Fix a commutative ring k. Definition 1. A twisted commutative algebra is a graded associative unital k-L algebra A =∞n=0Anequipped with an action of the symmetric group Snon An such that the following two conditions hold: (1) The multiplication map An×Am→ An+mis Sn×Sm⊂ Sn+mequivariant. (2) (Twisted commutativity.) Given x∈ Anand y∈ Am, we have yx = τ (xy) where τ∈ Sn+mis the element given by τ (i) = i + m for 1≤ i ≤ n and τ (i) = i− n for n + 1 ≤ i ≤ n + m. Definition 2. Let FB be the groupoid of finite sets and bijections. A tca is then a lax symmetric monoidal functor A : FB→ Modk, where the monoidal structure on the source is disjoint union and on the target is tensor product. Precisely, this means A is a functor assigning to every finite set S a k-module AStogether with maps AS⊗ AT→ AS∐T(this is the “lax monoidal” part) such that the diagram AS⊗ AT//AS∐T  AT⊗ AS// AT∐S commutes, where the vertical maps are the given isomorphisms (this is the “symmetric” part). The commutativity of this diagram corresponds to the twisted commutativity axiom in Definition 1. Definition 3. A representation of S∗is a sequence M = (Mn)n≥0where Mn is a representation of Snover k. If M and N are two representations of S∗, we define their tensor product to be the representation of S∗given by M (M⊗ N)n=IndSSni×Sj(Mi⊗kNj). i+j=n There is a natural isomorphism M⊗ N → N ⊗ M; this makes use of the element τ in Definition 1. In this way, the category Repk(S∗) of representations of S∗ has a symmetric monoidal structure. A tca is just a commutative algebra object in this tensor category; that is, it is an object A of Repk(S∗) equipped with a multiplication map A⊗ A → A and a unit map k → A satisfying the usual axioms. We note that there is a notion of module over a tca. From the perspective of Definition 3, a module is just a module object in the general sense of tensor categories. Examples. We now give some examples of the definitions: (1) Let A be the graded k-algebra k[t], where t has degree 1. We regard this as a tca by letting Snact trivially on each graded piece. An A-module is a representation M of S∗equipped with maps Mn→ Mn+1(multiplication Topology of Arrangements and Representation Stability79 by t). This looks a lot like what one gets from an FI-module, and, in fact, one can show that the category of A-modules is equivalent to the category of FI-modules. (2) Let V be a k-module and let A be the tensor algebra on V , equipped with its usual grading. We let Snact on An= V⊗nby permuting tensor factors. Then A is a tca. If we regard V as a representation of S∗concentrated in degree 1 then A is in fact the symmetric algebra on V (in the tensor category Repk(S∗)). If V is a free module of rank d then A-modules are equivalent to FId-modules. (3) Taking the perspective of Definition 2, let ASbe the free k-module on the set of matchings on S. (A matching is an undirected graph in which each vertex belongs to precisely one edge.) The multiplication map AS⊗ AT→ AS∐Tis induced by taking the disjonit union of matchings. This is a tca. In fact, it is the symmetric algebra on the trivial representation of S2, regarded as an object of Repk(S∗) concentrated in degree 2. Modules for this tca are equivalent to FIM-modules. 3. Structure theory One of the main problems in the subject of tca’s is to understand the structure of module categories. Some examples of the kinds of problems and results that have been studied: (1) One of the main problems is noetherianity: if A is a finitely generated tca over a noetherian coefficient ring, is the module category ModAlocally noetherian? This is known for the three examples given above (although only in characteristic 0 for the third example), and this nearly exhausts the list of known results. Draisma [2] has proven a topological version of noetherianity in general, which strongly suggests that ModAis locally noetherian in general. (2) If M is a module over a tca (and k is a field), its Hilbert series is definedP to be HM(t) =n≥0dim(Mn)tn!n. One would like to know the form of this series. Much is known for modules over the three example tca’s: for example, for Example 2 the Hilbert series is a polynomial in t and et. For more general tca’s, not much is yet known. (3) There are a whole manner of finer structural results for modules over the three example tca’s, many of which are analogous to classical results in commutative algebra. For example, there is a theory of local cohomology and depth. See [8, 9]. References
[193] Thomas Church, Jordan Ellenberg, Benson Farb. FI-modules and stability for representations of symmetric groups. Duke Math. J. 164 (2015), no. 9, 1833–1910. http://arxiv.org/ abs/1204.4533v3 · Zbl 1339.55004
[194] Jan Draisma. Topological Noetherianity of polynomial functors. http://arxiv.org/abs/ 1705.01419 80Oberwolfach Report 2/2018
[195] Nicholas J. Kuhn. Generic Representations of the Finite General Linear Groups and the Steenrod Algebra: I. Amer. J. Math. 116 (1994), no. 2, 327–360. · Zbl 0813.20049
[196] Jeremy Miller, Jennifer Wilson. Higher order representation stability and ordered configuration spaces of manifolds. http://arxiv.org/abs/1611.01920
[197] Andrew Putman, Steven V Sam, Andrew Snowden. Stability in the homology of unipotent groups. http://arxiv.org/abs/1711.11080 · Zbl 1319.05146
[198] Eric Ramos. Configuration spaces of graphs with certain permitted collisions. http://arxiv. org/abs/1703.05535
[199] Steven V Sam, Andrew Snowden. Stability patterns in representation theory. Forum. Math. Sigma 3 (2015), e11, 108 pp. http://arxiv.org/abs/1302.5859v2 · Zbl 1319.05146
[200] Steven V Sam, Andrew Snowden. GL-equivariant modules over polynomial rings in infinitely many variables. Trans. Amer. Math. Soc. 368 (2016), 1097–1158. http://arxiv.org/abs/ 1206.2233 · Zbl 1436.13012
[201] Steven V Sam, Andrew Snowden. GL-equivariant modules over polynomial rings in infinitely many variables. II http://arxiv.org/abs/1703.04516 · Zbl 1436.13012
[202] Andrew Snowden. Syzygies of Segre embeddings and ∆-modules. Duke Math. J. 162 (2013), no. 2, 225–277. http://arxiv.org/abs/1006.5248
[203] Philip Tosteson. Stability in the homology of Deligne-Mumford compactifications. http: //arxiv.org/abs/1801.03894 Configuration space in a product John D. Wiltshire-Gordon This talk explains how to compute the homotopy type of Conf(n, X× Y ) using the homotopy types of certain configuration spaces in X and Y separately. First, note that the homotopy types of Conf(n, X) and Conf(n, Y ) alone will not be enough to recover the homotopy type of Conf(n, X× Y ). For example, when X =0, 1 and Y = R, the inclusion X ⊂ Y induces an equivalence Conf(2,0, 1) ≃ Conf(2, R) but Conf(2, R× R) 6≃ Conf(2, 0, 1 × 0, 1). To avoid this pitfall, our theorem relies on a richer kind of configuration space. If X is a space and Γ is a graph, define the graphical configuration space Conf(Γ, X) =f : V (Γ) → X so that a ∼Γb =⇒ f(a) 6= f(b) . Writing GI for the category of finite graphs with injections, we have a functor Conf(−, X): GIop→ Top given by relabeling vertices and forgetting both vertices and edges. The restriction of this functor along the complete graph functor K : FI→ GI recovers the usual FIop-structure on configuration space. We introduce a new category called GI2to help with configuration space in a product. It is defined as the full subcategory of GI×GI spanned by pairs of graphs (Γ′, Γ′′) for which V (Γ′) = V (Γ′′). In other words, an object of GI2is a pair of graph structures on the same underlying set of nodes. The union functor U : GI2→ GI Topology of Arrangements and Representation Stability81 sends a pair (Γ′, Γ′′) to the graph on the same vertex set as Γ′and Γ′′that includes every edge that appears in either graph. Theorem 1. The natural map from the homotopy left Kan extension along Uop L(Uop)!Conf(−, X) × Conf(−, Y )→ Conf(−, X × Y ) to the configuration space in the product is a pointwise weak equivalence. Since the left hand side of the map in Theorem 1 only depends on homotopytheoretic information, we have found the desired description of Conf(n, X× Y ), and could even iterate to find Conf(n, X× Y × Z) for example. The rest of the talk switches to the following reformulation of Theorem 1. Theorem 2. IfP(Γ) is the poset of pairs of subgraphs Γ′, Γ′′⊆ Γ so that Γ = U (Γ′, Γ′′), then the natural map  hocolimConf(Γ′, X)× Conf(Γ′′, Y )→ Conf(Γ, X × Y ) (Γ′,Γ′′)∈P(Γ)op is a weak equivalence. Braid matroid Kazhdan–Lusztig polynomials Max Wakefield Kazhdan–Lusztig polynomials for matroids mimic the classical Kazhdan–Lusztig polynomials (originally defined in [4]) in many ways. Both these polynomials fit into the wider combinatorial theory of Kazhdan-Lusztig-Stanley polynomials defined in [7] and refined in [1]. In the case of matroids there is a significant amount of combinatorial machinery one can use to interpret these polynomials. From many perspectives braid matroids are the most important family of matroids. It is an open problem to find a simple closed formula for the braid matroid Kazhdan– Lusztig polynomials. In this note we will briefly survey matroid Kazhdan–Lusztig polynomials and discuss some recent results on computing them for braid matroids. We will focus on the combinatorial perspective, however a rich algebraic view was recently taken by Proudfoot and Young in [6] which yielded some crucial information about the generating functions of the coefficients for braid matroid Kazhdan–Lusztig polynomials. LetA = H1, . . . , Hn be an arrangement of hyperplanes in CSℓwith linear forms ker(αi) = Hi. The complex complement UA= Cℓ\Hiis an important manifold. In the case whereA is the braid arrangement (i.e. all the hyperplanes of the formxi− xj= 0) the manifold UAis the configuration space of ℓ-points in C. The Zariski closure in Cnof the compositions of the maps f : UA→ (C∗)n and g : (C∗)n→ (C∗)ndefined by f ( x) = (αi( x)) and g(xi) = (x−1i) respectively is called the reciprocal plane which we denote by XA= g◦ f(UA). The coordinate ring of XAis the Orlik-Terao algebra OT (A) = C[α−11, . . . , α−1n]. Let X P (XA, t) =dim(IH2i(XA))ti 82Oberwolfach Report 2/2018 where IH(XA) is the topological intersection cohomology over C. This Poincar´e polynomial was the motivation for the study matroid Kazhdan–Lusztig polynomials. Let M be a matroid with lattice of flats L(M ). If M is realizable with arrangementA then we write M(A) for the matroid and L(A) for its lattice of flats. For F∈ L(M) we call L(M)F=X ∈ L(M)|X ≤ F the localization of L(M) at F and L(M )F=X ∈ L(M)|X ≥ F the restriction of L(M) at F . Then we will denote MFand MFthe matroids associated to the lattices L(M )Fand L(M )F respectively. Now we can define the matroid Kazhdan–Lusztig polynomials. Definition 1 (Theorem 2.2 in [2]). There is a unique way to assign to each matroid M a polynomial P (M, t)∈ Z[t] such that the following conditions are satisfied: (1) If rkM = 0, then P (M, t) = 1. (2) If rkM > 0, then deg P (M, t) <X12rkM . (3) For every M , trkMP (M, t−1) =χ(MF, t)P (MF, t). F In this definition χ(M, t) is the characteristic polynomial of the matroid. This combinatorial definition turns out to give the Poincar´e polynomial of the intersection cohomology. Theorem 1 (Theorem 3.10 in [2]). P (XA, t) = P (M (A), t) Theorem 1 gives a few nice corollaries. First it shows that the Betti numbers of the intersection cohomology are combinatorial. Second it shows that for representable matroids the polynomials P (M, t) have non-negative coefficients. So, one can naturally conjecture that these polynomials have non-negative coefficients for all matroids. A question remains: does this combinatorial definition help us compute these polynomials for infinite families of matroids? The most important family of matroids is the so called braid matroids, denoted here as Bn, associated to the configuration spaces, type A Coxeter groups, and complete graphs. Conveniently, the lattice of flats of the braid matroid is the set partition lattice. Unfortunately, there are no known closed formulas for any coefficients for these polynomials and we do not even have a conjectural formula except for the top coefficient. Fortunately, using Stirling numbers we can obtain a few different formulas for the braid matroid Kazhdan–Lusztig polynomials. Definition 2. Suppose rk(L(M )) = N and I = (i1, i2, . . . , ir) has 0≤ i1≤ i2≤ · · · ≤ ir≤ N. The set of partial flags of L(M) associated to I is L(M )I=(X1, X2, . . . , Xr)∈ L(M)r| rk(Xj) = ij, and X1≤ X2≤ · · · ≤ Xr. The partial flag (also called multi-indexed) Whitney numbers of the second kind are WI=|L(M)I|. Using these partial flag Whitney numbers once can define an index set Siand two functions si: Si→ Z and t: Si→ 2Z[N ]constructed in [8]. Topology of Arrangements and Representation Stability83 Theorem 2 (Theorem 11 in [8]). For any finite, ranked lattice P such that rk(P ) = N , the degree i coefficient of the Kazhdan–Lusztig polynomial of P with 1≤ i < N/2 is X (−1)si(I)(Wt(I)(P )− WI(P )). I∈Si A very similar result was given in [5]. Theorem 3 (Theorem 3.3 in [5]). For all i > 0, the degree i coefficient of the matroid Kazhdan–Lusztig Polynomial, Cifor a matroid of rank N is XiXX (−1)|D|W(N−(a+a tr (S)r+1),...,N−(at1(S)+a0)) r=1D⊂[r](am) where W is the multi-indexed Whitney number of the sequence of integers (am) such that a0= 0, ar= i, ar+1= rk(M )− i, a0< a1<· · · < ar< ar+1, and tj(S) = mink | k ≥ j and k 6∈ S ∈ [r + 1]. Now both of these theorems use flag Whitney numbers of the second kind. For braid matroids these flag Whitney numbers are products of Stirling numbers of the second kind S(n, k) = the number of set partitions of a set with size n with k blocks. For I = (i1, i2, . . . , ik), kY−1 WI(Bn) =S(n− ij, n− ij+1). j=0 Since both Theorem 2 and Theorem 3 use Stirling numbers of the second kind for braid matroids wouldn’t it be nice to have a Stirling number of the first kind formula. This was the subject of recent work by Trevor Karn and the author in
[204] . To state the theorem, we need a little notation. Definition 3 (Theorem 3.3 in [3]). Let n≥ 2 and i <n−12. DefineKn,ito be the set of all triples (Λ, A, Ξ) where Λ = [λ1, . . . , λq] is a sequence of number partitions, and A = [α1, . . . , αq] and Ξ = [ξ1, . . . , ξq] are sequences of integers which satisfy: (1) λ1⊢ n (2) λj⊢ ℓ(λj−1) for all 1 < j≤ q (3) α1+ ξ1= n− 1 − i (4) αj+ ξj= ℓ(λj−1)− 1 − ξj−1for j > 1 (5) 0≤ αj≤ |λj| − ℓ(λj) for all j (6) ξj= 0 when ℓ(λj) = 1 (7) 0≤ ξj<ℓ(λj2)−1when ℓ(λj)≥ 2 (8) ξj= 0 if and only if q = j. UsingKn,ias an index set, we can state a Stirling number of the first kind formula for the braid matroid Kazhdan–Lusztig polynomials. 84Oberwolfach Report 2/2018 Theorem 4. For n≥ 2 and i <n−12,  XYqXℓ(λYj) P (Bn, t)i=m(λj)s(bjk, djk) , (Λ,A,Ξ)j=1(dj)k=1 k where (Λ, A, Ξ) = ([λ1, . . . , λq], [α1, . . . , αq], [ξ1, . . . , ξq])∈ Kn,i, bjkis the kthblock of λj, and the last sum is over all sequences (djk) = (dj1, . . . , djℓ(λ) satisfying j) Pℓ(λj) k=1djk= αj+ ℓ(λj) and 1≤ djk≤ bjk. Many questions remain open for braid matroid Kazhdan–Lusztig polynomials. (1) Do Theorems 2, 3, and 4 imply anything about the complexity of these coefficients? (2) Could Theorems 2, 3, and 4 help in computing the top coefficient of P (Bn, t), conjectured in [2]? References
[205] F. Brenti. P -kernels, IC bases and Kazhdan-Lusztig polynomials. J. Algebra, 259(2):613– 627, 2003. · Zbl 1022.20015
[206] B. Elias, N. Proudfoot, and M. Wakefield. The Kazhdan-Lusztig polynomial of a matroid. Adv. Math., 299:36–70, 2016. · Zbl 1341.05250
[207] T. Karn and M. Wakefield. Stirling numbers in braid matroid kazhdan-lusztig polynomials. Preprint. · Zbl 1402.05031
[208] D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53(2):165–184, 1979. · Zbl 0499.20035
[209] Proudfoot, Nicholas, and Xu, Yuan, and Young, Benjamin. The Z-polynomial of a matroid arXiv:math.CO/1706.05575v1. · Zbl 1380.05022
[210] N. Proudfoot and B. Young. Configuration spaces, FSop-modules, and Kazhdan–Lusztig polynomials of braid matroids. New York J. Math., 23:813–832, 2017. · Zbl 06747482
[211] R. P. Stanley. Subdivisions and local h-vectors. J. Amer. Math. Soc., 5(4):805–851, 1992. · Zbl 0768.05100
[212] M. Wakefield. A flag Whitney number formula for matroid Kazhdan–Lusztig polynomials. To appear in the Electronic Journal of Combinatorics. Resolvent Degree, Hilbert’s 13th Problem and Geometry Jesse Wolfson (joint work with Benson Farb) We start with a problem central to classical (and modern) mathematics. Problem 1. Find and understand formulas for the roots of a polynomial P (z) = zn+ a1zn−1+· · · + an in terms of the coefficients a1, . . . , an. It is well known that if n≥ 5 then no formula exists using only radicals and arithmetic operations in the coefficients ai.1Less known is Bring’s 1786 theorem [Bri] that any quintic can be reduced via radicals to a quintic of the form Q(z) = z5+ az + 1 (see [CHM] for a contemporary translation). In 1836, Hamilton [Ham] 1 This was claimed by Ruffini in 1799; a complete proof was given by Abel in 1824. Topology of Arrangements and Representation Stability85 extended Bring’s results to higher degrees, showing, for example, that any sextic can be reduced via radicals to Q(z) = z6+az2+bz+1, that any degree 7 polynomial can be reduced via radicals to one of the form (1)Q(z) = z7+ az3+ bz2+ cz + 1. and that any degree 8 polynomial can be reduced via radicals to one of the form Q(z) = z8+ az4+ bz3+ cz2+ d + 1. Hilbert conjectured explicitly that one cannot do better: solving a sextic (resp. septic, resp. octic) is fundamentally a 2-parameter (resp. 3-parameter, resp. 4-parameter) problem. More precisely, we have the following invariant, first introduced by Brauer [Bra]. Definition 2 (Resolvent degree). Fix a field k. Let eX→ X be a generically finite dominant map of k-varieties. The resolvent degree RD( eX→ X) is the smallest d≥ 0 with the following property: there is a chain of generically finite dominant maps Xr→ Xr−1→ · · · → X0= X such that Xr→ X factors through a dominant map to eX and such that for each i there is a variety Y with dim(Y )≤ d so that Xi+1→ Xiis a pullback Xei+1→ eY  yy Xi→ Y Example 3. LetPndenote the space of monic degree n polynomials (i.e. Ank/Sn), let fPndenote the space of monic degree n polynomials with a choice of root (i.e. Ank/Sn−1), and let fPn→ Pnbe the finite map obtained by forgetting a root. In this language, the classical results on reduction of parameters can be restated succinctly as: RD( ePn→ Pn) = 1∀n ≤ 5,andRD( ePn→ Pn)≤ n − 4 ∀n > 5. Buhler-Reichstein, Merkujev and others have developed a beautiful and widely applicable theory of essential dimension ed( eX→ X), where one forces r = 1 in Definition 2; see Reichstein’s 2010 ICM paper [Re] for a survey. This disallowing of so-called “accessory irrationalities” captures more of the arithmetic of the ground field k(X), whereas RD captures more of the intrinsic complexity of the branched cover. Hilbert’s problems. As already noted by Brauer [Bra], Hilbert’s conjecture (explicitly asked by Hilbert in [Hi1, p.424] and [Hi2, p.247]) that Hamilton’s reduction of parameters for the general polynomial of degree 6, 7, or 8 is optimal, can now be stated precisely, as can the problem for all degrees. Both Klein and Hilbert worked on this general problem for decades (see [Kl1, Hi1, Hi2]). Problem 4 (Klein, Hilbert, Brauer). Compute RD( ePn→ Pn). In particular: Hilbert’s Sextic Conjecture ([Hi2], p.247): RD( eP6→ P6) = 2. Hilbert’s 13th Problem ([Hi1],p.424): RD( eP7→ P7) = 3. 86Oberwolfach Report 2/2018 Hilbert’s Octic Conjecture ([Hi2], p.247): RD( eP8→ P8) = 4. Beyond these, we have the following, which are implicitly due to Hilbert, and probably Brauer. Conjecture 5. There exists an example with RD( eX→ X) > 1. Conjecture 6. RD( ePn→ Pn)→ ∞ as n → ∞. Along with the Hilbert Sextic Conjecture and Hilbert 13, these are clearly among the most important conjectures in this area. Amazingly, no progress has been made on these conjectures or on either of the three special cases above since Hilbert stated them. In 1957, Arnol’d and Kolmogorov proved (see [Ar]) that there is no local topological obstruction to reducing the number of variables; however, as Arnol’d and many others have noted, the global problem remains open. While these conjectures provide the primary challenges for the field, the explicit study of resolvent degree is already yielding improvements on old theorems, and striking relationships between seemingly different problems. Two sample theorems provide an indication of what to expect. First, we expect rapid improvement should be possible on existing upper bounds on resolvent degree. Brauer in 1975 proved that for n≥ 4, RD( ePn→ Pn)≤ n − r once n > B(r) := (r− 1)!. In [FW2], we prove the following. Let RD(r, N) denote the resolvent degree of finding an r-dimensional linear subspace on a cubic hypersurface in PN. A dimension count shows that the function RD(r, N ) grows at most polynomially in r and N . Theorem 7 (Farb-W). There exist a pair of polynomial functions f, g : N× N → Nsuch that, for n≥(d+k)!d!, RD( ePn→ Pn)≤ maxn − (d + k + 1), RD(f(d, k), g(d, k)). Corollary 8. There exist monotone increasing functions F W, ϕ : N→ N s.t. - For n > F W (r), RD( ePn→ Pn)≤ n − r, - For all d≥ 0, r ≥ ϕ(d), then B(r)/F W (r) ≥ d!. This improvement over Brauer uses ideas of Hilbert [Hi2], who used lines on cubic surfaces to simplify the degree 9 polynomial. We are confident that further improvements will follow from incorporating ideas of Hamilton and Sylvester. While such theorems do not address the fundamental questions of lower bounds above, they provide a testing ground for new methods and give us evidence as to the eventual shape of the function RD( ePn→ Pn). For a second example of the type of theorems we expect to follow from renewed interest in resolvent degree, we prove the following in [FW1]. Theorem 9 (Farb-W.). The following statements are equivalent: (1) Hilbert’s Sextic Conjecture is true: RD( eP6→ P6) = 2. (2) RD = 2 for the problem of finding the 27 lines on a cubic, given a “double six” set of lines. Topology of Arrangements and Representation Stability87 (3) RD = 2 for the problem of finding a fixed point for the hyperelliptic involution on a genus 2 curve. In fact, the resolvent degrees of all of the above problems coincide. We also prove similar reformulations for Hilbert’s 13th Problem and Hilbert’s Octic Conjecture. While we make no definite progress toward proving nontrivial lower bounds for RD, we hope that with renewed attention to Hilbert’s conjectures and to resolvent degree, future progress may be more forthcoming. References [Ar]V.I. Arnol’d, On the representation of continuous functions of three variables by superpositions of continuous functions of two variables, Mat. Sb. (n.S.) 48 (90):1 (1959), 3–74. [Bra]R. Brauer, On the resolvent problem, Ann. Mat. Pura Appl. (4) 102 (1975), 4555. [Bri]E. Bring, Meletemata quædam Mathematica circa Transformationem Æquationum Algebraicarum (“Some Selected Mathematics on the Transformation of Algebraic Equations”), Lund, 1786. [CHM] A. Chen, Y.-H. He, and J. McKay, Erland Samuel Bring’s “Transformation of algebraic equations”, arXiv:1711.09253v1. [FW1] B. Farb and J. Wolfson, Resolvent degree, Hilbert’s 13th Problem and geometry, preprint. [FW2] B. Farb and J. Wolfson, General upper bounds for the resolvent degree of roots of polynomials, in preparation. [Ham] W. Hamilton, Inquiry into the validity of a method recently proposed by George B. Jerrard, esq., for transforming and resolving equations of elevated degrees, Report of the Sixth Meeting of the British Association for the Advancement of Science (1836), Bristol, 295–348. [Hi1]D. Hilbert, Mathematical Problems, from Proceedings of the 1900 ICM, English translation reprinted in Bull. AMS, Vol. 37, No. 4 (2000), 407–436. [Hi2]D. Hilbert, ¨Uber die Gleichung neunten Grades, Math. Ann. 97 (1927), no. 1, 243–250. [Kl1]F. Klein, Sur la r´esolution, par fonctions hyperelliptique, de l’´equation du vingt-septi‘eme degr´e, de laquelle d´epend la d´etermination des vingt-sept droites d’une surface cubique, Jour. de math. pure et appl. (4) vol. 4, 1888. [Re]Z. Reichstein, Proceedings of the 2010 ICM, Vol. 2, 162–188. On the Johnson homomorphisms of the automorphism groups of free groups Takao Satoh In the 1980s, Dennis Johnson established a remarkable method to investigate the group structure of the mapping class groups of surfaces in a series of his works. In particular, he constructed a certain homomorphism τ to determine the abelianization of the Torelli group. Today, his homomorphism τ is called the first Johnson homomorphism. Over the last three decades, good progress was made in the study of the Johnson homomorphisms of mapping class groups through the works of a large number of authors, including Morita [13], Hain [9], Cohen-Pakianathan [3, 4] and Farb [8] as pioneer works. In addition to this, we have a lot of interesting and remarkable works given by participants of this workshop, including Brendle [2], Papadima–Suciu [15], Patzt [16], Djament–Vespa [7]. 88Oberwolfach Report 2/2018 The definition of the Johnson homomorphisms can be naturally generalized to the automorphism groups of free groups. Let Fnbe a free group of rank n, H the abelianization of Fn, and Aut Fnthe automorphism group of Fn. The kernel of the homomorphism Aut Fn→ GL(n, Z) induced from the action of Aut Fnon H, is called the IA-automorphism group of Fn, and is denoted by IAn. In 1965, Andreadakis [1] introduced a descending central filtration IAn=An(1)⊃ An(2)⊃ · · · of IAn, and showed that each graded quotient grk(An) :=An(k)/An(k+1) is a free abelian group of finite rank. We call the above filtration the Andreadakis–Johnson filtration of Aut Fn. Johnson studied this kind of filtration for the mapping class groups in the 1980s. In order to investigate the structure of grk(An), we consider the k-th Johnson homomorphism τk: grk(An)→ H∗⊗ZLn(k + 1). Each of τkis GL(n, Z)-equivariant and injective. Based on our previous work and a recent remarkable work by Darn´e [5], the stable cokernel of τkhas been determined as Coker(τk) ∼= H⊗k/(Cyclic Permutation) (n≥ k + 2). On the other hand, the mapping class group case is much more difficult and mysterious, and the image of the Johnson homomorphisms are not determined yet. One of the motivations to study of the Johnson homomorphisms is to consider applications to twisted cohomology groups. Kawazumi [12] extended the first Johnson homomorphism to Aut Fnas a crossed homomorphism. We [19] computed H1(Aut Fn, (H∗⊗Λ2H)Z⊗Q) = Q⊗2, and described generators with the extension of τ1. So far, there are only a few computations of stable twisted cohomology groups, including those by Hatcher–Wahl [10], Djament–Vespa [6] and RandalWilliams–Wahl [18]. Pettet [17] determined the GL(n, Q)-decomposition of the image of the cup product∪: Λ2H1(IAn, Q)→ H2(IAn, Q). Based on her results, recently we showed that H2(Aut Fn, (Im(∪))∗)⊃ Q⊕dnwhere dnis the number of the irreducible components of Im(∪). Recently, we discovered that the framework of the theory of the Johnson homomorphisms can be applied to the ring of complex functions on SL(2, C)-representations of Fn. Let R(Fn) be the set of all SL(2, C)-representations of Fn, andF(Fn) the set of all complex-valued functions on R(Fn). ThenF(Fn) naturally has the C-algebra structure by the pointwise sum and product. Furthermore, Aut Fnnaturally acts onF(Fn) from the right. For any x∈ Fnand any 1≤ i, j ≤ 2, define aij(x) ofF(Fn) to be (aij(x))(ρ) := (i, j)-component of ρ(x) for any ρ∈ R(Fn). Let RQ(Fn) be the Q-subalgebra ofF(Fn) generated by all aij(x) for x∈ Fnand 1≤ i, j ≤ 2. Let J be the ideal of RQ(Fn) defined by J := (aij(x)− δij| x ∈ Fn, 1≤ i, j ≤ 2) ⊂ RQ(Fn) Topology of Arrangements and Representation Stability89 where δijis Kronecker’s delta. Then, we have the descending filtration J⊃ J2⊃ J3⊃ · · · , and each graded quotient grk(J) := Jk/Jk+1is an Aut Fn-invariant finite dimensional Q-vector space. Set HQ:= H⊗ZQ. For n≥ 3, we [20] obtained the following. T (1)k≥1Jk=0. M (2) For any k≥ 1, grk(J) ∼=Se11HQ⊗QSe12HQ⊗QSe21HQ. e11+e12+e21=k (3) RQ(Fn) is an integral domain, and is isomorphic to the universal SL2representation ring of Fn. Now, for any k≥ 1, let Dn(k) be the kernel of the homomorphism Aut Fn→ Aut(J/Jk+1) induced from the action of Aut Fnon J/Jk+1. Then the groups Dn(k) define a descending filtrationDn(1)⊃ Dn(2)⊃ · · · of Aut Fn. This is an SL(2, C)-representation analogue of the Andreadakis-Johnson filtration. For n≥ 3, we [20] showed (1) [Dn(k),Dn(l)]⊂ Dn(k + l) for any k, l≥ 1. (2)An(k)⊂ Dn(k) for any k≥ 1. Furthermore, this is equal for 1 ≤ k ≤ 4. From Part (1), we see that the graded quotients grk(Dn) :=Dn(k)/Dn(k + 1) are abelian groups for any k≥ 1. In order to study the structure of grk(Dn), we have introduced the homomorphisms ηk: grk(Dn)→ HomQ(gr1(J), grk+1(J)) defined by  σ(modDn(k + 1))7→f(mod J2)7→ fσ− f (mod Jk+1). The homomorphisms ηkis SL(2, C)-representation analogues of the Johnson homomorphisms. In [20], we showed that each ηkis Aut Fn/Dn(1)-equivariant injective homomorphism. This implies that each of grk(Dn) is torsion-free, and that dimQ(grk(Dn)⊗ZQ) <∞. Now, we conjecture that An(k) =Dn(k) for any k≥ 1. Finally we remark that in [21], we consider the above framework for SL(m, C)representations of Fnfor any m≥ 2, and obtained similar results as a part of the above. References
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[233] T. Satoh, On the graded quotients of the SL(m, C)-representation algebras of groups, preprint, arXiv:math.GT/1607.05411. Polynomial behaviour for stable homology of congruence groups Aur´elien Djament An ideal in a (unital) ring is the same as a ring without unit: such a (non-unital) ring I can be seen as the two-sided ideal given by the kernel of the augmentation Z ⋉I ։ Z, where Z ⋉ I is the unital ring obtained by formally adding a unit to I. The framework in the preprint [6], on which this talk is reporting, is more general, but most of the ideas and applications are already in this classical setting. The congruence groups associated to I are defined by Γn(I) := Ker (GLn(Z ⋉ I) ։ GLn(Z)). We look for qualitative properties of the homology of these groups. As in the case of usual linear groups, we have obvious stabilisation maps H∗(Γn(I); Z)→ H∗(Γn+1(I); Z): we will deal only with stable properties (as in algebraic K-theory), that is, properties of the colimit of this sequence of graded abelian groups. But we have also a richer structure: H∗(Γn(I); Z) in endowed with a natural action of GLn(Z) (induced by the conjugation action) which is generally not trivial (even stably). We will later express these structures (and their compatibility properties) in a functorial setting. Topology of Arrangements and Representation Stability91 Earlier known results Suslin [12] proved the following striking Theorem (which improves the rational result that he got with Wodzicki in [13], with a different method). Theorem 1 (Suslin 1995). Let d > 0 be an integer. (1) The following statements are equivalent. (a) Stably in n, the action of GLn(Z) on Hi(Γn(I); Z) is trivial for i < d; (b) I is excisive for algebraic K-theory in homological degree < d; (c) TorZ⋉iI(Z, Z) = 0 for 0 < i < d. (2) There is a natural map Hd(Γn(I); Z)→ glnTorZ⋉dI(Z, Z)(where gln(M ) denotes the n×n matrices with entries in M) which is GLn(Z)-equivariant, compatible with stabilisation in n, and whose kernel and cokernel bear a trivial GLn(Z)-action stably in n if the previous conditions are fulfilled. (Note that TorZ⋉1I(Z, Z)≃ I/I2, so the conditions are only seldom fulfiled for d > 1; for d = 1, the last statement is classical and not hard.) Other known results give informations on H∗(Γn(I)) for each homological degree, but only for particular non-unital rings I. In [1], Calegari proved the following asymptotic polynomial behaviour for homology of classical congruence groups. Theorem 2 (Calegari 2015). Let p be a prime number and k, i be non-negative integers. Then dim Hk(Γn(piZ); Fp) =n2k+ O(n2k−2). k! Another important recent result (whose methods are completely independent from the ones used to prove both previous Theorems) is due to Putman [9], with an input given by an older work by Charney [4]. This result was quickly improved by the systematic use of functorial methods that we will remind now. Statements in terms of polynomial functors Let (C, +, 0) be a small symmetric monoidal category whose unit 0 is an initial object. For convenience we will assume that the objects ofC are the natural integers and that + is the usual sum on objects. The precomposition by− + 1 is an exact endofunctor, denoted by τ , of the categoryC-Mod of functors from C to abelian groups; with Vespa we studied in [7] the quotient category St(C-Mod) of C-Mod obtained by killing the functors which are stably zero, that is, by quotienting out the localising subcategory ofC-Mod generated by functors F such that the canonical map F→ τ(F ) is zero (equivalently, a functor F is stably zero if and only if colimF (n) = 0). n∈N We introduced two notions of polynomial functor of degree d: a strong one, which captures also unstable phenomena, and a weak one, which depends only of the isomorphism class of the functor in St(C-Mod). For example, a functor inC-Mod is weakly polynomial of degree ≤ 0 if and only if it is isomorphic 92Oberwolfach Report 2/2018 in St(C-Mod) to a constant functor. For the definition of strongly and weakly polynomial functors and properties, we refer to [7] or to the talk by Vespa in this meeting. Weakly polynomial functors of (weak) degree≤ d (or more precisely, their images in St(C-Mod)) form a localising subcategory of St(C-Mod) denoted byPold(C-Mod). For example, gl•(M ) is a strongly polynomial functor of degree 2 in S(Z)-Mod (where S(Z) is defined just below), for any abelian group M . We are interested here in the following monoidal categoriesC with the previous properties: the category FI for which FI(n, m) is the set of injections from n := 1, . . . , n to m (the monoidal structure being given by disjoint union) and the category S(R), where R is a unital ring, for which S(R)(n, m) :=(f, g) ∈ HomR(Rn, Rm)× HomR(Rm, Rn)| g ◦ f = Id (the monoidal structure being given by direct sum). These categories are also homogeneous categories in the sense of Randal-Williams and Wahl [10] (a very general framework which is related to the one used at the beginning of [8]). For any unital ring R, n7→ GLn(R) defines a functor GL•(R) from S(R) to the category of groups. If I is a non-unital ring, n7→ Γn(I) is a subfunctor of GL•(Z⋉ I). By taking the homology, we get a functor Hd(Γ•(I)) in St(S(Z ⋉ I)-Mod) for each d, which lives indeed in S(Z)-Mod (because inner automorphisms act trivially in homology). By restricting it along the canonical monoidal functor FI→ S(Z), several authors, improving Putman [9], showed that Hd(Γ•(I)) is strongly polynomial for each d if the ring I is nice enough—see Church–Ellenberg–Farb– Nagpal [3] of Church–Ellenberg [2]. Recently, Church–Miller–Nagpal–Reinhold [5] obtained the following result, always by using FI-modules. Theorem 3 (Church–Miller–Nagpal–Reinhold, preprint 2017). If I is an ideal in a unital ring R satisfying Bass condition (SRr+2), then for each non-negative integer d, Hd(Γ•(I); Z) is a weakly polynomial functor of (weak) degree≤ 2d + r. In [6], the following stronger result is proven. Theorem 4.Let I be a ring without unit and e > 0 an integer such that TorZ⋉iI(Z, Z) = 0 for 0 < i < e (for example, e = 1). (1) For each integers r, d≥ 0 and each object F in Polr(S(Z ⋉ I)-Mod), the functor Hd(Γ•(I); F ) belongs toPol2[d/e]+r(S(Z)-Mod) (where the brackets denote the floor function). (2) If e is odd (respectively even), then for each integer n≥ 0, Hne(Γ•(I); F ) is isomorphicinthequotientcategory Pol2n(S(Z)-Mod)/Pol2n−2(S(Z)-Mod) to Λn(gl•(TorZ⋉eI(Z, Z))) (resp. Sn(gl•(TorZ⋉eI(Z, Z)))), where Λn(resp. Sn) denotes the n-th exterior (resp. symmetric) power (over the integers). For n = 1, the second part of this theorem is equivalent to Suslin’s Theorem 1. Ingredients of the proof The input of the proof of Theorem 4 is a version in degree 0 with twisted coefficients: one has an (easy) stable natural isomorphism H0(Γ•(I); F )≃ Φ∗(F ) in Topology of Arrangements and Representation Stability93 St(S(Z)-Mod) for any functor F in S(Z ⋉ I)-Mod, where Φ : S(Z ⋉ I)→ S(Z) denotes the reduction modulo the ideal I and Φ∗the left Kan extension along Φ. One can then derive this isomorphism (even in a quite more general framework) to get a stable spectral sequence  Ei,j2= Li(− ⊗Hj(Γ•(I); Z))◦ Φ∗(F )⇒ Hi+j(Γ•(I); F ) ⊕ where⊗: (S(Z)-Mod)× (S(Z)-Mod) → S(Z)-Mod is the composition of the ⊕ external tensor product with the left Kan extension along the direct sum functor S(Z)× S(Z) → S(Z). When F factorises through Φ : S(Z ⋉ I)→ S(Z), the abutment H∗(Γ•(I); F ) of the spectral sequence can be expressed simply from F and H∗(Γ•(I); Z), thanks to the universal coefficients exact sequence for group homology. So the spectral sequence gives informations on H∗(Γ•(I); Z). One needs several steps to show the wished result with this program, especially: • a comparison theorem of stable (in the sense of categories St introduced above!) derived categories of S(Z)-Mod and F(Z)-Mod, where F(Z) denotes Quillen’s category of factorizations of free abelian groups of finite rank, on (weakly) polynomial functors. This is inspired by Scorichenko’s thesis [11]; • A study of the left derived functors of Φ∗on polynomial functors (using the first step); • a study of the tensor product ⊗and its left derivatives on polynomial ⊕ functors (also using the first step); • a concrete argument of triangular groups inspired by Suslin–Wodzicki [13]; • some functorialities of the above spectral sequence. References
[234] F. Calegari, The stable homology of congruence subgroups, Geometry & Topology 19 (2015), 3149–3191. · Zbl 1336.11045
[235] T. Church and J. Ellenberg, Homology of FI-modules, Geometry & Topology 21 (2017), 2373–2418 · Zbl 1371.18012
[236] T. Church, J. Ellenberg, B. Farb and R. Nagpal, FI-modules over Noetherian rings, Geometry & Topology 18 (2014), 2951–2984. · Zbl 1344.20016
[237] R. Charney, On the problem of homology stability for congruence subgroups, Communications in Algebra 12 (1984), 2081–2123. · Zbl 0542.20023
[238] T. Church, J. Miller, R. Nagpal and J. Reinhold, Linear and quadratic ranges in representation stability, Preprint arXiv:1706.03845. · Zbl 1392.15030
[239] A. Djament, De l’homologie stable des groupes de congruence, preprint available at https://hal.archives-ouvertes.fr/hal-01565891.
[240] A. Djament and C. Vespa, Foncteurs faiblement polynomiaux, to appear in IMRN, available at https://hal.archives-ouvertes.fr/hal-00851869. · Zbl 1346.20070
[241] A. Djament and C. Vespa, Sur l’homologie des groupes orthogonaux et symplectiques ‘a coefficients tordus, Ann. Sci. ´Ec. Norm. Sup´er. 43 (2010), 395–459. · Zbl 1221.20036
[242] A. Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), 987–1027. 94Oberwolfach Report 2/2018 · Zbl 1334.20045
[243] O. Randal-Williams and N. Wahl, Homological stability for automorphism groups, Advances in Mathematics 318 (2017), 534–626. · Zbl 1393.18006
[244] A. Scorichenko, Stable K-theory and functor homology over a ring, PhD thesis, Evanston (2000).
[245] A. Suslin, Excision in integer algebraic K-theory, Trudy Mat. Inst. Steklov 208 (1995), 290–317.
[246] A. Suslin and M. Wodzicki, Excision in algebraic K-theory, Ann. of Math. 136 (1992), 51–122. Secondary representation stability for configuration spaces Jeremy Miller (joint work with Jennifer Wilson) 1. Indecomposables of an FI-module Throughout, we work over a commutative unital base ring R. All homology groups will be understood to have coefficients in R, all tensor products will be over R, and the term FI-module will mean a functor from the category of finite sets and injections to the category of R-modules. One of the most popular notions of representation stability is the notion of finite generation. The following is a measure of the generators of an FI-module. Following the notation of [2], we make the following definition. Definition 1.1. Let V be an FI-module and S be a set. Let  M H0FI(V )S= cokerVT→ VS . T⊂S,T 6=S We say that V has generation degree≤ d if VS= 0 for all sets S of cardinality strictly larger than d. Note that if the modules VSare finitely generated R-modules for all S, then an FI-module is finitely generated as an FI-module if and only if it has finite generation degree. In general, the groups H0FI(V ) should not be thought of as the generators of V as an FI-module but only a measure of how large a minimal generating set must be. Generators are naturally subobjects while the groups H0FI(V ) are a quotient. Although not standard terminology, it would be reasonable to call the groups H0FI(V ) indecomposables as they are analogous to the indecomposables of a graded algebra. The subscript 0 in H0FI(V ) is used because often one considers higher left derived functors of H0FIwhich are denoted by HiFI. Note that if V is an FI ♯-module in the sense of [3], then HiFI(V ) ∼= 0 for all i > 0. Moreover, we can recover V from H0FI(V ) by an explicit formula given in [3]. For this and other reasons, FI ♯-modules are often called induced or Topology of Arrangements and Representation Stability95 free. In this case, H0FI(V )Snaturally sit as a submodule of VSand can be more reasonably thought of as generators. 2. Configuration spaces Definition 2.1. For S a set and M a space, let ConfS(M ) denote the space of injections of the set S into the space M . Topologize ConfS(M ) with the subspace topology inside the space of all maps from S to M equipped with the compact open topology and with S equipped with the discrete topology. Let [k] =1, . . . , k and denote Conf[k](M ) by Confk(M ). We call Confk(M ) the configuration space of k ordered points in M . One of the most intensely studied families of FI-modules are the cohomology of ordered configuration spaces of points in a manifold. See [1] [3], [4], [5]. Here, we will instead study the homology of configuration spaces. For this to make sense, we must restrict to a certain class of manifolds. We will assume that M is a connected, non-compact n-dimensional manifold with n > 1. In this case, there exists an embedding e : Rn⊔ M ֒→ M which we will fix once and for all. If M has multiple ends, then the isotopy class of e will not be unique. This embedding induces a map ConfS(Rn)× ConfT(M )→ ConfS⊔T(M ) which in turn induces a map on homology Hi(ConfS(Rn))⊗ Hj(ConfT(M ))→ Hi+j(ConfS⊔T(M )). For fixed α∈ Hi(ConfS(Rn)), we denote the induced map by tα: Hj(ConfT(M ))→ Hi+j(ConfS⊔T(M )) and call it the stabilization map associated to the homology class α. Let p denote the class of a point in H0(Conf1(Rn)). Implicit in [3] is the fact that the maps tp: Hi(Confk(M ))→ Hi(Confk+1(M )) induce an FI-module structure on the functor S7→ Hi(ConfS(M )). We denote this FI-module by Hi(Conf(M )). In [3], Church–Ellenberg–Farb proved the following (also see [8]). Theorem 2.2 (Church–Ellenberg–Farb). The FI-modules Hi(Conf(M )) have a natural FI ♯-module structure and have generation degree≤ 2i. 96Oberwolfach Report 2/2018 3. Secondary stability From now on, we assume M is a surface.Let l denote a generator of H1(Conf2(R2)) ∼= R. This class induces a map tl: Hi(Confk(M ))→ Hi+1(Confk+2(M )). One can check that this induces a map on indecomposables tl: H0FI(Hi(Confk(M )))→ H0FI(Hi+1(Confk+2(M ))). Use the convention that fractional dimensional homology groups are zero and define W (M, i)k= H0FI(Hi+k/2(Confk(M ))). We get a sequence of symmetric group representations and equivariant maps tltltltl W (M, i)0−→ W (M, i)2−→ W (M, i)4−→ W (M, i)6−→ . . . and W (M, i)1−→ W (M, i)3−→ W (M, i)tl5−→ W (M, i)tl7−→ . . .tl Note that one of these sequences will be zero depending on the parity of i. Denote the other sequence by W (M, i). The sequence W (M, i) consists of the indecomposables of the FI-module H∗(Conf(M )) which lie a distance i above the stable range. There does not seem to be an interesting FI-module structure on W (M, i). However, the maps tVldo give the sequences W (M, i) the structure of aSym2R-module. HereVSym2R is the free twisted skew commutative algebra on the trivial one dimensional representationV in degree two. See [9] for a definition ofSym2R. The main theorem of [8] is the following. Theorem 3.1 (M.–Wilson). If R has characteristic zero and M is finite type, thenV the sequences W (M, i) are finitely generated for all i as modules overSym2R. We call this phenomenon secondary representation stability as it involves a stability pattern outside the classical stable range which only manifests itself after one appropriately accounts for the primary stability pattern. This was inspired by secondary homological stability which is a similar pattern discovered by Galatius– Kupers–Randal-Williams [6] and is also present in the work of Hepworth [7]. The need to work in characteristic zero stems from the fact that currently, the categoryV ofSym2R is only known to be locally Noetherian if R is a ring of characteristic zero, a result of Nagpal–Sam–Snowden [9]. References
[247] Thomas Church, Homological stability for configuration spaces of manifolds, Invent. Math. 188-2 (2012), 465–504. · Zbl 1244.55012
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[250] Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014) 2951–2984. · Zbl 1344.20016
[251] Jeremy Miller, Thomas Church, Rohit Nagpal, and Jens Reinhold, Linear and quadratic ranges in representation stability. · Zbl 1392.15030
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[253] Richard Hepworth, On the edge of the stable range.
[254] Jeremy Miller and Jennifer C. H. Wilson, Higher order representation stability and ordered configuration spaces of manifolds.
[255] Rohit Nagpal, Steven Sam, and Andrew Snowden, Noetherianity of some degree two twisted skew-commutative algebras. Combinatorics of Abelian arrangements Emanuele Delucchi 1. Abelian arrangements Let G be one of the groups C, C∗or E (an elliptic curve), and let Λ be a free abelian group of rank rank d. Any choice of elements α1, . . . , αn∈ Λ determines a family of homomorphisms αi: Hom(Λ, G)→ G and thus an abelian arrangement in Gd≃ Hom(Λ, G): A=H1, . . . , Hn, where Hi:= ker αi. From a topological point of view, the object of interest is the complement M (A ) := Gd\SA. As the combinatorial data associated to an abelian arrangement we consider – the dimension function δA: 2[n]→ N, I 7→ dim(∩i∈IHi) = corankhαi: i∈ Ii; – the multiplicity function mA: 2[n]→ N, I 7→ β0(∩i∈IHi); – the poset of layersC(A ), the set of all connected components of intersections of the Hi, partially ordered by reverse inclusion. Algebraic models for the cohomology of M (A ) have been given by Bibby [3] and by Dupont [16], who also addressed formality questions [17]. Local systems cohomology has been studied by Levin and Varchenko [18] and Denham, Suciu and Yuzvinsky [15]. We aim at a structural understanding of the combinatorial structures, and motivate our approach with a review of some special cases. 1.1. Weyl arrangements. Let Φlbe a rank l root system of type ABCD. Taking αi := Φlas a subset of the associated coroot lattics Λ :=hΦ∨li, we obtain the Weyl arrangements AΦl. Bibby [3] proved representation stability for the action of the Weyl groups W (Φl) on M (AΦl) by way of explicit descriptions of the posets of layers. Using this we can prove that all posetsC(AΦl) are EL-shellable [12]. 98Oberwolfach Report 2/2018 1.2. Linear arrangements: G = C. We refer to the contribution of Michael Falk in this volume for a detailed overview of the theory in this case. Here we stress that, from a combinatorial point of view, the function δAand the posetC(A ) encode equivalent combinatorial data: by knowing one of them it is possible to reconstruct the other. (Notice that, in this case, mAis constant equal to 1, hence it does not add any information.) As an abstract poset,C(A ) has the structure of a geometric lattice. The class of geometric lattices is larger than that of intersection posets of arrangements, but still corresponds to a class of functions defined by some of the combinatorial properties of δA. Every such abstract function – respectively, every geometric lattice – defines a matroid [20]. 1.3. Toric arrangements: G = C∗. As an update to the overview given in [7, Introduction] we mention the study of the ring H∗(M (A ), Z) in [6] and [21] and De Concini and Gaiffi’s work constructing projective wonderful models for M (A ) and computing their cohomology [10, 11]. Combinatorial aspects of toric arrangements appeared in many contexts - see the introduction to [13]. In particular, d’Adderio and Moci [9] and Br¨and´en and Moci [5], devised a theory of arithmetic matroids designed to underpin some properties of Moci’s arithmetic Tutte polynomial [19] X (1)TA(x, y) :=mA(S)(x− 1)δ(S)(y− 1)|S|+δ(S)−δ(S). S⊆A 2. Group actions on semimatroids We propose a combinatorial theory based on the observation that abelian arrangements are quotients of periodic arrangements of affine hyperplanes. In fact, to any locally finite1set of affine hyperplanes fAin a vectorspace we can associate the posetL of all intersection subspaces ordered by reverse inclusion and a function δ : 2A→ Z, δ(X) := dim(∩X), where we set dim(∅) := −1. Posets of the form L are geometric semilattices, and the function δ satisfies the axioms for a semimatroid with fAas its ground set. Semimatroids and geometric semilattices are abstractly equivalent in the sense explained in§1.2, see [1, 13]. Definition 1. Let G be a group. A G-semimatroid S is given by a δ-preserving action of G on the ground set E of a semimatroid or, equivalently, an action of G by poset automorphisms on the associated geometric semilatticeL. We can define – the posetPS:=L/G of orbits; – a function δS: 2E/G→ Z induced by δ (see [13, Definition 3.2]); – an “orbit-counting” function mS: 2E/G→ N, mS(X) =|p ∈ PS| p is a supremum of X /G| – a polynomial TS(x, y) defined from δSand mSas in Equation (1). Definition 2. Call the G-semimatroid S 1 I.e., every point of the space has a neighbourhood that meets only finitely many hyperplanes. Topology of Arrangements and Representation Stability99 – translative ifL/G is a finite poset and, for all x ∈ L and every g ∈ G, the existence of any y∈ L with y ≥ x and y ≥ gx implies x = gx. – refined if it is translative, the group G is finitely generated free abelian and, for all x∈ L, stab(x) is a free direct summand of G of rank δ(x). Every abelian arrangement A gives rise to a (refined) G-semimatroid S, with PS≃ C(A ), mA= mS, TA(x, y) = TS(x, y). Many well-known properties of matroids can be generalized, as the following sample of [8, 13] shows. Theorem 1. If S is translative, then TS(x, y) satisfies deletion-contraction and χPS(t) = (−1)δ(∅)TS(1− t, 0). Moreover, if S is refined, then eHi( b∆(PS), Z) = 0 for i < dim( b∆(PS)). Notice that there are translative, nonrefined, representable actions for which the topological claim of the theorem fails. See [13] for a discussion of representability and of conditions under which mS satisfies the axioms of arithmetic matroids. However, we do not know whether every arithmetic matroid arises from a G-semimatroid. References
[256] F. Ardila, Semimatroids and their Tutte polynomials, Rev. Colombiana Mat. 41 (2007), no. 1, 39–66. · Zbl 1136.05008
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[259] C. Bibby, Representation stability for the cohomology of arrangements associated to root systems, ArXiv e-prints, March 2016.
[260] P. Br¨and´en, L. Moci, The multivariate arithmetic Tutte polynomial. Trans. Amer. Math. Soc. 366 (2014), no. 10, 5523–5540. · Zbl 1300.05133
[261] F. Callegaro, E. Delucchi,The integer cohomology algebra of toric arrangements, Adv. Math. 313 (2017), 746–802. · Zbl 1401.32022
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[263] A. d.Al‘ı, E. Delucchi, Stanley-Reisner rings for abelian arrangements, in preparation.
[264] M. D’Adderio, L. Moci, Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math. 232 (2013), 335–367. · Zbl 1256.05039
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[266] C. De Concini, G. Gaiffi, Cohomology rings of compactifications of toric arrangements, ArXiv e-prints, January 2018.
[267] E. Delucchi, N. Girard, G. Paolini, Shellability of posets of labeled partitions and arrangements defined by root systems, ArXiv e-prints, June 2017.
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[271] C. Dupont, The Orlik-Solomon model for hypersurface arrangements, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2507–2545. · Zbl 1332.14015
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[276] R. Pagaria, Combinatorics of toric arrangements, ArXiv e-prints, October 2017. Solomon-Terao algebra of hyperplane arrangements Takuro Abe (joint work with Toshiaki Maeno, Satoshi Murai and Yasuhide Numata) This is a short report on the forthcoming paper [2]. Let V = CℓandA a hyperplane arrangement in V , i.e., a finite set of linear hyperplanes in V . For each H∈ A fix a linear form αH∈ V∗such that ker(αH) = H. Let S = C[x1, . . . , xℓ] the coordinate ring of V and DerS :=Lℓi=1S∂xithe free S-module of S-derivations. The most important algebra of the hyperplane arrangement is the logarithmic derivation module defined as follows: D(A) := θ ∈ DerS | θ(αH)∈ SαH(∀H ∈ A). A is free with exponents exp(A) = (d1, . . . , dℓ) if D(A) is a free S-module with homogeneous basis θ1, . . . , θℓ∈ D(A) with deg(θi) = di(i = 1, . . . , ℓ). The most important consequence of the freeness is the following factorization theorem due to Terao in [5]: [Yℓ π(A; t) := Poin(Cℓ\H; t) =(1 + dit). H∈Ai=1 Based on logarithmic derivation modules, Solomon and Terao introduced a polynomial Ψ(A; x, t) ∈ Q[x, t] and proved that Ψ(A; 1, t) = π(A; t). For details, see
[277] and [2]. Our purpose is to consider the other specialization Ψ(A; x, 1). First let us define the new algebra ST (A, η) of A and a homogeneous polynomial η of degree d > 0 as follows: ST (A, η) := S/a(A, η), here a(A, η) := θ(η) | θ ∈ D(A) is the Solomon-Terao ideal, and ST (A, η) the Solomon-Terao algebra with respect toA and η. Let us introduce an example on the above objects. LetA be the arrangement defined by x1x2(x21− x21) = 0 in C2. Then we can compute D(A) = hx1∂x1+ x2∂x2, x31∂x1+ x32∂x2iS, thusA is free with exponents (1, 3). When η = x21+x22, the Solomon-Terao algebra is a(A, η) = (x21+x22, x41+x42). Thus ST (A, η) coincides with the coinvariant algebra Topology of Arrangements and Representation Stability101 of the type B2. This observation is true for all the other Weyl arrangements, which justifies our definition. To justify this definition more, let us recall two results. The first one is by Solomon and Terao in [4], asserting that for each d > 0, there exists a non-empty Zariski open set Ud(A) of the homogeneous polynomial of degree d such that dimCST (A, η) < ∞ for all η ∈ Ud(A). Hence for a generic η, ST (A, η) is Artinian. Second one is due to the first author, Horiguchi, Masuda, the third author and Sato in [1]. To state it, let us introduce a notation. Let W be the Weyl group acting on V , Φ the corresponding root system and Φ+a fixed positive system. Let α1, . . . , αℓbe the simple system. A subset IP⊂ Φ+is a lower ideal if α∈ I, β ∈ Φ+ satisfy α− β ∈ℓi=1Z≥0αi, then β∈ I. For a lower ideal I, we can defie the ideal arrangementAIas the set of all reflecting hyperplanes corresponding to the roots in I. Also, for I, we can define the regular nilpotent Hessenberg varierty X(I), see [1] for details. Then ST (AI, P1)≃ H∗(X(I), C), where P1is the lowest degree W -invariant polynomial. Hence the Solomon–Terao algebra could be a cohomology ring of certain algebraic varieties. Also, whenA satisfies a generic condition called the tameness, it is essentially shown in [4] (see also [2]) that Hilb(ST (A, η); x) = Ψ(A; x, 1) for η∈ U2(A). Thus for an ideal arrangement AI, it holds that √ Ψ(A; x, 1) = Hilb(ST (A, P1); x) = Poin(X;x). Thus the Solomon–Terao algebra gives an algebraic counter part of Ψ(A; x, 1) with possible nice geometric interpretations. Moreover, as an algebra itself, we can show the following main result about the Solomon–Terao algebra in [2]. Theorem ([3], [2]). ST (A, η) is a complete intersection ring if and only if A is free for all d > 0 and all η∈ Ud(A). Note that the above theorem is shown by Epure and Schulze independently in
[278] when d = 2 with more general setup, i.e., for hypersurface singularities. Hence we can give another characterization of the freeness in terms of the complete intersections of the Solomon–Terao algebra. Since Solomon–Terao algebras provide a way to construct Artinian algebras from hyperplane arrangements, we can consider several problems related to Artinian rings. For example, the following questions are important. Problem ([2]). Let η∈ S2. Then when is ST (A, η) Gorenstein? If we want ST (A, η) to be a cohomology ring of some varieties, then it is necessary for ST (A.η) to be Gorenstein. However, we have no example of a non-free arrangementA such that ST (A, η) is Gorenstein. 102Oberwolfach Report 2/2018 References
[279] T. Abe, T. Horiguchi, M. Masuda, S. Murai and T. Sato, Hessenberg varieties and hyperplane arrangements. arXiv:1611.00269.
[280] T. Abe, T. Maeno, S. Murai and Y. Numata, Solomon–Terao algebra of hyperplane arrangements. arXiv:1802.04056.
[281] R. Epure and M. Schulze, A Saito criterion for holonomic divisors. arXiv:1711.10259.
[282] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement. Adv. in Math. 64 (1987), no.3, 305–325. · Zbl 0625.05001
[283] H. Terao, Generalized exponents of a free arrangement of hyperplanes and Shephard–Todd– Brieskorn formula. Invent. Math. 63 (1981), 159–179. Configuration spaces of graphs Eric Ramos Definition 1. A graph is a connected and compact 1-dimensional CW complex. We call the 0-cells of a graph G the vertices of G, while the 1-cells are referred to as the edges of G. The number of edges adjacent to a vertex v will be called the degree of v, and will be denoted µ(v). Given a graph G, we write Confn(G) to denote the n-stranded configuration space ofG, Confn(G) =(x1, . . . , xn)∈ Gn| xi6= xj. We will write UConfn(G) to denote the unordered n-stranded configuration space ofG UConfn(G) = Confn(G)/Sn In the first part of this survey, we will briefly recount the main structural theorems about the spaces Confn(G) and UConfn(G). One of the primary techniques in the study of graph configuration spaces is the use of certain cellular models for Confn(G) and UConfn(G). The first such model was proposed by Abrams in [Ab]. Using this model, he proved the following. Theorem 1 (Abrams [Ab], Theorem 3.10). Let G be a graph. Then Confn(G) and UConfn(G) are aspherical. Other cellular models have been constructed by Ghrist [Gh], Swiatkowski [Sw], Farley and Sabalka [FS], and L¨utgehetmann [Lu]. Each of these models has proven to be useful in different circumstances. For instance, the models of Ghrist and Swiatkowski each showed the following. Theorem 2 (Ghrist [Gh], Theorem 3.3; Swiatkowski [Sw], Theorem 0.1). Let G be a graph which is not homeomorphic to a circle. Then Confn(G) is homotopy equivalent to a CW complex whose dimension is at most the number of vertices of G of degree≥ 3. The same statement is true for UConfn(G). As a consequence of the above, we immediately obtain that the homological dimensions of Confn(G) and UConfn(G) are bounded independently of n. This behavior seems to be largely unique in the study of configuration spaces. We Topology of Arrangements and Representation Stability103 will also see that, in combination with the following theorem of Gal, it leads to problems when trying to compute the relevant homology groups. Theorem 3 (Gal [Ga], Theorem 2). Let G be a graph with e edges, and let e(t) denote the power series Xtn e(t) =χ(Confn(G)). n! n≥0 Then, 1Y e(t) =(1 + (1− µ(v))t)) (1− t)e where the product is over the vertices of G. The following theorem was proven using the Farley-Sabalka model for UConfn(G). Theorem 4 (Ko and Park [KP], Theorem 3.5). Let G be a graph.Then H1(UConfn(G)) is torsion-free if and only if G is planar. If H1(UConfn(G)) has torsion, then it must be 2-torsion. In contrast to the above, it is conjectured that Hi(Confn(G)) is always torsionfree [CL]. In fact, this has been proven for trees. Theorem 5 (Chettih and L¨utgehetmann [CL], Theorem A). If G is a tree, then Hq(Confn(G)) is torsion-free for all q≥ 0. To conclude, we outline the work that has been done towards applying techniques from representation stability theory to understand the homology groups of Confn(G) and UConfn(G). One should observe that Theorems 2 and 3 imply that at least one of the homology groups Hq(Confn(G)) has Betti numbers which grow at least factorially in n. While this would seem to preclude the usual FI-module techniques, there are still some conclusions one can draw. Theorem 6. Let G be a graph, and let SGdenote the integral polynomial ring whose variables are labeled by the edges of G. Then: (1) [An, Drummond-Cole, Knudsen [ADK], Theorem 4.5] For all qL≥ 0, the abelian groupnHq(UConfn(G)) can be equipped with an action of SG, turning it into a finitely generated graded SG-module. (2) [Ramos [Ra], Theorem D] If G is a tree, then the SG-module L nHq(UConfn(G)) decomposes as a direct sum of graded twists of squarefree monomial ideals. Moreover, this decomposition only depends on q and the degree sequence of G. Note that Maciazek and Sawicki independently proved the statement about the homology groups only depending on the degree sequence of the tree independently of the author [MS, Theorem V.3]. Theorem 7 (L¨utgehetmann [Lu2], Theorem I). Let G be a 3-connected graph with at least 4 vertices of degree≥ 3. Then the FI-module H1(Confn(G); Q) is finitely generated. 104Oberwolfach Report 2/2018 Fundamentally, one of the main difficulties with configuration spaces of graphs is that it is difficult for points to move around one another. Therefore, allowing the number of points being configured to grow leads to very unstable behaviors. One way to combat this is to fix the number of points being configured and instead allow the graph itself to vary. LetT denote the category of trees and injective maps, and letG denote the category of graphs and injective maps. The following theorem is to appear in future work. Theorem 8 (L¨utgehetmann and Ramos). For all k, q≥ 0, The functor T 7→ Hq(Confk(T )) fromT to the category of abelian groups is finitely generated. The same is true of the functor G7→ Hq(Confk(G)) fromG to the category of abelian groups, so long as q≤ 1. It is unknown whether the second half of the above theorem can be expanded to include all q≥ 0. Another theorem in the same vein is the above is the following. Note that for any injection of sets f :1, . . . , n ֒→ 1, . . . , m, one obtains a map of graphs Kn→ Kmbetween complete graphs. This induces an FI-module structure on the homology groups Hq(Confk(Kn)) and Hq(UConfk(Kn)), for each fixed k, q≥ 0, where we allow n to vary. Theorem9(Ramos and White [RW],Theorem G). The FI-modules Hq(Confk(Kn)) and Hq(UConfk(Kn)) are finitely generated for all choices of q and k. In fact, the above theorem will hold whenever Knis replaced by any vertexstable FI-graph (see [RW] for definitions). Acknowledgements: The author was supported by NSF grants DMS-1704811 and DMS-1641185. References [Ab]A. Abrams, Configuration spaces and braid groups of graphs, Ph.D thesis. B. Knudsen, Subdivisional spaces and graph braid groups, Preprint. [ADK] B. H. An, G. C. Drummond-Cole and B. Knudsen, Subdivisional spaces and graph braid groups, Preprint. [BF]K. Barnett and M. Farber, Topology of configuration space of two particles on a graph. I., Algebr. Geom. Topol. 9 (2009), no. 1, 593–624. [CL]S. Chettih and D. L¨utgehetmann, The Homology of Configuration Spaces of Graphs, Preprint. [FS]D. Farley and L. Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005), 1075–1109. [Ga]S. R. Gal , Euler characteristic of the configuration space of a complex, Colloq. Math. 89 (2001), 61–67. [Gh]R. Ghrist, Configuration spaces and braid groups on graphs in robotics, Knots, braids, and mapping class groups - papers dedicated to Joan S. Birman (New York, 1998), AMS/IP Stud. Adv. Math. 24, Amer. Math. Soc., Providence, RI (2001), 29–40. [KP]K. H. Ko, and H. W. Park, Characteristics of graph braid groups, Discrete Comput Geom 48(2012), no. 4, 915–963. [Lu]D. L¨utgehetmann, Representation Stability for Configuration Spaces of Graphs, Preprint. Topology of Arrangements and Representation Stability105 [Lu2]D. L¨utgehetmann, Representation Stability for Configuration Spaces of Graphs, Ph.D. Thesis. [MS]T. Maciazek and A. Sawicki, Homology groups for particles on one-connected graphs, Journal of Mathematical Physics 58 (2017), no. 6, 062103, 24pp. [Ra]E. Ramos, Stability phenomena in the homology of tree braid groups, Preprint. [RW]E. Ramos, Families of nested graphs with compatible symmetric-group actions, Preprint. [Sw]J. Swiatkowski, Estimates for homological dimension of configuration spaces of graphs Colloquium Mathematicum 89.1 (2001): 69–79. Freeness of multi-reflection arrangements for complex reflection groups Gerhard R¨ohrle (joint work with Torsten Hoge, Toshiyuki Mano, Christian Stump) In his seminal work [13], Ziegler introduced the concept of multi-arrangements generalizing the notion of hyperplane arrangements. In [10], Terao showed that every reflection multi-arrangement of a real reflection group with a constant multiplicity is free. The aim of the joint work [4], reported on in this talk, is to generalize this result from real reflection groups to unitary reflection groups. Before reporting on our generalizations of [10, Thm. 1.1] in [4], I want to briefly recall the background and motivation for Terao’s work. In [2, Conj. 3.3], Edelman and Reiner conjectured that the cones over the extended Shi arrangements and the extended Catalan arrangements are free with prescribed exponents. Edelman and Reiner were able to prove their conjecture in case of the extended Catalan arrangement for the underlying root system of type A in loc. cit. If the above conjecture is true, then Ziegler’s theorem [13, Thm. 11] implies the freeness of the multi-arrangements of the underlying Weyl arrangements with constant multiplicity at every hyperplane (with exponents derived from the conjecture). Terao’s theorem [10, Thm. 1.1] confirms this consequence of the conjecture. Ultimately, the conjecture of Edelman and Reiner was proved by Yoshinaga in [12] by combining [10, Thm. 1.1] with a local criterion for freeness, [12, Thm. 2.5]. From now on suppose that W is an irreducible unitary reflection group with reflection representation V ∼= Cℓ. Denote the set of reflections of W byR = R(W ), and the associated reflection arrangement in V by A = A (W ). Following [3], the Coxeter number of W is given by 1X1 ℓeH=ℓ|R| + |A |, H∈A generalizing the usual Coxeter number of a real reflection group to irreducible unitary reflection groups. Let Irr(W ) denote the irreducible complex representations of W up to isomorphism. For U in Irr(W ) of dimension d, denote by  expU(W ) :=n1(U )≤ . . . ≤ nd(U ) 106Oberwolfach Report 2/2018 the U -exponents of W given by the d homogeneous degrees in the coinvariant algebra of W in which U appears. In particular, the exponents of W are  exp(W ) := expV(W ) =n1(V )≤ . . . ≤ nℓ(V ) and the coexponents of W are  coexp(W ) := expV∗(W ) =n1(V∗)≤ . . . ≤ nℓ(V∗). The group W is well-generated if ni(V ) + nℓ+1−i(V∗) = h, e.g., see [7, 6, 1]. For H∈ A , let eHdenote the order of the point-wise stabilizer of H in W . Consider the order multiplicity function ω : A→ N, ω(H) = eH for each hyperplane H∈ A . For m ∈ N let mω and mω+1 denote the multiplicities defined by mω(H) = meHand mω(H) + 1 = meH+ 1 for H∈ A , respectively. The following is [4, Thm. 1.1], generalizing [10, Thm. 1.1] to the case of wellgenerated finite unitary reflection groups. Theorem 1. Let W be an irreducible, well-generated unitary reflection group with reflection arrangement A . Let ω : A→ N given by ω(H) = eH, and let m∈ N. Then (i) the reflection multi-arrangement (A , mω) is free with exponents  exp(A , mω) =mh, . . . , mh, (ii) the reflection multi-arrangement (A , mω + 1) is free with exponents  exp(A , mω + 1) =mh + n1(V∗), . . . , mh + nℓ(V∗).  Note from above that coexp(W ) = expV∗(W ) =n1(V∗), . . . , nℓ(V∗). In the special case when W is a Coxeter group, Theorem 1 recovers [10, Thm. 1.1], as then ω≡ 2 and coexp(W ) = exp(W ). In [4, Thm. 4.20], we prove a more general version of Theorem 1 based on a generalization of Yoshinaga’s approach [11] to [10, Thm. 1.1]. More precisely, we first extend Yoshinaga’s construction of a basis of the module of derivations and of Saito’s Hodge filtration to well-generated unitary reflection groups by using recent developments of flat systems of invariants in the context of isomonodromic deformations and differential equations of Okubo type due to Kato, Mano and Sekiguchi [5]. Our second main result [4, Thm. 1.2] extends Theorem 1 further to the infinite three-parameter family W = G(r, p, ℓ) of imprimitive reflection groups. It turns out that the corresponding multi-arrangements are also free. However, the description of the exponents is considerably more involved and depends on the representation theory of the Hecke algebra associated to the group W . To this end, let Ψ denote the permutation on Irr(W ) introduced by Malle in [6, Sec. 6C], Topology of Arrangements and Representation Stability107 having the semi-palindromic property on the fake degrees of W . This is, for any U in Irr(W ) of dimension d, we have ni(U ) + nd+1−i(Ψ(U∗)) = hU, P where hU=|A | −r∈Rχ(r)/χ(1), where χ is the character of U . A direct calculation shows that hV= h is the Coxeter number of W . Theorem 2. Let W = G(r, p, ℓ) with reflection arrangement A . Let ω : A→ N given by ω(H) = eH, and let m∈ N. Then (i) the reflection multi-arrangement (A , mω) is free with exponents  exp(A , mω) =mh, . . . , mh, (ii) the reflection multi-arrangement (A , mω + 1) is free with exponents  exp(A , mω + 1) =mh + n1(Ψ−m(V∗)), . . . , mh + nℓ(Ψ−m(V∗)).  Note this time that expΨ−m(V∗)(W ) =n1(Ψ−m(V∗)), . . . , nℓ(Ψ−m(V∗)). Remarks 3. (i). Observe that the group G(r, p, ℓ) is well-generated if and only if p∈ 1, r. Moreover, Ψ(V∗) = V∗if and only if W is well-generated [6, Cor. 4.9]. Thus, Theorem 2 extends Theorem 1 to the class of imprimitive reflection groups that are not well-generated. (ii). While the reflection arrangements of the reflection groups G(r, 1, ℓ) and G(r, p, ℓ) for 1 < p < r coincide, the multi-arrangements in Theorem 2 depend on the structure of the underlying group. (iii). Computational evidence for small values for m suggest that Theorem 2 extends to the remaining eight irreducible complex reflection groups of exceptional type that are not well-generated. References
[284] D. Bessis, Finite complex reflection arrangements are K(π, 1). Ann. of Math., 181 (2015), 809–904. · Zbl 1372.20036
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[288] M. Kato, T. Mano and J. Sekiguchi, Flat structure on the space of isomonodromic deformations. https://arxiv.org/abs/1511.01608 (2015).
[289] G. Malle, On the rationality and fake degrees of characters of cyclotomic algebras. J. Math. Sci. Univ. Tokyo 6 (1999), no. 4, 647–677. · Zbl 0964.20003
[290] P. Orlik and L. Solomon, Unitary reflection groups and cohomology, Invent. Math. 59, (1980), 77–94. · Zbl 0452.20050
[291] H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320. · Zbl 0509.14006
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[296] G. Ziegler, Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986), 345–359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989. Homology of Artin groups: A combinatorial group theoretic approach Ye Liu (joint work with Toshiyuki Akita) Given an arbitrary Coxeter system (W, S), or equivalently a Coxeter graph Γ, the Artin group A = A(Γ) associated to Γ is obtained from the standard Coxeter presentation of W = W (Γ) by dropping the relations s2= 1 for s∈ S. The celebrated K(π, 1) conjecture asserts that A admits a nice K(π, 1) space by realizing W as a reflection group acting on a Tits cone [6]. Homology of Artin group A(Γ) can be computed from the conjectural space if the K(π, 1) conjecture is proved. However the K(π, 1) conjecture has only been proved for certain classes of Artin groups (see [1] or [6]) and remained open in general. In this talk, we start a combinatorial group theoretic approach to the computation of homology of Artin groups, without assuming that the K(π, 1) conjecture holds. The first step is the following easy observation. Theorem 1. For an arbitrary Coxeter graph Γ, we have H1(A(Γ); Z) ∼= Zc(Γ), where c(Γ) is the number of connected components of Γodd, the induced subgraph of Γ obtained by deleting edges with even number label and edges with∞ label. This result follows from the fact that H1(A; Z) is the abelianization of A, and the latter is obtained by imposing commutating relations for each pair of standard generators. Our next result is less trivial. Let us define the following numbers associated to a Coxeter graph Γ. Denote by P (Γ) the set of pairs of non-adjacent vertices of Γ. We say thats, t ≡ s, t′ in P (Γ) if t, t′∈ S are joined by an edge with odd number label (i.e. m(t, t′) is odd). Let∼ be the equivalence relation in P (Γ) generated by≡. Now define n1(Γ) = # (P (Γ)/∼), n2(Γ) = #s, t ⊂ S | m(s, t)≥ 4, m(s, t) is even and n3(Γ) = rank H1(Γodd; Z). Theorem 2 ([1]). For an arbitrary Coxeter graph Γ, we have H2(A(Γ); Z2) ∼= Zn(Γ)2, where n(Γ) = n1(Γ) + n2(Γ) + n3(Γ). The idea of proof is to use the Hopf’s formula. Topology of Arrangements and Representation Stability109 Theorem 3 (Hopf’s formula). If a group G has a presentationhS | Ri, then N∩ [F, F ] H2(G; Z) ∼=, [F, N ] where F = F (S) is the free group generated by S and N = N (R) is the normal closure of R. Applying Hopf’s formula to Coxeter groups and Artin groups with their standard presentations, we may construct explicitly second homology classes as cosets x[F, N ] with x∈ N ∩[F, F ] as above. We manage to find a set Ω(W ) of generators of H2(W ; Z) and a set Ω(A) of generators of H2(A; Z) such that the homomorphism p∗: H2(A; Z)→ H2(W ; Z) induced by the natural map p : A→ W maps Ω(A) onto Ω(W ). Moreover we have by construction #Ω(W ) = n(Γ). On the other hand, Howlett proved the following. Theorem 4 ([5]). For an arbitrary Coxeter graph Γ, we have H2(W (Γ); Z) ∼= Zn(Γ)2. Hence we know that Ω(W ) is a basis of H2(W ; Z), and we have proved that p∗ is surjective. Theorem 2 follows without difficulties. We expect that the above computation extends to higher homology of Artin groups. In fact, we have the similar ingredients: H3(W ; Z) has been computed in
[297] and [2], the higher Hopf formulae have been studied in [3]. References
[298] T. Akita, Y. Liu, Second mod 2 homology of Artin groups, Algebr. Geom. Topol. 18 (2018), 547–568. · Zbl 06828013
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[303] L. Paris, K(π, 1) conjecture for Artin groups, Ann. Fac. Sci. Toulouse Math. 23 (2014), 361–415. Milnor fiber complexes and some representations Alexander R. Miller H. O. Foulkes discovered some amazing characters for the symmetric group Snby summing Specht modules of certain ribbon shapes according to height [7]. These characters have some remarkable properties and have been the subject of many investigations, most recently because of connections with adding random numbers, shuffling cards, the Veronese embedding, and combinatorial Hopf algebras, see [2, 5, 6, 9, 17]. We give a new approach to these characters which works for a wide variety of reflection groups. The approach is geometric and based on an object 110Oberwolfach Report 2/2018 called the Milnor fiber complex. It gives new results and it unifies, explains, and extends previously known (type A) ones. This work appears in [12, 13, 14, 15]. Coxeter and Shephard groups. Let V be an ℓ-dimensional vector space over C, and let G be a finite group with presentation (1)h r1, r2, . . . , rℓ| ripi= 1, rirjri. . .= rjrirj. . .i6= j i | z | z mijtermsmjiterms where pi≥ 2, mij= mji≥ 2, and pi= pjwhen mijis odd. Write R =r1, . . . , rℓ. Finite Coxeter groups are the ones where each piis 2. In general G has a Coxeterlike diagram Γ and a canonical faithful representation G⊂ GL(V ) as a (complex) reflection group in which the generators riact on V as reflections in the sense that they have finite order and the fixed spaces ker(1− ri) are hyperplanes [10]. The group is identified with its canonical representation as a reflection group and called irreducible if it acts irreducibly on V . Being irreducible is equivalent to the diagram having exactly one connected component. Finite groups with presentation (1) were classified in [10]. The irreducible ones are precisely the finite irreducible Coxeter groups and the groups known as Shephard groups (symmetry groups of objects called regular complex polytopes [3] studied by Shephard and Coxeter). Milnor fiber complex. Associated to G is an abstract simplicial complex ∆ with simplices (labeled by) cosets ghIi of standard parabolic subgroups hIi (I ⊂ R) with face relation “ghIi is a face of hhJi” ⇔ ghIi ⊃ hhJi, and with G acting by left translation. If G is a Coxeter group, then this is the classical abstract description of the Coxeter complex [26]. See [22, 19, 12, 15] for details, geometry, and history. Foulkes characters. Each type-selected subcomplex ∆S(S⊂ R) is a bouquet of spheres, and we call the CG-module on the top reduced homology group H|S|−1(∆S) a ribbon representation, see [12]. Its character ρSis an alternating sum of characters induced by principal characters of parabolic subgroups [12]. The (generalized) Foulkes characters defined in [13] are X (2)φk=ρS(k = 0, 1, . . . , ℓ). S⊂R |S|=k An immediate benefit of this approach is the following formula [13, Theorem 1] Xℓ (3)φk(g) =(−1)k−iℓ− if k− ii−1(∆g) i=0 where ∆g=σ ∈ ∆ : gσ = σ and fk(Σ) is the number of k-simplices in Σ. The face numbers fk(Σ) can be computed with a formula of Orlik and Solomon. Assume G irreducible. Let L be the set of all intersections of reflecting hyperplanes ordered by reverse inclusion, and let µ be the M¨obius function. For X∈ L define BX(t) = (−1)dim XPY≥Xµ(X, Y )(−t)dim Y. Let d1≤ d2≤ . . . ≤ dℓbe the basic degrees of G. Then Orlik [22] (after Orlik–Solomon in the Coxeter case) proved X (4)fi−1(∆g) =BY(d1− 1) Y Topology of Arrangements and Representation Stability111 where the sum is over all i-dimensional subspaces Y above Vg= ker(1− g) in L. Elucidating and generalizing classical (type A) results. Our approach elucidates and extends the type A theory (due to Foulkes, Kerber–Th¨urlings, Diaconis– Fulman, and Isaacs), which previously rested on ad hoc proofs by induction. See [13]. For example, if G is the wreath product Zr≀ Sn(Zrcyclic of order r), then L is a Dowling lattice and the restrictions LXdepend only on the dimension of X∈ L, so that by (3) and (4) the φi’s depend only on fixed-space dimension in the sense that φi(g) = φi(h) whenever dim Vg= dim Vh. The r = 1 case of this is the classical fact that the Foulkes characters φi(g) of Sndepend only on the number of cycles of g. The only previous proof of this for Snis the original one due to Foulkes [7] which uses the Murnaghan–Nakayama rule and induction. Adding random numbers. Interestingly, these generalized Foulkes characters have recently been connected to adding random numbers in other number systems. Persi Diaconis and Jason Fulman [6] connected the hyperoctahedral ones (type B) to adding random numbers in balanced ternary and other number systems that minimize carries, and Nakano–Sadahiro [16] connected the Foulkes characters for Zr≀ Snto a generalized carries process and riffle shuffles. New phenomena. If G is the wreath product Zr≀ Sn, then the Foulkes characters form a basis for the space of class functions χ(g) of G that depend only on length ℓ(g) = mink : g = t1t2. . . tk, tia reflection, see [13, 14]. Danny Goldstein, Robert M. Guralnick, and Eric M. Rains together made the remarkable experimental observation [18] that in fact the hyperoctahedral Foulkes characters play the role of irreducibles among the hyperoctahedral characters that depend only on length, in the sense that the characters of the hyperoctahedral group Bnthat depend only on length are precisely the N-linear combinations of the hyperoctahedral Foulkes characters. We prove this conjecture in [14]. In fact we prove that the same is true for all wreath products Zr≀ Snwith r > 1, not just r = 2. It is an open problem to give a nice description of the characters χ(g) for Sn (r = 1) that depend only on ℓ(g), or in other words, that depend only on the number of cycles of g. Kerber [9, p. 306] noticed that already for S5the N-linear combinations of Foulkes character do not account for all the characters of S5that depend only on length. In [14] we prove that this is always the case for symmetric groups Snwith n≥ 3. Note: This line of investigation makes sense for any finite group with given set of generators closed under conjugation. Curious classification. In [13] we determined all the irreducible cases of G where the φi’s depend only on fixed-space dimension. This led to a curious classification with 11 equivalent conditions [13, Thm. 14]. For example, we find that the φi’s depend only on fixed-space dimension if and only if the sequence of basic degrees d1, d2, . . . , dℓis arithmetic. Another equivalent condition is that the diagram of G contains no subdiagram of type D4, F4, or H4. We recently found this condition in [1] Abramenko’s answer to a geometric problem: In which Coxeter complexes ∆ are all walls ∆r(r a reflection) Coxeter complexes? In [15] we extend Abramenko’s 112Oberwolfach Report 2/2018 result to Milnor fiber complexes in two ways and find another equivalent condition for the Foulkes characters to depend only on fixed-space dimension. In the course of that work we also discovered a beautiful enumerative condition [15, Thm. 11]: if G is irreducible, then the diagram contains no subdiagram of type D4, F4, or H4 if and only if for each g∈ G the number of top cells in ∆gis given by (5)fp−1(∆g) = d1d2· · · dp,p = dim Vg. References
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[329] J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386, 1974. Problem Session Nate Harman, Aur´elien Djament, Roberto Pagaria, Jeremy Miller, Weiyan Chen, Jesse Wolfson, Masahiko Yoshinaga, Alexander R. Miller, Graham Denham, Dan Petersen, Michael Falk We hosted two evening problem sessions during the workshop. Various participants from very different backgrounds proposed open questions to the audience. The sessions stimulated active discussions among the participants. The problems are collected below in the order that they were proposed. 1. Nate Harman (University of Chicago). The following standard theorem from representation theory of the symmetric groups roughly says that “FI-modules see all representations of polynomial growth”: Theorem 1. Suppose Vnis a sequence of irreducible representations of Sn. If there exists a constant d such that dim Vn< ndfor all n≫ 0, then either Vnor Vn⊗ sign is a factor of an FI-module generated in degree at most d. With the motivation to understand low dimensional representations of the braid group, we ask the following question: Question 1. Is there an analog if we replace Snby the braid group Bn? Conjecture 1. All representations of Bnwith slow growth come from finitely generated modules over certain category. Ivan Marin remarked that the conjecture is known for linear growth, e.g., in the case when the dimension is n− 1. (see [5]) 2. Aur´elien Djament (CNRS, Nantes). Let k be a maximal ordered field (e.g., k = R). Let’s consider the following monomorphisms between orthogonal groups, for all n and i: On(k)× Oi(k) ֒→ On,i(k). Question 2. Does this map induces an isomorphism of Hd(−, Z) for n ≫ d, i? The homology here is understood as the group homology of discrete groups. For i = 1, a theorem of B¨okstedt, Brun, and Dupont [2] shows that the answer is yes for d < n. 114Oberwolfach Report 2/2018 3. Roberto Pagaria (SNS, Pisa). LetA be a central toric arrangement in the torus T , letL be its poset of layers, and let M(A) be the complement in T . Consider the cohomology algebra with rational coefficients H•(M (A)) and its associated graded algebra with respect to the Leray filtration, gr H•(M (A)). Theorem 2 ([9, Theorem 4.6]). The poset of layersL determines the algebra gr H•(M (A)). Moreover, a stronger statement holds: the posetL describes the cohomology algebra H•(M (A)). We ask whether the converse holds: Question 3. Does the cohomology algebra H•(M (A)) determine the poset of layersL? The analogous statement in the setting of hyperplane arrangements has a negative answer (see [4] or [3]), but the combinatorics of toric arrangements is richer than combinatorics of hyperplane arrangements. In order to solve this problem, a deep study of characteristic varieties of toric arrangements could be useful. 4. Jeremy Miller (Purdue University). It was previously known that (by Proposition A.1. of [10]) H2(Out(Fn), Zn) = Z/(n− 1)Zfor n≥ 9. Question 4. Is Hi(Out(Fn), (Z/pZ)n) eventually periodic as n increases? If yes, we ask the same question replacing the twisted coefficient (Z/p)nby a polynomial functor, or a VIC-module. Andrew Snowden remarked that this question is motivated by Rohit Nagpal’s work (Theorem C of [8]), and the subsequent works by various authors. 5. Weiyan Chen (University of Minnesota, Twin Cities). Definition 1. The Schwarz genus of a cover f : Y→ X is the minimum number k such that there exists a cover of X by connected open subsets U1, U2, ..., Ukwhere the restriction of f to each open subset is trivial. Alex Suciu remarked that Schwarz genus is a special case of a more general concept called “sectional category” which can be defined for any fiberation. Consider the following spaces. X :=Smooth complex homogeneous polynomials F (x, y, z) of degree 3/C× Y :=(F, p) ∈ X × CP2: p is a flex on the smooth cubic curve F = 0. Then Y is a covering space of X of degree 9 (since there are 9 flexes on every smooth cubic curve). Question 5. What is the Schwarz genus of the flex cover Y→ X as defined above? Topology of Arrangements and Representation Stability115 Weiyan Chen remarked that he was able to bound the number to be no smaller than 3 and no larger than 9. This question is motivated by the work of Smale
[330] , who first interpreted the Schwarz genus as a lower bound for the topological complexity of any algorithm solving certain problems. In a similar way, an answer to the question above gives a lower bound for any algorithm that finds a flex for any given smooth cubic curve. One can ask the similar question for many enumerate problems. 6. Jesse Wolfson (University of California, Irvine). Theorem 3 (Jacobi, 1850). Every smooth quartic plane curve has 28 bitangent lines. Let H4,2denote the space of smooth quartic curves in CP2. Precisely, let 4+2 H4,2= (P (2) Σ)/PGL3(C) where Σ denote the discriminant locus containing singular homogeneous quartic polynomials that give singular quartic curves. Let H4,2(1) denote the space of smooth quartic curves equipped with a bitangent line. Jacobi’s theorem tells us that H4,2(1) is a degree 28 cover of H4,2. Other classical covers of interest are H4,2(S), H4,2(A), and H4,2(C), the moduli of quartics equipped with a Steiner complex, Aronhold set, and Cayley octad respectively. These give degree 63, 288, and 36 covers of H4,2. Question 6. What is H∗(H4,2(1), Q)? Similarly, what is H∗(H4,2(X), Q) for X = S, A or C? Weiyan Chen commented that the computation of H∗(H4,2(1), Q) can be found in a paper by Orsola Tommasi [13]. Dan Petersen commented that work of Olof Bergvall [1] is also relevant. 7. Masahiko Yoshinaga (Hokkaido University). Definition 2. f : Z→ C is quasi-polynomial if there exists ρ > 0 and g1(t), ..., gρ(t)∈ C[t] such that  g1(n),if n = 1 mod ρ  g2(n),if n = 2 mod ρ f (n) =  · · · gρ(n),if n = ρmod ρ. Furthermore, f has the GCD-property if gi(t) = gj(t)if (i, ρ) = (j, ρ). In other words, f has the GCD-property if the constituent depends only on the GCD with the period ρ. Question 7. Which rational polytope has Ehrhart quasi-polynomial with GCDproperty? 116Oberwolfach Report 2/2018 Example 1. Let P1= [0, 1/3].  n+33,if n = 0mod 3 LP1(n) =n+2,if n = 1mod 3 n+13 3,if n = 2mod 3 Hence the Ehrhart quasi-polynomial of P1does not have GCD-property. Example 2. Let P2= [1/3, 4/3]. ( n + 1,if n = 0mod 3 LP2(n) = n,if n = 1, 2 mod 3 Hence the Ehrhart quasi-polynomial of P2has GCD-property. Example 3. The Ehrhart quasi-polynomial of the fundamental alcove of a root system has GCD-property. [14] Question 8. Let P be an integral zonotope in Zℓ. Let a∈ Qℓ. Does the translated zonotope P′= a + P have the Ehrhart quasi-polynomial with GCD-property? 8. Alexander R. Miller (Universit¨at Wien). For λ, µ partitions of n, let χλ denote the character of the irreducible Sn-representation corresponding to λ. Here is an observation: as n→ ∞, (1)Prob(χλ(g) = 0 for λ⊢ n and g ∈ Sn)−→ 1. Conjecture 2. As n→ ∞, (2)Prob(χλ(µ) = 0mod 2 for λ, µ⊢ n) −→ 1. In other words, the conjecture says that an entry chosen uniformly at random from the character table of Snis even with probability→ 1 as n → ∞. The probability measures in (1) and in (2) are different: the former is uniform over elements in Sn, while the latter is uniform over conjugacy classes of Sn. Alexander Miller remarked that he has done some computer experiments which suggest that the probability in (2) converges to 1 following the graph of p 2π−1arctan(n/2− 1). It was also remarked that similar unexpected behavior occurs for other primes. 9. Graham Denham (University of Western Ontario). If we set Y Q(x1, . . . , xn) :=(xi− xj), 1≤i<j≤n then the ordered configuration space becomes Conf(n, C) =(x1, . . . , xn) : Q6= 0. Consider the Milnor fiber Fn:=(x1, . . . , xn) : Q2= 1. Topology of Arrangements and Representation Stability117 Fncarries commuting actions of two groups: an action of Snby permuting the coordinates, and an action of the group µn(n−1)of n(n− 1)-th roots of unity by the diagonal multiplication. Question 9. What is H∗(Fn, Q) as a (Sn× µn(n−1))-module? A stability phenomenon was discovered by Simona Settepanella: the action of µn(n−1)on Hi(Fn, Q) is trivial when n≫ i. Question 10. Is there a categoryC similar to F I such that Hi(Fn, Q) becomes a finitely generatedC-module? Notice that there is no obvious way to make Hi(Fn, Q) an FI-module, since there is no natural map between Fnand Fn+1. Kevin Casto remarked that it may be fruitful to study the cohomology of the maximal abelian cover of Conf(n, C), which is also the classifying space of the commutator subgroup of the pure braid group, and whose cohomology is naturally an FI-module. 10. Dan Petersen (Stockholm University). Theorem 4 (Randal-Williams–Wahl, [11]). If Vnis a sequence of polynomial coefficient system for the braid group Bn, then H∗(Bn, Vn) stabilizes as n→ ∞. Question 11. Let Pnbe the pure braid groups. The collection H∗(Pn, Vn) is an FI-module. Is it finitely generated? Notice that the theorem of Randal-Williams–Wahl above implies that H∗(Pn, Vn) satisfies multiplicity stability. One can also prove that the homology grows polynomially in n. Since the (co)homology of pure braid groups was the original motivating example for the theory of representation stability, it is surprising that this is not known. 11. Michael Falk (Northern Arizona University). Let Md,ndenote the space of unlabeled affine arrangements of n hyperplanes in general position in Cd. Problem 1. Find a presentation for π1(Md,n) that specializes to Artin’s presentation of the full braid group for d = 1. There has been some work on the space of labeled affine arrangements in general position [6, 7], but one would expect a nicer presentation for the unlabeled version, since that is the case for d = 1: there are fewer generators and more symmetric relations in Artin’s presentation of the full braid group, than in the standard presentation of the pure braid group. After a brief dicussion, Alex Suciu asked the following question: Question 12. Does Md,nhave a nice compactification? 118Oberwolfach Report 2/2018 12. Dan Petersen (Stockholm University). Suppose E and V are finite dimensional vector spaces over C. Define a commutative algebra A := C⊕ E where elements in E multiply to 0 (a square zero extension). Let g := Lie(V ), a free Lie algebra. Question 13. What is the Lie algebra homology H∗(g⊗ A)? The answer should be expressed as a sum of polynomial functors in E and V . When E is 1-dimensional, an answer can be calculated by hand (even this case is not obvious). A complete answer to the question in its general form will help us understand the cohomology of the “link” of the tropical moduli space of curves M2,n. References
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[344] M. Yoshinaga, Worpitzky partitions for root systems and characteristic quasi-polynomials, to appear in Tohoku Math. J. 70 (2018), arXiv:1501.04955 Reporter: Christin Bibby Topology of Arrangements and Representation Stability119 Participants Prof. Dr. Takuro Abe Institute of Mathematics for Industry Kyushu University Fukuoka 819-0395 JAPAN Dr. Christin N. Bibby Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109-1043 UNITED STATES Dr. Rachael Jane Boyd Department of Mathematical Sciences University of Aberdeen King’s College Aberdeen AB24 3UE UNITED KINGDOM Prof. Dr. Tara Brendle School of Mathematics and Statistics University of Glasgow 15 University Gardens Glasgow G12 8QW UNITED KINGDOM Dr. Filippo Callegaro Dipartimento di Matematica ”L.Tonelli” Universita di Pisa Largo Bruno Pontecorvo, 5 56127 Pisa ITALY Luigi Caputi Fakult¨at f¨ur Mathematik Universit¨at Regensburg Universit¨atsstrasse 31 93053 Regensburg GERMANY Kevin P. Casto Department of Mathematics The University of Chicago 5734 South University Avenue Chicago, IL 60637-1514 UNITED STATES Dr. Weiyan Chen Department of Mathematics University of Minnesota 127 Vincent Hall 206 Church Street S. E. Minneapolis, MN 55455 UNITED STATES Prof. Dr. Daniel C. Cohen Department of Mathematics Louisiana State University Baton Rouge LA 70803-4918 UNITED STATES Dr. Zsuzsanna Dancso School of Mathematics and Statistics The University of Sydney Sydney NSW 2006 AUSTRALIA Prof. Dr. Emanuele Delucchi D´epartement de Math´ematiques Universit´e de Fribourg Perolles Chemin du Musee 23 1700 Fribourg SWITZERLAND Prof. Dr. Graham Denham Department of Mathematics Middlesex College University of Western Ontario London ON N6A 5B7 CANADA 120Oberwolfach Report 2/2018
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