Recent zbMATH articles in MSC 32 https://zbmath.org/atom/cc/32 2022-09-13T20:28:31.338867Z Werkzeug On polynomials counting essentially irreducible maps https://zbmath.org/1491.05064 2022-09-13T20:28:31.338867Z "Budd, Timothy" https://zbmath.org/authors/?q=ai:budd.timothy-g Summary: We consider maps on genus-$$g$$ surfaces with $$n$$ (labeled) faces of prescribed even degrees. It is known since work of \textit{P. Norbury} [Math. Res. Lett. 17, No. 3, 467--481 (2010; Zbl 1225.32023)] that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$$g$$ curves with $$n$$ labeled points and is given by a symmetric polynomial $$N_{g,n} (\ell_1, \ldots, \ell_n)$$ in the face degrees $$2\ell_1, \ldots, 2\ell_n$$. We generalize this by restricting to genus-$$g$$ maps that are essentially $$2b$$-irreducible for $$b\geqslant 0$$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $$2b$$ and each such cycle of length $$2b$$ is required to bound a face of degree $$2b$$. The enumeration of such maps is shown to be again given by a symmetric polynomial $$\hat{N}_{g,n}^{(b)}(\ell_1, \ldots, \ell_n)$$ in the face degrees with a polynomial dependence on $$b$$. These polynomials satisfy (generalized) string and dilaton equations, which for $$g\leqslant 1$$ uniquely determine them. The proofs rely heavily on a substitution approach by \textit{J. Bouttier} and \textit{E. Guitter} [Electron. J. Comb. 21, No. 1, Research Paper P1.23, 18 p. (2014; Zbl 1300.05070)] and the enumeration of planar maps on genus-$$g$$ surfaces. Two algorithms for computing the general component of jet scheme and applications https://zbmath.org/1491.13036 2022-09-13T20:28:31.338867Z "Cañón, Mario Morán" https://zbmath.org/authors/?q=ai:canon.mario-moran "Sebag, Julien" https://zbmath.org/authors/?q=ai:sebag.julien Let $$X$$ be an integral variety over a perfect field $$k$$, $$\mathcal L_m(X)$$ its jet scheme of level $$m\in \mathbb N$$ and $$\mathcal L_{\infty}(X)$$ its arc scheme. The general component $$\mathcal G_m(X)$$ of $$\mathcal L_m(X)$$ is the Zariski closure of $$\mathcal L_m(\mathrm{Reg}(X))$$. If $$X$$ is smooth on $$k$$ then the geometry and topology of $$\mathcal L_m(X)$$ are well understood. In this paper the authors consider the case $$X$$ is not smooth and study some properties of the general component $$\mathcal G_m(X)$$ by means of a smooth birational model of $$X$$. Indeed, under the further hypothesis that $$X$$ is affine embedded in $$\mathbb A^N_k$$, the authors prove that a birational model of $$X$$ provides a description of $$\mathcal G_m(X)$$ that gives rice to an algorithm which computes a Groebner basis of the defining ideal of $$\mathcal G_m(X)$$ in $$\mathbb A^N_k$$ as a subscheme of $$\mathcal L_m(X)$$ (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7. The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and which encodes the geometry of all $$\mathcal G_m(X)$$'' (Sections 8 and 9). Reviewer: Francesca Cioffi (Napoli) Equisingularity of families of functions on isolated determinantal singularities https://zbmath.org/1491.14003 2022-09-13T20:28:31.338867Z "Carvalho, R. S." https://zbmath.org/authors/?q=ai:carvalho.r-s "Nuño-Ballesteros, J. J." https://zbmath.org/authors/?q=ai:nuno-ballesteros.juan-jose "Oréfice-Okamoto, B." https://zbmath.org/authors/?q=ai:orefice-okamoto.b "Tomazella, J. N." https://zbmath.org/authors/?q=ai:tomazella.joao-nivaldo In [\textit{J. J. Nuño-Ballesteros} et al., Math. Z. 289, No. 3--4, 1409--1425 (2018; Zbl 1400.32015)], some of the authors showed that the family of varieties $$\left \{(X_t, 0) \right \}_{t \in D}$$ is Whitney equisingular if and only if it is good and all the polar multiplicities $$m_i(X_t, 0)$$, $$i = 0, \dots, d$$ are constant on $$t$$. In this paper, the authors also characterize the Whitney equisingularity for analytic families of function germs $$F=\left \{f_t : (X_t, 0) \to (\mathbb{C}, 0)\right \}_{t \in D}$$ with isolated critical points, where $$(X_t, 0)$$ are $$d$$-dimensional isolated determinantal singularities. For this, the authors introduce the $$(d-1)$$th polar multiplicity of the fiber $$Y:=f^{-1}(0) \subset X$$ of a function germ $$f:(X,0) \to (\mathbb{C},0)$$ with isolated singularity. The main result is: the family $$F$$ is Whitney equisingular if only if $$(\mathcal{X},0)=\left \{(X_t, 0) \right \}_{t \in D}$$ is a good family, $$m_i(X_t,0)$$, $$i=0, \dots, d$$ and $$m_k(Y_t,0)$$, $$k=0,\dots, d-1$$ are constant on $$t\in D$$. Reviewer: Daiane Alice Henrique Ament (Lavras) Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities https://zbmath.org/1491.14004 2022-09-13T20:28:31.338867Z "Hussain, Naveed" https://zbmath.org/authors/?q=ai:hussain.naveed "Yau, Stephen S.-T." https://zbmath.org/authors/?q=ai:yau.stephen-shing-toung "Zuo, Huaiqing" https://zbmath.org/authors/?q=ai:zuo.huaiqing It is well known that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but solvable and nilpotent Lie algebras are not. In the paper under review, the authors give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. In particular, they get the correspondence between the nilpotent Lie algebras of dimension less than or equal to $$7$$ and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to $$1$$. Reviewer: Rong Du (Shanghai) Multiplicity, regularity and blow-spherical equivalence of real analytic sets https://zbmath.org/1491.14005 2022-09-13T20:28:31.338867Z "Sampaio, José Edson" https://zbmath.org/authors/?q=ai:sampaio.j-edson The author is interested in studying multiplicity and regularity on an analytic setting and, in this context, he introduces a weaker variation of the concept of blow-spherical equivalence introduced by \textit{A. Fernandes} et al. [Indiana Univ. Math. J. 66, No. 2, 547--557 (2017; Zbl 1366.14051)], which he also names blow-spherical equivalence or equivalence under a blow-spherical homeomorphism. Equivalence under this concept lives strictly between topological equivalence and sub analytic bi-Lipschitz equivalence and, also, between topological equivalence and differential equivalence. Recently, the author conjectured that if $$X$$ and $$Y$$ are two real analytic sets included in $$\mathbb{R}^n$$ and there is a bi-Lipschitz homeomorphism: $$\varphi: (\mathbb{R}^n, X, 0) \rightarrow (\mathbb{R}^n, Y, 0)$$, then the multiplicities at origin of $$X$$ and $$Y$$ satisfy $$m(X) \equiv m(Y) \bmod 2$$. He proved that this conjecture is true when $$n=3$$. In this paper, the author proposes the same conjecture but replacing bi-Lipschitz'' with blow-spherical''. Proposition 5.2, Theorems 5.5, 5.7 and 5.16 and Corollary 5.18 in the paper prove that the conjecture holds whenever $$n \leq 3$$ or whenever $$\varphi$$ is also image arc-analytic. The author also gives a real analogue of the Gau-Lipman's Theorem. Finally, and concerning regularity of complex analytic sets, the author proves that if a real analytic set $$X \subseteq \mathbb{R}^n$$ is blow-spherical regular at $$0 \in X$$, then $$X$$ is $$C^1$$ smooth at $$0$$ if and only if the dimension $$d$$ of $$X$$ is $$1$$. Reviewer: Carlos Galindo (Castellón) Dimension of the moduli space of a germ of curve in $$\mathbb{C}^2$$ https://zbmath.org/1491.14006 2022-09-13T20:28:31.338867Z "Genzmer, Yohann" https://zbmath.org/authors/?q=ai:genzmer.yohann The paper studies the moduli space $$\mathbb{M}(S) := \mathrm{Top}(S,0)/\mathrm{Diff}(\mathbb{C}^{2},0)$$ of the topological class of a curve germ $$(S, 0)\subset (\mathbb{C}^{2},0)$$ by the action of the group $$\mathrm{Diff}(\mathbb{C}^{2},0)$$. The authors prove an explicit formula for the generic dimension of the moduli space in terms of the minimal resolution of the curve S, by using technics from the theory of holomorphic foliations. Reviewer: Mihai-Marius Tibar (Lille) A-Hilbert schemes for $$\displaystyle\frac{1}{r}(1^{n-1},a)$$ https://zbmath.org/1491.14008 2022-09-13T20:28:31.338867Z "Jung, Seung-Jo" https://zbmath.org/authors/?q=ai:jung.seung-jo Summary: For a finite group $$G \subset \mathrm{GL}(n, \mathbb{C})$$, the $$G$$-Hilbert scheme is a fine moduli space of $$G$$-clusters, which are 0-dimensional $$G$$-invariant subschemes $$Z$$ with $$H^0(\mathcal{O}_Z)$$ isomorphic to $$\mathbb{C}[G]$$. In many cases, the $$G$$-Hilbert scheme provides a good resolution of the quotient singularity $$\mathbb{C}^n/G$$, but in general it can be very singular. In this note, we prove that for a cyclic group $$A \subset \mathrm{GL}(n, \mathbb{C})$$ of type $$\frac{1}{r}(1, \dots, 1, a)$$ with $$r$$ coprime to $$a$$, $$A$$-Hilbert Scheme is smooth and irreducible. Local dynamics of non-invertible maps near normal surface singularities https://zbmath.org/1491.14055 2022-09-13T20:28:31.338867Z "Gignac, William" https://zbmath.org/authors/?q=ai:gignac.william "Ruggiero, Matteo" https://zbmath.org/authors/?q=ai:ruggiero.matteo In the study of the dynamics of a dominant non-invertible holomorphic map germ $$f:(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)$$, a successful approach consists in investigating the dynamics of $$f$$ on modifications of $$(\mathbb{C}^2,0)$$. Here a modification $$\pi:X_\pi \to (\mathbb{C}^2,0)$$ is a proper holomorphic map that is an isomorphism over $$\mathbb{C}^2 \setminus \{0\}$$, and one studies then the dynamics of the induced (meromorphic) map $$f_\pi:X_\pi \dashrightarrow X_\pi$$ on the exceptional set $$\pi^{-1}\{0\}$$. In this memoir the authors generalize many results to the singular case, namely replacing $$(\mathbb{C}^2,0)$$ by the germ of a normal surface singularity $$(X,x_0)$$. A first main result is that the problematic situation of indeterminacy points for infinitely many powers $$f_\pi^n$$ cannot occur (except in the very special case of a finite germ at a cusp singularity). Namely, for any modification $$\pi:X_\pi \to (X,x_0)$$ one can find a modification $$\pi':X_{\pi'} \to (X,x_0)$$ dominating $$\pi$$ such that, if $$E$$ is an exceptional divisor of $$\pi'$$, then $$f_{\pi'}^n (E)$$ is an indeterminacy point of the lift $$f_{\pi'}: X_{\pi'} \dashrightarrow X_{\pi'}$$ for at most finitely many $$n$$. Moreover $$X_{\pi'}$$ can be chosen to have at most cyclic quotient singularities. As in the smooth case, the strategy to prove this is analyzing the dynamics on a space that encodes all such modifications simultaneously, namely a suitable space $$\mathcal V_X$$ of centered, rank one semivaluations on the local ring $$\mathcal O_{X,x_0}$$, with the induced $$f_*:\mathcal V_X \to \mathcal V_X$$. The result is roughly as follows: there is a subset $$S\subset \mathcal V_X$$, homeomorphic to either a point, a closed interval or a circle, such that $$f_*(S) =S$$ and for any quasimonomial valuation $$v\in \mathcal V_X$$ we have that $$f_*^n(v) \to S$$ as $$n \to \infty$$. Its proof is the core of the paper, with main technical tool the construction of a suitable distance on $$\mathcal V_X$$ and the study of its non-expanding properties. Further, the authors derive three applications. The first is an asymptotic functoriality' result, partially controlling the fact that the pull-back on the group of exceptional divisors of a modification $$\pi$$ is in general not functorial. The second treats the sequence of attraction rates of a quasimonomial $$v\in \mathcal V_X$$: it eventually satisfies an integral linear recursion relation (with a similar exception as before). The third says that the first dynamical degree of $$f$$ is a quadratic integer. It is also worth mentioning that, since the used techniques are valuative (rather than complex analytic), all results are in fact valid over an arbitrary field of characteristic zero, and that some results are even valid in positive characteristic. Reviewer: Wim Veys (Leuven) On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties https://zbmath.org/1491.14059 2022-09-13T20:28:31.338867Z "Felisetti, Camilla" https://zbmath.org/authors/?q=ai:felisetti.camilla "Shen, Junliang" https://zbmath.org/authors/?q=ai:shen.junliang "Yin, Qizheng" https://zbmath.org/authors/?q=ai:yin.qizheng Irreducible holomorphic symplectic manifolds, that is simply connected compact Kähler manifolds $$M$$ such that there exists a symplectic form $$\sigma \in H^0(M, \Omega_M^2)$$ that generates all the other holomorphic forms, have been studied intensively since their introduction as part of the Beauville-Bogomolov decomposition. Since the generalisation of the decomposition theorem to klt spaces, cf. [\textit{A. Höring} and \textit{T. Peternell}, Invent. Math. 216, No. 2, 395--419 (2019; Zbl 07061101)], singular irreducible symplectic varieties appear as a natural extension of this important class of manifolds. In this paper the authors consider irreducible symplectic projective varieties that admit a Lagrangian fibration $$\pi: M \rightarrow B$$, i.e. a fibration onto a normal projective variety $$B$$ such that the general fibre is an abelian variety of dimension $$\frac{1}{2}\dim M$$. In this case the perverse $$t$$-structure on the constructible derived category of $$B$$ induces a filtration on the intersection cohomology $$\mbox{IH}^*(M, \mathbb C)$$, and one denotes by $$^{\mathfrak p}\mbox{Ih}^{i,j}(\pi)$$ the perverse numbers determined by the graded pieces of the filtration. The first main result of this paper is that the perverse numbers $$^{\mathfrak p}\mbox{Ih}^{i,j}(\pi)$$ are a deformation invariant of Lagrangian fibrations if the second Betti number of the variety $$M$$ is at least five. Moreover the authors show that $^{\mathfrak p}\mbox{Ih}^{0,d}(\pi) = \ ^{\mathfrak p}\mbox{Ih}^{d,0}(\pi)$ vanishes for $$d$$ odd and is equal to one for $$d$$ even. As a consequence one obtains that the intersection cohomology $$\mbox{IH}^*(B, \mathbb C)$$ on the base $$B$$ is isomorphic to the cohomology of the projective space $$H^*(\mathbb P^{\frac{1}{2}\dim M}, \mathbb C)$$. Note that while for smooth symplectic manifolds one expects that the base $$B$$ of the Lagrangian fibration is actually a projective space, cp. [\textit{J.-M. Hwang}, Invent. Math. 174, No. 3, 625--644 (2008; Zbl 1161.14029)], there are examples of singular irreducible symplectic varieties where this is not the case [\textit{D. Matsushita}, Sci. China, Math. 58, No. 3, 531--542 (2015; Zbl 1317.14023)]. The authors also show that the image of the restriction map $\mbox{IH}^*(M, \mathbb C) \rightarrow H^*(M_b, \mathbb C)$ to a smooth fibre $$M_b$$ of the Lagrangian fibration is generated by a relatively ample divisor class. \newline The second part of the paper concerns the relation between the perverse numbers and the Hodge numbers $$\mbox{Ih}^{i,j}(M)$$ defined by the pure Hodge structure on $$\mbox{IH}^*(M, \mathbb C)$$. If the variety $$M$$ admits a resolution of singularities $$M' \rightarrow M$$ such that the symplectic form $$\sigma$$ pulls-back to a {\em symplectic} form on $$M'$$, the authors show that the numbers coincide, i.e. one has $\mbox{Ih}^{i,j}(M) = \ ^{\mathfrak p}\mbox{Ih}^{i,j}(\pi)$ for all $$i,j$$. As part of the proof they show that for symplectic varieties $$M$$ admitting such a symplectic resolution the LLV algebra [\textit{E. Looijenga} and \textit{V. A. Lunts}, Invent. Math. 129, No. 2, 361--412 (1997; Zbl 0890.53030)] associated with the intersection cohomology is isomorphic to $$\mathfrak{so}(b_2(M)+2)$$. Reviewer: Andreas Höring (Nice) Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups https://zbmath.org/1491.14060 2022-09-13T20:28:31.338867Z "Ito, Atsushi" https://zbmath.org/authors/?q=ai:ito.atsushi-m Summary: It is known that the automorphism group of a K-polystable Fano manifold is reductive. \textit{G. Codogni} and \textit{R. Dervan} [Ann. Inst. Fourier 66, No. 5, 1895--1921 (2016; Zbl 1370.32010)] constructed a canonical filtration of the section ring, called Loewy filtration, and conjectured that the filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a counterexample to their conjecture. Delta invariants of projective bundles and projective cones of Fano type https://zbmath.org/1491.14062 2022-09-13T20:28:31.338867Z "Zhang, Kewei" https://zbmath.org/authors/?q=ai:zhang.kewei "Zhou, Chuyu" https://zbmath.org/authors/?q=ai:zhou.chuyu The $$\delta$$-invariant (also known as the stability threshold) of a Fano variety characterizes its K-stability. It was proved by \textit{K. Fujita} and \textit{Y. Odaka} [Tohoku Math. J. (2) 70, No. 4, 511--521 (2018; Zbl 1422.14047)] and \textit{H. Blum} and \textit{M. Jonsson} [Adv. Math. 365, Article ID 107062, 57 p. (2020; Zbl 1441.14137)] that a Fano variety $$X$$ is K-semistable (resp. uniformly K-stable) if and only if $$\delta(X) > 1$$ (resp. $$\delta(X) \geq 1$$). If $$X$$ is not uniformly K-stable, the $$\delta$$-invariant equals the greatest Ricci lower bound as shown in [\textit{I. A. Cheltsov} et al., Sel. Math., New Ser. 25, No. 2, Paper No. 34, 36 p. (2019; Zbl 1418.32015); \textit{R. J. Berman} et al., J. Am. Math. Soc. 34, No. 3, 605--652 (2021; Zbl 1487.32141)]. In this article, the authors compute $$\delta$$-invariant of $$\mathbb{P}^1$$-bundles and projective cones over a Fano variety $$V$$. Suppose $$V$$ is a Fano variety of dimension $$n$$ and Fano index $$\geq 2$$. Let $$L = -\frac{1}{r}K_V$$ be an ample line bundle for some rational number $$r>1$$. Let $$\tilde{Y}:=\mathbb{P}_V(L^{-1} \oplus\mathcal{O}_V)$$ be the $$\mathbb{P}^1$$-bundle as the compactification of the total space of $$L^{-1}$$. Let $\beta_0:= \left(\frac{n+1}{n+2}\cdot \frac{(r+1)^{n+2} - (r-1)^{n+2}}{(r+1)^{n+1} - (r-1)^{n+1}} - (r-1)\right)^{-1}.$ Theorem 1.1 states that $\delta(\tilde{Y}) = \min \left\{\frac{\delta(V)r\beta_0}{1+\beta_0(r-1)}, \beta_0\right\}.$ Let $$Y$$ be the projective cone over $$V$$ with polarization $$L$$. Then $$\tilde{Y}\to Y$$ is a birational morphism that contracts the zero section $$V_0$$ to the cone point in $$Y$$. Theorem 1.4 implies that $\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}\min\left\{1, \delta(V), \frac{n+1}{r}\right\}.$ In particular, if $$V$$ is K-semistable then $$r\leq n+1$$ by \textit{K. Fujita} [Am. J. Math. 140, No. 2, 391--414 (2018; Zbl 1400.14105)], so we have $$\delta(Y) = \frac{(n+2)r}{(n+1)(r+1)}$$. Similar computations of $$\delta$$-invariants are also done for log Fano pairs $$(\tilde{Y}, aV_0+bV_\infty)$$ and $$(Y, cV_\infty)$$. Applications are included for computations of $$\delta$$-invariants of certain singular hypersurfaces, and the existence of conical Kähler-Einstein metrics on projective spaces with certain cone angle along a smooth Fano hypersurface. Reviewer: Yuchen Liu (Evanston) Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings https://zbmath.org/1491.14080 2022-09-13T20:28:31.338867Z "Đinh, Sĩ Tiệp" https://zbmath.org/authors/?q=ai:dinh.si-tiep "Jelonek, Zbigniew" https://zbmath.org/authors/?q=ai:jelonek.zbigniew "Phạm, Tiến Sơn" https://zbmath.org/authors/?q=ai:pham-tien-son. Summary: Given a closed semi-algebraic set $$X\subset\mathbb{R}^n$$ and a continuous semi-algebraic mapping $$G: X\to\mathbb{R}^m$$, it will be shown that there exists an open dense semi-algebraic subset $$\mathscr{U}$$ of $$L(\mathbb{R}^n,\mathbb{R}^m)$$, the space of all linear mappings from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$, such that for all $$F\in\mathscr{U}$$, the image $$(F+G)(X)$$ is a closed (semi-algebraic) set in $$\mathbb{R}^m$$. To do this, we study the tangent cone at infinity $$C_\infty X$$ and the set $$E_\infty X\subset C_\infty X$$ of (unit) exceptional directions at infinity of $$X$$. Specifically we show that the set $$E_\infty X$$ is nowhere dense in $$C_\infty X\cap\mathbb{S}^{n-1}$$. On finiteness theorems of polynomial functions https://zbmath.org/1491.14082 2022-09-13T20:28:31.338867Z "Koike, Satoshi" https://zbmath.org/authors/?q=ai:koike.satoshi "Paunescu, Laurentiu" https://zbmath.org/authors/?q=ai:paunescu.laurentiu Summary: Let $$d$$ be a positive integer. We show a finiteness theorem for semialgebraic $$\mathscr{RL}$$ triviality of a Nash family of Nash functions defined on a Nash manifold, generalising Benedetti-Shiota's finiteness theorem for semialgebraic $$\mathscr{RL}$$ equivalence classes appearing in the space of real polynomial functions of degree not exceeding $$d$$. We also prove Fukuda's claim, Theorem 1.3, and its semialgebraic version Theorem 1.4, on the finiteness of the local $${\mathscr{R}}$$ types appearing in the space of real polynomial functions of degree not exceeding $$d$$. Calculus of multilinear differential operators, operator $$L_\infty$$-algebras and $$IBL_\infty$$-algebras https://zbmath.org/1491.18023 2022-09-13T20:28:31.338867Z "Bashkirov, Denis" https://zbmath.org/authors/?q=ai:bashkirov.denis "Markl, Martin" https://zbmath.org/authors/?q=ai:markl.martin Summary: We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie-admissible and \texttt{Lie}$$_\infty$$-structures. As a main application, and the original motivation for the present work, we show how a certain generalization of the well-known big bracket construction of Lecomte-Roger and Kosmann-Schwarzbach encompassing the case of homotopy involutive Lie bialgebras can be obtained. Analytic extensions of representations of $$^*$$-subsemigroups without polar decomposition https://zbmath.org/1491.22003 2022-09-13T20:28:31.338867Z "Oeh, Daniel" https://zbmath.org/authors/?q=ai:oeh.daniel The problem of analytic extensions for a pair $$(G, \tau)$$ of a Lie group $$G$$ with involution $$\tau$$ is related to decomposition $$\mathfrak g=\mathfrak h\oplus \mathfrak q$$ of the Lie algebra $$\mathfrak g$$ of $$G$$ into the ($$\pm 1$$)-eigenspaces and extending a strongly continuous representation $$\pi$$ of an open $$*$$-subsemigroup $$S$$ of $$G$$ on a complex Hilbert space $$H$$ to a strongly continuous unitary representation $$\pi^c : G^c \to U(H)$$ of the 1-connected Lie group $$G^c$$ with Lie algebra $$\mathfrak g^c =\mathfrak h \oplus i\mathfrak q$$. A typical example is the one-parameter semigroup $$\mathbb R^+$$ inside the real Lie group $$\mathbb R$$ with involution $$\tau=-\mathrm{id}$$, where the corresponding infinitesimal generator $$A$$ of $$\pi$$ is self-adjoint and by functional calculus, $$\pi^c(it) := e^{itA}, t\in\mathbb R$$ gives an analytic extension on $$\mathbb R^c = i\mathbb R$$. More generally, the celeberated Lüscher-Mack theorem states that for the Lie subgroup $$H$$ of the Lie algebra $$\mathfrak h$$ and nonempty open convex cone $$C\subseteq \mathfrak q$$, invariant under the adjoint action of $$H,$$ and the $$*$$-semigroup $$\Gamma(C)$$ generated by $$H\exp(C)$$, every contraction representation $$\pi: \Gamma(C)\to B(H)$$ can be analytically continued to a strongly continuous unitary representation $$\pi^c$$ of $$G^c$$, which means that the infinitesimal generators of the one-parameter (semi)-groups of elements in $$\mathfrak h + C$$ coincide up to multiplication with $$i$$. However, since the proof of the this theorem relies on the existence of coordinates of the second kind (in the sense of Hilgert and Neeb), it only works for finite dimensional (real) Lie groups. The only known infinite dimensional extensions of the Lüscher-Mack theorem for Banach-Lie groups is when the open subsemigroup $$S$$ to start with is an Olshanski semigroup, that is, a semigroups of the form $$\Gamma(C)$$ with additional condition that the polar map $$:H \times C \to \Gamma(C);\ (h, x)\mapsto h\exp(x)$$, is a diffeomorphism, or it satisfies $$SH = S$$. In the paper under review, the analytic extension problem (to certain diffirent 1-connected Lie groups) in finite-dimensional case is discussed. The authors use Simon's exponentiation theorem to prove the existence of analytic extension on the 1-connected Lie group $$G_1^c$$, with Lie algebra $$\mathfrak g_1^c = [\mathfrak q, \mathfrak q] \oplus i\mathfrak q$$ (under certain nondegeneracy condition, known to hold for Olshanski semigroups). In their argument, they use Fröhlich's theorem as a criteria for the essential self-adjointness of unbounded operators on Hilbert spaces. Reviewer: Massoud Amini (Tehran) Effective counting of simple closed geodesics on hyperbolic surfaces https://zbmath.org/1491.37033 2022-09-13T20:28:31.338867Z "Eskin, Alex" https://zbmath.org/authors/?q=ai:eskin.alex "Mirzakhani, Maryam" https://zbmath.org/authors/?q=ai:mirzakhani.maryam "Mohammadi, Amir" https://zbmath.org/authors/?q=ai:mohammadi.amir Given a compact hyperbolic surface $$S$$, the problem of counting closed geodesics of at most a certain length has a long history, going back to J. Delsarte, with important contributions by A. Huber, G. Margulis, S. J. Patterson, and many others. The number of such closed geodesics is known to grow exponentially, and this asymptotics can be made effective by studying the exponential rate of mixing of the geodesic flow on the unit tangent bundle of the surface. The problem of counting \textit{simple} closed geodesics, that is, those which do not intersect themselves, is subtle, and building on polynomial upper and lower bounds shown by \textit{M. Rees} [Ergodic Theory Dyn. Syst. 1, 461--488 (1981; Zbl 0539.58018)], \textit{M. Mirzakhani} proved in her thesis [Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves, PhD thesis, Harvard Univ. (2004)] an asymptotic growth rate for the number of simple closed geodesics of a given topological type on a hyperbolic surface, by relating it to a lattice point counting problem on the space of measured laminations $$\mathcal{ML}(S)$$. This paper makes effective the counting of simple closed geodsics by effectivizing the lattice point counting on $$\mathcal{ML}(S)$$, using the exponential mixing of the geodesic flow on the unit tangent bundle to the \textit{moduli space} of the surface $$S$$, as shown by \textit{A. Avila} et al. [Publ. Math., Inst. Hautes Étud. Sci. 104, 143--211 (2006; Zbl 1263.37051)]. Reviewer: Jayadev Athreya (Seattle) Analytic normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold https://zbmath.org/1491.37048 2022-09-13T20:28:31.338867Z "Rousseau, Christiane" https://zbmath.org/authors/?q=ai:rousseau.christiane "Teyssier, Loïc" https://zbmath.org/authors/?q=ai:teyssier.loic This paper is concerned with local holomorphic vector fields on the complex plane, say $$(\mathbb C^2,0)$$, at a saddle-node singular point: the vector field vanishes at 0, and its linear part has only one non zero eigenvalue. For a general deformation of the vector field, the singular point bifurcates into $$k$$ distinct singular points (with non-vanishing eigenvalues) where $$k\ge2$$ is the Milnor number of the saddle-node. The goal of the paper is to provide the analytic classification of such deformations and is a continuation of the paper [Mosc. Math. J. 8, No. 3, 547--614 (2008; Zbl 1165.37016)] by the same authors. The analytic classification of saddle-nodes (without deformation) with respect to orbital conjugacy (i.e., classification of foliations rather than vector fields) goes back to [\textit{J. Martinet} and \textit{J.-P. Ramis}, Publ. Math., Inst. Hautes Étud. Sci. 55, 63--164 (1982; Zbl 0546.58038)] and gives rise to a functional moduli (the moduli space identifies with the space of convergent power series $$\mathbb C\{z\}$$ for short). These can be viewed as a non linear version of Stokes matrices for irregular singular points of differential systems of linear ordinary differential equations. The classification up to analytic conjugacy (for vector fields) has been done in [\textit{L. Teyssier}, C. R., Math., Acad. Sci. Paris 336, No. 8, 619--624 (2003; Zbl 1257.37031); \textit{S. M. Voronin} and \textit{Yu. I. Meshcheryakova}, Russ. Math. 46, No. 1, 11--14 (2002; Zbl 1223.37062)] and gives rise to some additional moduli. Combining ideas of \textit{R. Oudkerk} [Contemp. Math. 303, 79--105 (2002; Zbl 1025.37028)] and \textit{A. A. Glutsyuk} [Trans. Mosc. Math. Soc. 2001, 49--95 (2001; Zbl 1004.34081)], the present authors provide in their previous work [loc. cit.] a list of invariants for the analytic classification of unfoldings up to analytic equivalence. The present paper characterizes which invariants are realized in the special case of unfoldings with a central manifold, therefore ending the analytic classification for such unfoldings. Let us be more precise. A deformation $$\varepsilon\mapsto Z_\varepsilon$$ is defined by a holomorphic vector field $$Z=A(\varepsilon,x,y)\frac{\partial}{\partial x}+B(\varepsilon,x,y)\frac{\partial}{\partial y}$$, where $$A,B$$ are holomorphic on $$(\mathbb C^{k+2},0)$$, and $$\varepsilon\in(\mathbb C^{k},0)$$ is the parameter space. The goal of this paper is to study some class of deformations up to change of coordinates of the form $$(\varepsilon,(x,y))\mapsto(\phi(\varepsilon),\Psi_\varepsilon(x,y))$$. The orbital classification is considered, where one also allows to multiply $$Z$$ by a function germ, therefore changing the time. In this paper, it is assumed that $$y=0$$ is an invariant curve for the vector field $$Z_\varepsilon$$ for all $$\varepsilon$$, meaning that the coefficient $$B$$ factors as $$B=y\tilde B$$. Unfolding a saddle node means that we get a saddle node for $$\varepsilon=0$$ (its linear part has a vanishing eigenvalue along $$y=0$$, and non vanishing in the other direction) which splits into non degenerate singular points for $$\varepsilon\not=0$$ (i.e., with non-vanishing eigenvalues). After discussing the formal classification (i.e., with formal power series $$\Psi_\varepsilon$$) of such unfoldings, the authors focus on unfoldings formally conjugated to $\hat{Z}= u_\varepsilon(x)\left(\underbrace{(x^{k+1}+\varepsilon_{k-1}x^{k-1}+\cdots+\varepsilon_1x+\varepsilon_0)}_{P_\varepsilon(x)}\frac{\partial}{\partial x}+y(1+\mu_\varepsilon x^k)\frac{\partial}{\partial y}\right),$ which is a kind of formal normal form. Here $$\varepsilon\mapsto u_\varepsilon,\mu_\varepsilon$$ are holomorphic function germs, which are the formal invariants of the deformation. The first result is an almost unique normal form for those unfoldings $$Z$$ formally conjugated to $$\hat{Z}$$ with respect to analytic change of coordinate. We state here the case where $$\mu(0)\not\in\mathbb R_{\le0}$$ for simplicity: $$Z$$ is analytically equivalent to a normal form $\frac{1}{1+u(\varepsilon)Q(\varepsilon,x,y)}\left(\hat{Z}+R(\varepsilon,x,y)\frac{\partial}{\partial y}\right),$ where $$Q,R\in\mathbb C\{\varepsilon,y\}[x]$$ are degree $$k$$ polynomials in $$x$$-variable taking the form $$y\left(\sum_{j=1}^k f_j(\varepsilon,y)x^j\right)$$. Moreover, this normalization is unique up to a change of coordinate of the form $$(\varepsilon,x,y)\mapsto(\varepsilon,x,c(\varepsilon)y),$$ where $$c\in\mathbb C\{\varepsilon\}^\times$$. The proof partly follows the case of a single saddle node done by the reviewer in [\textit{F. Loray}, Astérisque 297, 167--187 (2004; Zbl 1083.32027)]; however, there are additional difficulties to deal with the parametric setting. In the case $$\mu(0)\in\mathbb R_{\le0}$$, the authors also provide almost unique normal forms. The second part deals with another approach by nonlinear Stokes analysis for the orbital analytic classification. Namely, in their previous work, the authors associate to the formal model $$\hat Z$$ a cell-decomposition of the parameter space $$(\mathbb C^k,0)\ni\varepsilon$$ by $$\frac{1}{k+1}\begin{pmatrix}2k\\ k\end{pmatrix}$$ open cells $$\mathcal E_l$$. Then, to each unfolding $$Z$$ formally conjugated to $$\hat Z$$, they associate $$k$$ functions $$\phi_l^j(\varepsilon,h)$$ holomorphic on $$(\mathbb C^k,0)\times(\mathbb C,0)$$ that can be interpreted as nonlinear Stokes maps: they characterize the orbital analytic class of the unfolding. In the second part of the paper, the authors provide necessary and sufficient condition on a collection $$(\phi_l^j)_{j,l}$$ to be the Stokes maps of an unfolding of $$\hat Z$$. This provides a description of the moduli space with respect to orbital analytic classification. The paper is carefully written with many other interesting results and nice pictures. Reviewer: Frank Loray (Rennes) A look into homomorphisms between uniform algebras over a Hilbert space https://zbmath.org/1491.46045 2022-09-13T20:28:31.338867Z "Dimant, Verónica" https://zbmath.org/authors/?q=ai:dimant.veronica "Singer, Joaquín" https://zbmath.org/authors/?q=ai:singer.joaquin The authors continue their study of homomorphisms between Banach algebras of holomorphic functions on the open unit ball $$B$$ of $$\ell_2.$$ (Earlier work appeared in [Math. Nachr. 293, No. 7, 1328--1344 (2020; Zbl 07317209); J. Geom. Anal. 31, No. 6, 6171--6194 (2021; Zbl 07379201)].) The focus here is on non-zero homomorphisms $$\Phi:\mathcal A_u(B) \to \mathcal H^\infty(B),$$ where $$\mathcal H^\infty(B)$$ is the Banach algebra of bounded $$\mathbb C$$-valued holomorphic functions on $$B$$ and $$\mathcal A_u(B)$$ is the subalgebra consisting of uniformly continuous functions on $$B.$$ The main interest is in the structure of the set $$\mathcal M$$ of such homomorphisms. The authors first define a mapping $\xi:\mathcal M \to H^\infty(B,\ell_2),\ \ \xi(\Phi) \equiv [y \in B \leadsto [x^\ast \in A_u(B) \leadsto \Phi(x^\ast)(y)]].$ It is observed that $$\xi$$ is a projection, having as range the closed unit ball $$\overline{B}_{\mathcal H^\infty(B,\ell_2)}.$$ For any $$g \in \overline{B}_{\mathcal H^\infty(B,\ell_2)},$$ the {\em fiber} over $$g$$ is the set $$\mathcal F(g) \equiv \xi^{-1}(g).$$ (This is intentionally reminiscent of the classical situation of fibers in the maximal ideal space $$\mathcal M(\mathcal H^\infty(\mathbb D))$$ over the closed unit disc $$\overline{\mathbb D}.$$) The authors identify three types of fibers in $$\mathcal M(\mathcal H^\infty(\mathbb D)),$$ studying special properties of each. Motivated by the classical paper of \textit{I. J. Schark} [J. Math. Mech. 10, 735--746 (1961; Zbl 0139.30402)], vector-valued cluster sets are also considered. Techniques and results are based on work of \textit{B. J. Cole} et al. [Mich. Math. J. 39, No. 3, 551--569 (1992; Zbl 0792.46016)] as well as more recent work of \textit{R. M. Aron} et al. [Math. Ann. 353, No. 2, 293--303 (2012; Zbl 1254.46057)]. Reviewer: Richard M. Aron (Kent) Geometric Arveson-Douglas conjecture and holomorphic extension https://zbmath.org/1491.47006 2022-09-13T20:28:31.338867Z "Douglas, Ronald George" https://zbmath.org/authors/?q=ai:douglas.ronald-george "Wang, Yi" https://zbmath.org/authors/?q=ai:wang.yi.7|wang.yi.9|wang.yi.5|wang.yi.8|wang.yi.10|wang.yi.4|wang.yi.6|wang.yi.3|wang.yi.1 Summary: In this paper, we introduce techniques from complex harmonic analysis to prove a weaker version of the geometric Arveson-Douglas conjecture on the Bergman space for a complex analytic subset that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $$\mathcal{T}(L^{\infty})$$, which implies the essential normality of the quotient module. Combining some other techniques, we actually obtain the $$p$$-essential normality for $$p>2d$$, where $$d$$ is the complex dimension of the analytic subset. Finally, we show that our results apply to the closure of a radical polynomial ideal $$I$$ whose zero variety satisfies the above conditions. A~key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset, generalizing the result of \textit{F. Beatrous jun.}'s paper [Mich. Math. J. 32, 361--380 (1985; Zbl 0584.32024)]. Essential normality -- a unified approach in terms of local decompositions https://zbmath.org/1491.47007 2022-09-13T20:28:31.338867Z "Wang, Yi" https://zbmath.org/authors/?q=ai:wang.yi.3|wang.yi.7|wang.yi.6|wang.yi.8|wang.yi.4|wang.yi.9|wang.yi.5|wang.yi.1|wang.yi.10|wang.yi.2 Summary: In this paper, we define the asymptotic stable division property for submodules of $$L_a ^2 (\mathbb{B}_n)$$. We show that under a mild condition, a~submodule with the asymptotic stable division property is $$p$$-essentially normal for all $$p > n$$. A~new technique is developed to show that certain submodules have the asymptotic stable division property. This leads to a unified proof of most known results on essential normality of submodules as well as new results. In particular, we show that an ideal defines a $$p$$-essentially normal submodule of $$L_a ^2 (\mathbb B_n )$$, $$\forall p > n$$, if its associated primary ideals are powers of prime ideals whose zero loci satisfy standard regularity conditions near the sphere. Semigroups of composition operators and integral operators in BMOA-type spaces https://zbmath.org/1491.47030 2022-09-13T20:28:31.338867Z "Daskalogiannis, V." https://zbmath.org/authors/?q=ai:daskalogiannis.v "Galanopoulos, P." https://zbmath.org/authors/?q=ai:galanopoulos.petros Summary: The aim of this article is to study semigroups of composition operators $$T_t=f\circ \phi_t$$ on the BMOA-type spaces $$BMOA_p$$, and on their little oh'' analogues $$VMOA_p$$. The spaces $$BMOA_p$$ were introduced by \textit{R.-H. Zhao} [On a general family of function spaces. Helsinki: Suomalainen Tiedeakatemia (1996; Zbl 0851.30017)] as part of the large family of $$F(p, q, s)$$ spaces, and are the Möbius invariant subspaces of the Dirichlet spaces $$D^p_{p-1}$$. We study the maximal subspace $$[\phi_t, BMOA_p]$$ of strong continuity, providing a sufficient condition on the infinitesimal generator of $$\{\phi_t\}$$, under which $$[\phi_t, BMOA_p]= VMOA_p$$, and a related necessary condition in the case where the Denjoy-Wolff point of the semigroup is in $$\mathbb{D}$$. Further, we characterize those semigroups, for which $$[\phi_t, BMOA_p]= VMOA_p$$, in terms of the resolvent operator of the infinitesimal generator of $$(T_t|_{VMOA_p})$$. In addition, we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators $$T_g$$. We characterize the symbols $$g$$ for which $$T_g: BMOA \rightarrow BMOA_1$$ is bounded or compact, thus extending a related result from [\textit{C. Yuan} and \textit{C.-Z. Tong}, Complex Anal. Oper. Theory 12, No. 8, 1845--1875 (2018; Zbl 1403.30022)] to the case $$p=1$$. We also prove that, for $$1<p<2$$, compactness of $$T_g$$ on $$BMOA_p$$ is equivalent to weak compactness. Twisting non-shearing congruences of null geodesics, almost CR structures and Einstein metrics in even dimensions https://zbmath.org/1491.53077 2022-09-13T20:28:31.338867Z "Taghavi-Chabert, Arman" https://zbmath.org/authors/?q=ai:taghavi-chabert.arman This paper is devoted to the study of the geometry of twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. The author gives a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, i.e., such that the screen bundle of the congruence is equipped with a bundle complex structure. In this case the leaf space of the congruence locally acquires a partially integrable contact almost CR structure of positive definite signature. Further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure and for the flatness of the latter are given. It is shown that under certain natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR-Einstein structure on the leaf space of the congruence. These metrics depend on three paramenters and include the Feffermann-Einstein metric and Taub-NUT-(A)dS metric in the integrable case. In the non-integrable case, new solutions to the Einstein field equations are obtained. These can be constructed from strictly almost Kähler-Einstein manifolds. Reviewer: María Ferreiro-Subrido (Santiago de Compostela) Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions https://zbmath.org/1491.53087 2022-09-13T20:28:31.338867Z "Paoletti, Roberto" https://zbmath.org/authors/?q=ai:paoletti.roberto From the paper's abstract: Suppose that a compact and connected Lie group $$G$$ acts on a complex Hodge manifold $$M$$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $$A$$ on $$M$$. Then there is an induced unitary representation on the associated Hardy space and, if the moment of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behaviour of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.'' The theorems obtained in this paper extend the asymptotic results obtained by the author previously in a series of papers (some jointly with Galasso), from the case when $$G$$ is a torus or $$\mathrm{SU}(2)$$ or $$\mathrm{U}(2)$$ to the case when $$G$$ is a compact connected Lie group which is `acceptable'' (this condition is satisfied, in particular, when $$G$$ is $$\mathrm{U}(n)$$ for some $$n\ge 1$$ or when $$G$$ is a connected and simply connected compact semisimple Lie group). Reviewer: Tatyana Barron (London, Ontario) Strict quantization of coadjoint orbits https://zbmath.org/1491.53094 2022-09-13T20:28:31.338867Z "Schmitt, Philipp" https://zbmath.org/authors/?q=ai:schmitt.philipp Consider the complexification $$\hat{\mathcal{O}}$$ of a co-adjoint $$\mathcal{O}$$ orbit of a connected semisimple Lie group. Equivariant formal deformation quantizations of $$\hat{\mathcal{O}}$$ were constructed by Alekseev-Lachowska from the Shapovalov pairing, see [\textit{A. Alekseev} and \textit{A. Lachowska}, Comment. Math. Helv. 80, No. 4, 795--810 (2005; Zbl 1162.53327)]. The first result of the paper is an explicit formula for the canonical element of the Shapovalov pairing. Furthermore, it is proved that there is a countable set of poles $$P$$, and for every $$\hbar\in \mathbb{C}\setminus P$$ there is a $$\hat{G}$$-invariant product which is also continuous with respect to the topology of locally uniform convergence on the space of holomorphic functions on $$\hat{\mathcal{O}}$$. The dependence of the star products $$\hat{\star}_{\hbar}$$ on $$\hbar$$ is holomorphic. The author mainly works in the complex setting and restricts to the real setting only in the very end. This provides examples of quantizations in a Fréchet-algebraic setting. Reviewer: Andrea Galasso (Taipei) The first eigenvalue of the Laplacian on orientable surfaces https://zbmath.org/1491.58012 2022-09-13T20:28:31.338867Z "Karpukhin, Mikhail" https://zbmath.org/authors/?q=ai:karpukhin.mikhail-a "Vinokurov, Denis" https://zbmath.org/authors/?q=ai:vinokurov.denis Summary: The famous Yang-Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus $$\gamma$$ and the area. Its proof relies on the existence of holomorphic maps to $$\mathbb{CP}^1$$ of low degree. Very recently, Ros was able to use certain holomorphic maps to $$\mathbb{CP}^2$$ in order to give a quantitative improvement of the Yang-Yau inequality for $$\gamma =3$$. In the present paper, we generalize Ros' argument to make use of holomorphic maps to $$\mathbb{CP}^n$$ for any $$n>0$$. As an application, we obtain a quantitative improvement of the Yang-Yau inequality for all genera except for $$\gamma = 4,6,8,10,14$$. Liftable vector fields, unfoldings and augmentations https://zbmath.org/1491.58014 2022-09-13T20:28:31.338867Z "Nuño-Ballesteros, J. J." https://zbmath.org/authors/?q=ai:nuno-ballesteros.juan-jose "Oset Sinha, R." https://zbmath.org/authors/?q=ai:sinha.raul-oset The authors show how to describe the module of liftable vector fields of any finite singularity type map-germ in terms of the liftable vector fields of a stable unfolding of the map. In fact, it is a generalization of the result from [\textit{T. Nishimura} et al., Math. Ann. 366, No. 1--2, 573--611 (2016; Zbl 1368.32016)]. As an instructive example, they compute explicitly a system of generators for the module of liftable vector fields associated with the family $$H_k$$ from Mond's list of simple germs [\textit{D. Mond}, Proc. Lond. Math. Soc. (3) 50, 333--369 (1985; Zbl 0557.58006)], then discuss in detail the complex case, describe a relation between the liftable vector fields of a one-parameter stable unfolding of a map-germ and its augmentations, etc. Reviewer: Aleksandr G. Aleksandrov (Moskva) Two-particle bound states at interfaces and corners https://zbmath.org/1491.81018 2022-09-13T20:28:31.338867Z "Roos, Barbara" https://zbmath.org/authors/?q=ai:roos.barbara "Seiringer, Robert" https://zbmath.org/authors/?q=ai:seiringer.robert Summary: We study two interacting quantum particles forming a bound state in $$d$$-dimensional free space, and constrain the particles in $$k$$ directions to $$( 0 , \infty )^k \times \mathbb{R}^{d - k}$$, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from $$k$$ to $$k + 1$$. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all $$k$$ the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of \textit{S. Egger} et al. [J. Spectr. Theory 10, No. 4, 1413--1444 (2020; Zbl 1469.81021)] to dimensions $$d > 1$$. Uncovering the impact of delay phenomenon on random walks in a family of weighted $$m$$-triangulation networks https://zbmath.org/1491.82018 2022-09-13T20:28:31.338867Z "Guo, Junhao" https://zbmath.org/authors/?q=ai:guo.junhao "Wu, Zikai" https://zbmath.org/authors/?q=ai:wu.zikai Probing multi-step electroweak phase transition with multi-peaked primordial gravitational waves spectra https://zbmath.org/1491.83013 2022-09-13T20:28:31.338867Z "Morais, António P." https://zbmath.org/authors/?q=ai:morais.antonio-p "Pasechnik, Roman" https://zbmath.org/authors/?q=ai:pasechnik.roman (no abstract)