Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512023-12-07T16:00:11.105023ZWerkzeugHyperbolic space for touristshttps://zbmath.org/1522.000042023-12-07T16:00:11.105023Z"Blåsjö, Viktor"https://zbmath.org/authors/?q=ai:blasjo.viktor(no abstract)Book review of: C. Alsina and R. B. Nelsen, A cornucopia of quadrilateralshttps://zbmath.org/1522.000162023-12-07T16:00:11.105023Z"Baxa, C."https://zbmath.org/authors/?q=ai:baxa.christophReview of [Zbl 1443.51001].Book review of: N. Sidoli and Y. Isahaya, Thābit ibn Qurra's restoration of Euclid's \textit{Data}. Text, translation, commentaryhttps://zbmath.org/1522.000202023-12-07T16:00:11.105023Z"Baxa, C."https://zbmath.org/authors/?q=ai:baxa.christophReview of [Zbl 1403.01039].Book review of: M. Dunajski, Geometry. A very short introductionhttps://zbmath.org/1522.000592023-12-07T16:00:11.105023Z"Huggett, Stephen"https://zbmath.org/authors/?q=ai:huggett.stephen-aReview of [Zbl 1486.00004].Book review of: G. van Brummelen, The doctrine of triangles. A history of modern trigonometryhttps://zbmath.org/1522.000832023-12-07T16:00:11.105023Z"Mansfield, Daniel"https://zbmath.org/authors/?q=ai:mansfield.daniel-fReview of [Zbl 1475.51001].Some thoughts on the Epicurean critique of mathematicshttps://zbmath.org/1522.001392023-12-07T16:00:11.105023Z"Aristidou, Michael"https://zbmath.org/authors/?q=ai:aristidou.michael(no abstract)Geometrical constructivism and modal relationalism: further aspects of the dynamical/geometrical debatehttps://zbmath.org/1522.001462023-12-07T16:00:11.105023Z"Read, James"https://zbmath.org/authors/?q=ai:read.james(no abstract)On the relationship between geometric objects and figures in Euclidean geometryhttps://zbmath.org/1522.010042023-12-07T16:00:11.105023Z"Bacelar Valente, Mario"https://zbmath.org/authors/?q=ai:bacelar-valente.marioSummary: In this paper, we will make explicit the relationship existing between geometric objects and figures in planar Euclidean geometry. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. It corresponds to a resemblance-like relationship between objects and figures. This relation is what enables figures to have a role in pure and applied geometry. That is, we can use a figure in pure geometry as a representation of geometric objects because of this relation. Moving beyond pure geometry, we will defend that there are two other `layers' of representation at play in applied geometry.
For the entire collection see [Zbl 1487.68008].Gapless lines and gapless proofs: intersections and continuity in Euclid's \textit{Elements}https://zbmath.org/1522.010052023-12-07T16:00:11.105023Z"De Risi, Vincenzo"https://zbmath.org/authors/?q=ai:de-risi.vincenzoThe question the author tries to answer is related to the reasons why Euclid believed that the two circles intersect and that their intersection is a point in \textit{Elements} I,1, and regarding why no ``gap'' was found by commentators during all of antiquity, only to surface for the first time during the 17th century. In fact, the first known criticism is attributed to the Epicurean philosopher Zeno of Sidon (2nd century BC), and the second is an objection rebutted by Alexander of Aphrodisias (3rd century AD), but none of them has anything to do with why the circles intersect, but rather with questions regarding the nature of the intersection (how do we know it is not a segment?) or of size (what if the radius of the circles corresponds to the diameter of the Universe?).
Two solutions are listed. One is that an Aristotelian form of continuity of lines and circles would have ensured the existence of the intersection, an interpretation that does not stand criticism based on an analysis of methods of proof in the whole corpus of ancient geometry, and the other is that Euclid drew ``diagrammatic inferences'', and that one can add some axioms that Euclid took for granted to provide a gapless development of Book I of \textit{The Elements} (this interpretation is rejected, given that ``continuity (i. e. completeness) cannot be a \textit{diagrammatic property} in the strict sense'' (p.\ 248)).
The author's own proposal, which vindicates Euclid's own approach, is the following: Euclid did infer diagrammatically that the two circles cross, but that diagrammatical inference did not ensure that the two circles intersect in a point. It is by using a definition -- ``The limits of a line are points'' -- as an axiom that Euclid ensures that the intersection is a point.
Reviewer: Victor V. Pambuccian (Glendale)Hero and the tradition of the circle segmenthttps://zbmath.org/1522.010092023-12-07T16:00:11.105023Z"Mendell, Henry"https://zbmath.org/authors/?q=ai:mendell.henry-rThis paper presents in great detail some answers to several questions raised by the four procedures for finding the approximate area of a circular segment Hero provides in his \textit{Metrica}. In particular, regarding the formula (what Hero refers to as ``the Ancient method''):
\[
\frac{1}{2}(b+h)h,\tag{\(\ast\)}
\]
where \(b\) denotes the base of the segment and \(h\) its height, the author establishes its use in Uruk, as documented by the late 5th century BC tablet W 23291-x, a tablet that is the background for the same method mentioned in the 3rd century BC Demotic Egyptian treatise assembled by \textit{R. A. Parker} [Demotic mathematical papyri. Providence, RI: Brown University Press; London: Lund Humphries (1972; Zbl 0283.01001)] from P. Cairo 89127-30, 89137-43. He also presents ``some grounds for why it would be seen as plausible'', given that the formula is very far from good for small values of \(h\), when compared with \(b\). The author also refers to ``some startling coincidences'' (p.\ 452). An algorithm in the Old Babylonian tablet BM 85194 is also considered relevant.
There are also attempts at reconstructing the exploratory path by which the procedures might have been arrived at. They are summarized in two theorems. With \textit{segment$_1$} and \textit{segment$_2$} standing for the areas of adjacent segments on a rectangle inscribed in a circle, $h_1$ and $b_1$ standing for the height and base of \textit{segment$_1$} and $h_2$ and $b_2$ for the height and base of \textit{segment$_2$}, it is shown that, (i) with $\pi=3$, \textit{segment$_1$}$+$ \textit{segment$_2$} $= \frac{1}{2}(b_1 + h_1)\cdot h_1 +\frac{1}{2}{b_2 + h_2}\cdot h_2$, and that (ii) \textit{segment$_1$}$+$ \textit{segment$_2$} $= \frac{1}{2}(b_1 + h_1)\cdot h_1 + (\frac{b_1}{2})^2 \cdot \frac{\pi-3}{2}+ \frac{1}{2}(b_2 + h_2)\cdot h_2 + (\frac{b_1}{2})^2 \cdot \frac{\pi-3}{2}$.
By taking $\pi$ to be $3\frac{1}{7}$, (ii) becomes \textit{segment$_1$}$+$ \textit{segment$_2$} $= \frac{1}{2}(b_1 + h_1)\cdot h_1+ \frac{(\frac{b_1}{2})^2}{14}+ \frac{1}{2}(b_2 + h_2)\cdot h_2 + \frac{(\frac{b_2}{2})^2}{14}$.
This allows the author to vindicate Hero, for he was right in stating that the Ancient method is tied to taking $\pi=3$ and that the Revised method to taking $\pi = 3\frac{1}{7}$.
``As to the Ancient method for finding the area of the segment, well, it works perfectly on a semicircle, perfectly on a side of a square, and fairly well on one figure in between, the equilateral triangle. Therefore, it works fairly well.'' (p.\ 469)
This positive tone is at odds with the language used by Van der Waerden, for whom the fact of the same \textit{inaccurate} formula appears in the Chinese \textit{Nine chapters}, in Hero's \textit{Metrica}, and in Egypt is assigned a great amount of significance for the hypothesis, put forward by \textit{A. Seidenberg} [Arch. Hist. Exact Sci. 18, 301--342 (1978; Zbl 0392.01002)], of a single origin for mathematics. We find, on page 40 of [\textit{B. L. van der Waerden}, Geometry and algebra in ancient civilizations. Berlin: Springer-Verlag (1983; Zbl 0534.01001)], the following (more on the matter of other area formulas found in the \textit{Metrica} can be found on pages 184--186 of the same book):
``We find it very curious that the same inaccurate formula \((\ast)\) also occurs in the ``Metrica'' of Heron of Alexandria [\(\ldots\)] and in a papyrus from Cairo written in the third century B.C. and published by R. A. Parker. [\(\ldots\)]
As a general principle, if one finds one and the same \textit{correct} rule of computation in several civilizations, one always has to take into account the possibility of independent invention, but if the rule is \textit{incorrect}, independent invention is next to impossible. Therefore we are bound to suppose that the Chinese formula [\(\ldots\)] and the equivalent formula \((\ast)\) used in the Cairo papyrus and mentioned by Heron, were derived from a common origin.
According to Parker, the Demotic papyrus shows clear traces of Babylonian influence. The problems and the solutions in the papyrus are just of the same kind as those we find in Babylonian problem texts. However, the false rule \((\ast)\) is not found in any extant Babylonian text, as far as I know.''
In this sense, the current paper answers the question left open by Van der Waerden, by detecting a Babylonian text in which \((\ast)\) can be found. This had not been noted before. \textit{J. Friberg} [Unexpected links between Egyptian and Babylonian mathematics. Hackensack, NJ: World Scientific (2005; Zbl 1128.01002)] found, not long ago, on page 133, that ``In Babylonian mathematics, the use of the rule is not documented''.
Reviewer: Victor V. Pambuccian (Glendale)Abscissas and ordinateshttps://zbmath.org/1522.010102023-12-07T16:00:11.105023Z"Pierce, David"https://zbmath.org/authors/?q=ai:pierce.david-austin(no abstract)On commensurability and symmetryhttps://zbmath.org/1522.010112023-12-07T16:00:11.105023Z"Pierce, David"https://zbmath.org/authors/?q=ai:pierce.david-austin(no abstract)Complexity of the isomorphism problem for computable free projective planes of finite rankhttps://zbmath.org/1522.031272023-12-07T16:00:11.105023Z"Kogabaev, N. T."https://zbmath.org/authors/?q=ai:kogabaev.nurlan-talgatovichSummary: Studying computable representations of projective planes, we prove that the isomorphism problem in the class of free projective planes of finite rank is an \(m\)-complete \(\Delta_3^0\)-set within the class.The \(\delta_\alpha^0\)-computable enumerations of the classes of projective planeshttps://zbmath.org/1522.031282023-12-07T16:00:11.105023Z"Voĭtov, A. K."https://zbmath.org/authors/?q=ai:voitov.a-kSummary: Studying computable representations of projective planes, for the classes \(K\) of pappian, desarguesian, and all projective planes, we prove that \(K^c/_\simeq\) admits no hyperarithmetical Friedberg enumeration and admits a Friedberg \(\Delta_{\alpha+3}^0\)-computable enumeration up to a \(\Delta_\alpha^0\)-computable isomorphism.Infinite families of \(t\)-designs from the binomial \(x^4 +x^3\) over \(\mathrm{GF}(2^n)\)https://zbmath.org/1522.050162023-12-07T16:00:11.105023Z"Ling, Xin"https://zbmath.org/authors/?q=ai:ling.xin"Xiang, Can"https://zbmath.org/authors/?q=ai:xiang.canSummary: Combinatorial \(t\)-designs have nice applications in coding theory, finite geometries and engineering areas. \(t\)-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs \(t\)-designs from the quadratic polynomial \(x^4 +x^3\) over \(\mathrm{GF}(2^n)\) and determine their parameters. We yield \(2\)-\(\left( 2^n, 3\cdot 2^{n-2},3\cdot 2^{n-2}\left( 3\cdot 2^{n-2}-1 \right) \right)\) designs for \(n\) even and \(3\)-\(\left( 2^n, 2^{n-1},2^{n-1}\left( 2^{n-2}-1 \right) \right)\) designs for \(n\) odd.How many cards should you lay out in a game of \textit{EvenQuads}: a detailed study of caps in \(\mathrm{AG}(n, 2)\)https://zbmath.org/1522.050252023-12-07T16:00:11.105023Z"Crager, Julia"https://zbmath.org/authors/?q=ai:crager.julia"Flores, Felicia"https://zbmath.org/authors/?q=ai:flores.felicia"Goldberg, Timothy E."https://zbmath.org/authors/?q=ai:goldberg.timothy-e"Rose, Lauren L."https://zbmath.org/authors/?q=ai:rose.lauren-l"Rose-Levine, Daniel"https://zbmath.org/authors/?q=ai:rose-levine.daniel"Thornburgh, Darrion"https://zbmath.org/authors/?q=ai:thornburgh.darrion"Walker, Raphael"https://zbmath.org/authors/?q=ai:walker.raphaelIn the paper under review, the authors consider sets \(\mathcal{O}\) of points in \(\mathrm{AG}(n, 2)\) such that any four points of \(\mathcal{O}\) are in general position. Equivalently, no affine plane is contained in \(\mathcal{O}\). All such sets of size at most \(9\) in \(\mathrm{AG}(n, 2)\) are classified, up to affine equivalence. Moreover, the complete and maximal sets of this type in \(\mathrm{AG}(n, 2)\), \(n \le 6\), are characterized.
Reviewer: Francesco Pavese (Bari)Graphs with no induced house nor induced hole have the de Bruijn-Erdős propertyhttps://zbmath.org/1522.050542023-12-07T16:00:11.105023Z"Aboulker, Pierre"https://zbmath.org/authors/?q=ai:aboulker.pierre"Beaudou, Laurent"https://zbmath.org/authors/?q=ai:beaudou.laurent"Matamala, Martín"https://zbmath.org/authors/?q=ai:matamala.martin"Zamora, José"https://zbmath.org/authors/?q=ai:zamora.joseSummary: A set of \(n\) points in the plane which are not all collinear defines at least \(n\) distinct lines. \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)] conjectured that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least 5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of `good pairs' that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs.
{{\copyright} 2022 Wiley Periodicals LLC}Simple lattices and free algebras of modular formshttps://zbmath.org/1522.110322023-12-07T16:00:11.105023Z"Wang, Haowu"https://zbmath.org/authors/?q=ai:wang.haowu"Williams, Brandon"https://zbmath.org/authors/?q=ai:williams.brandonLet \(L\) be a simple lattice of signature \((n,2)\) and \(L'\) be the dual lattice. Here, simple implies that \(L\) is an even integral lattice and the attached dual Weil representation admits no cusp forms of weight \(1+n/2\). Let \(\operatorname{O}^+(L)\) denote the subgroup of the integral orthogonal group \(\operatorname{O}(L)\) that preserves the type IV Hermitian symmetric domain \(\mathcal{D}(L)\). The discriminant kernel of \(L\), denoted by \(\widetilde{\operatorname{O}}^+(L)\), is the kernel of \(\operatorname{O}^+(L)\) on \(\mathbb{C}[L'/L]\). The main theorem concerns simple lattices \(L\) of signature \((n,2)\), with \(3\leq n\leq 10\), and consists of two parts. First, for the subgroup \(\operatorname{O}_r(L)\) generated by reflections in the orthogonal group of \(L\), it is found that the graded ring of modular forms \(M_{*}(\operatorname{O}_r(L))\) is freely generated. Second, for the subgroup \(\widetilde{\operatorname{O}}_r(L)\) generated by reflections in the discriminant kernel of \(L\), it is established that the graded ring of modular forms \(M_*(\widetilde{\operatorname{O}}_r(L))\) is freely generated for all but the five cases of \(2U(2)\oplus A_1(2)\), \(U\oplus U(2)\oplus A_1(2)\), and \(2U\oplus A_1(m)\) for \(m=2,3,4\).
The proof of the theorem utilizes work of [\textit{H. Wang}, Compos. Math. 157, No. 9, 2026--2045 (2021; Zbl 1482.11074)] and centers around the construction and analyzing of Jacobians. The potential Jacobians constructed are cusp forms which vanish precisely on all mirror reflections in an arithmetic group \(\Gamma\) with multiplicity one if and only if \(M_{*}(\Gamma)\) is freely generated by \(n+1\) modular forms, where \(n\) is the dimension of the symmetric domain. Indeed, one of the five exceptions above correlates to when \(\widetilde{\operatorname{O}}_r(L)\) is empty, while the other four coincide with cases when the potential Jacobians are not cusp forms.
As pointed out by the authors, some of the cases included in the main theorem have been developed elsewhere in the literature. Citations and tables displaying this information is provided. Meanwhile, the authors often include their own proofs which are sometimes simpler and also fit the narrative. The proof of the main theorem is established by case-by-case analysis, which is performed over the course of Sections 3--9. Individual theorems are provided for the specific cases, and these results include generators of the appropriate space of modular forms and a description of the Jacobian of the generators in terms of Borcherds products. Sections 1 and 2 are the introduction and preliminaries, respectively.
Reviewer: Matthew Krauel (Sacramento)From geometry to arithmetic of compact hyperbolic Coxeter polytopeshttps://zbmath.org/1522.201492023-12-07T16:00:11.105023Z"Bogachev, Nikolay"https://zbmath.org/authors/?q=ai:bogachev.nikolay-vThe author establishes some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and shows that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
Reviewer: Erich W. Ellers (Toronto)On basic invariants of some finite subgroups in \(\mathrm{SL}_3(\mathbf{C})\)https://zbmath.org/1522.201582023-12-07T16:00:11.105023Z"Rudnitskiĭ, Oleg Ivanovich"https://zbmath.org/authors/?q=ai:rudnitskii.oleg-ivanovichSummary: The work is devoted to the study of algebras of invariants of finite unitary groups \(G'=G\cap \mathrm{SL}_3(\mathbf{C})\), where \(G\) is a finite unitary irreducible primitive group generated by reflections in the unitary space \(U^3\). It is known that the system of invariants of the group \(G'\) that form an algebra is obtained from the system of invariants of the group \(G\) that form the algebra by adding all semi-invariants of the group \(G\) of a special form. In the paper, generators of the algebras of invariants of all the indicated groups \(G'\) are constructed.Classification and decomposition of quaternionic projective transformationshttps://zbmath.org/1522.201972023-12-07T16:00:11.105023Z"Dutta, Sandipan"https://zbmath.org/authors/?q=ai:dutta.sandipan"Gongopadhyay, Krishnendu"https://zbmath.org/authors/?q=ai:gongopadhyay.krishnendu"Lohan, Tejbir"https://zbmath.org/authors/?q=ai:lohan.tejbirSummary: We consider the projective linear group \(\mathrm{PSL}(3, \mathbb{H})\). We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of \(\mathrm{PSL}(3, \mathbb{H})\). We further decompose elements of \(\mathrm{SL}(3, \mathbb{H})\) as products of simple elements, where an element \(g\) in \(\mathrm{SL}(3, \mathbb{H})\) is called \textit{simple} if it is conjugate to an element of \(\mathrm{SL}(3, \mathbb{R})\). We have also revisited real projective transformations and following Goldman's ideas, have offered a complete classification for elements of \(\mathrm{SL}(3, \mathbb{R})\).Thrice-punctured sphere groups in hyperbolic \(4\)-spacehttps://zbmath.org/1522.300222023-12-07T16:00:11.105023Z"Kim, Youngju"https://zbmath.org/authors/?q=ai:kim.youngjuSummary: A thrice-punctured sphere group is a non-elementary group generated by two parabolic isometries whose product is a parabolic isometry. We prove that the deformation space of a thrice-punctured sphere group acting on hyperbolic \(4\)-space is \(7\)-dimensional. Among them, there is a \(5\)-dimensional parameter space of linked thrice-punctured sphere groups. In particular, there is a \(1\)-parameter family of discrete linked thrice-punctured sphere groups such that the rotation angles of the two parabolic generators and the product of the generators are fixed.The Newton polytope of the optimal differential operator for an algebraic curvehttps://zbmath.org/1522.320902023-12-07T16:00:11.105023Z"Krasikov, Vitaly A."https://zbmath.org/authors/?q=ai:krasikov.vitaly-a"Sadykov, Timur M."https://zbmath.org/authors/?q=ai:sadykov.t-mSummary: We investigate the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The main result is a description of the coefficients of this operator in terms of their Newton polytopes.Polygon recutting as a cluster integrable systemhttps://zbmath.org/1522.370762023-12-07T16:00:11.105023Z"Izosimov, Anton"https://zbmath.org/authors/?q=ai:izosimov.antonSummary: Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with opposite orientation. Recuttings along different diagonals generate an action of the affine symmetric group on the space of polygons. We show that this action is given by cluster transformations and is completely integrable. The integrability proof is based on interpretation of recutting as refactorization of quaternionic polynomials.Some results on isomorphisms of finite semifield planeshttps://zbmath.org/1522.510012023-12-07T16:00:11.105023Z"Kravtsova, Olga V."https://zbmath.org/authors/?q=ai:kravtsova.olga-vadimovna"Panov, Sergei V."https://zbmath.org/authors/?q=ai:panov.sergei-v"Shevelyova, Irina V."https://zbmath.org/authors/?q=ai:shevelyova.irina-vSummary: The authors extend an approach to construct and classify the semifield projective planes using the linear space and spread set. Follows results are given: an estimate of the order of autotopism group and a number of isomorphic semifield planes defined by fixed linear space.An incidence result for well-spaced atoms in all dimensionshttps://zbmath.org/1522.510022023-12-07T16:00:11.105023Z"Bradshaw, Peter J."https://zbmath.org/authors/?q=ai:bradshaw.peter-jIncidence geometry is concerned with counting incidences between geometric objects, points and lines. Given a finite set of lines \( L\) in \(\mathbb{R}^2\), a point is \(k\)-rich if between \(k\) and \(2k\) lines from \(L\) pass through it. The set of \(k\)-rich points is denoted by \(P_{k}(L)\). For \(k \geq2\), the classical Szemerédi-Trotter theorem [\textit{E. Szemerédi} and \textit{W. T. Trotter jun.}, Combinatorica 3, 381--392 (1983; Zbl 0541.05012)] bounds sharply \(|P_{k}(L)|\). By duality, the above bound also holds if the roles of points and lines are interchanged.
\textit{L. Guth} et al. [Geom. Funct. Anal. 29, No. 6, 1844--1863 (2019; Zbl 1460.52017)] proved an analogue of the Szemerédi-Trotter theorem for suitably well-spaced sets of tubes of thickness \(\delta\) in \([0, 1]^2\). Furthermore, they proved a similar result in \([0, 1]^3\) which is an analogue of the Guth-Katz bound [\textit{L. Guth} and \textit{N. H. Katz}, Ann. Math. (2) 181, No. 1, 155--190 (2015; Zbl 1310.52019)]. Both bounds are essentially sharp. The objects of interest of this article, as by Guth et al. [loc. cit.], are small \(\delta\)-atoms and thin \(\delta\)-tubes. A \(\delta\)-atom is a closed ball in \([0, 1]^d\) of diameter \(\delta\) and a \(\delta\)-tube is the set of all points in \([0, 1]^d\) which are within a distance \(\delta/2\) of some fixed line.
In this article, the author proves an incidence result counting the \(k\)-rich \(\delta\)-tubes induced by a well-spaced set of \(\delta\)-atoms. This result coincides with the bound that would be heuristically predicted by the Szemerédi-Trotter theorem and holds in all dimensions \(d \geq2\). In addition, as an application of his result the author proves an analogue of \textit{J. Beck}'s theorem [Combinatorica 3, 281--297 (1983; Zbl 0533.52004)] for \(\delta\)-atoms and \(\delta\)-tubes.
Reviewer: Boumédiène Et-Taoui (Mulhouse)On finite generalized quadrangles of even orderhttps://zbmath.org/1522.510032023-12-07T16:00:11.105023Z"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.taoIn this paper, the author proves two longstanding conjectures on automorphism groups of finite generalised quadrangles. The first one concerns the conjecture of \textit{S. E. Payne} [Conf. Comb. Graph Theor. Comput. 485--504 (1975; Zbl 0336.05016)] that every skew translation generalized quadrangle of even order is a(n ordinary) translation generalized quadrangle. To explain this result it is good to recall that in the classical quadrangles of order \(s\), either every point \(p\) is a point of symmetry (that is, the group of collineations fixing all points collinear to \(p\) has order \(s\)), or each line is an axis of symmetry (this is the dual of the previous). But if the characteristic of the underlying field is equal to \(2\), then the quadrangle is self dual and hence all points are centres of symmetry and all lines are axes of symmetry. Now it is an easy consequence of the classification of so-called Moufang quadrangles that, if all points of a quadrangle of order \(s\) are centres of symmetry, and \(s\) is even, then all lines are axes of symmetry (and the quadrangle is classical). The local version, however, is much more difficult to prove. It says that, if a point \(p\) of a generalised quadrangle of even order \(s\) is a point of symmetry, and the \(s\) collineations that fix all points collinear to \(p\) belong to a collineation group that acts simply-transitively on the set of points not collinear to \(p\), then all lines through \(p\) are axes of symmetry. That is exactly what the author proves.
The second conjecture of \textit{D. Ghinelli} [Geom. Dedicata 41, No. 2, 165--174 (1992; Zbl 0746.51011)] is easier to explain: the author shows that no generalised quadrangle of even order \(s\) admits a collineation group acting simply-transitively on the point set. In particular, there is no such thing as a Singer cycle for generalised quadrangles of even order.
The proofs of both results use new ideas (e.g. character theory, bounds for bipartite graphs using spectral theory) compared to the classical approaches.
Reviewer: Hendrik Van Maldeghem (Gent)Exponentially larger affine and projective capshttps://zbmath.org/1522.510042023-12-07T16:00:11.105023Z"Elsholtz, Christian"https://zbmath.org/authors/?q=ai:elsholtz.christian"Lipnik, Gabriel F."https://zbmath.org/authors/?q=ai:lipnik.gabriel-fSummary: In spite of a recent breakthrough on upper bounds of the size of cap sets (by \textit{E. Croot} et al. [Ann. Math. (2) 185, No. 1, 331--337 (2017; Zbl 1425.11019)] and by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)]), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus \(p\). Moreover, we show that for all primes \(p \equiv 5 \bmod 6\) with \(p \leqslant 41\), the new construction leads to an exponentially larger growth of the affine and projective caps in \(\mathrm{AG}(n,p)\) and \(\mathrm{PG}(n,p)\). For example, when \(p=23\), the existence of caps with growth \((8.0875\ldots)^n\) follows from a three-dimensional example of Bose, and the only improvement had been to \((8.0901\ldots)^n\) by Edel, based on a six-dimensional example. We improve this lower bound to \((9-o(1))^n\).
{{\copyright} 2022 The Authors. \textit{Mathematika} is copyright {\copyright} University College London and published by the London Mathematical Society on behalf of University College London.}On tight sets of hyperbolic quadricshttps://zbmath.org/1522.510052023-12-07T16:00:11.105023Z"Gavrilyuk, Alexander L."https://zbmath.org/authors/?q=ai:gavrilyuk.alexander-lA tight set \({\mathcal T} \) of points of a finite polar space \({\mathcal P}\) of rank \(r\) of \({\mathbb F}_q\) with \(\kappa\) the average number of points of \({\mathcal T}\) collinear to a given point satisfies \(\kappa = |{\mathcal T} | \frac{q^{r-1}-1}{q^r-1} + q^{r-1}.\) The author generalizes a previous result by the author and \textit{K. Metsch} [J. Comb. Theory, Ser. A 127, 224--242 (2014; Zbl 1302.51008)] to give that the parameter \(\chi\) of a tight set \({\mathcal T}\) of a hyperbolic quadric \(Q^+(2n+1,q)\) of odd rank \(n+1\) satisfies \(\binom{x}{2} + w(w-\chi) \equiv 0 \pmod{q+1},\) where \(w\) is the number of points of \({\mathcal T}\) in any generator of \(Q^+(2n+1,q).\)
Reviewer: Steven T. Dougherty (Scranton)A modular equality for \(m\)-ovoids of elliptic quadricshttps://zbmath.org/1522.510062023-12-07T16:00:11.105023Z"Gavrilyuk, Alexander L."https://zbmath.org/authors/?q=ai:gavrilyuk.alexander-l"Metsch, Klaus"https://zbmath.org/authors/?q=ai:metsch.klaus"Pavese, Francesco"https://zbmath.org/authors/?q=ai:pavese.francescoSummary: An \(m\)-ovoid of a finite polar space \(\mathcal{P}\) is a set \(\mathcal{O}\) of points such that every maximal subspace of \(\mathcal{P}\) contains exactly \(m\) points of \(\mathcal{O}\). In the case when \(\mathcal{P}\) is an elliptic quadric \(\mathcal{Q}^-(2r+1, q)\) of rank \(r\) in \(\mathbb{F}_q^{2r+2}\), we prove that an \(m\)-ovoid exists only if \(m\) satisfies a certain modular equality, which depends on \(q\) and \(r\). This condition rules out many of the possible values of \(m\). Previously, only a lower bound on \(m\) was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the \(m\)-ovoids of \(\mathcal{Q}^-(7,q)\) for \(q = 2\) and \((m, q) = (4, 3)\).
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaceshttps://zbmath.org/1522.510072023-12-07T16:00:11.105023Z"Lavrauw, Michel"https://zbmath.org/authors/?q=ai:lavrauw.michel"Lia, Stefano"https://zbmath.org/authors/?q=ai:lia.stefano"Pavese, Francesco"https://zbmath.org/authors/?q=ai:pavese.francescoFixing an odd prime power \(q\), in the article under review, the authors study the orbits of a particular action of the projective linear group \(\mathrm{PGL}(2,q^2)\) that arises from the Hermitian Veronese curve in projective three space \(\mathrm{PG}(3,q^2)\). In doing so, the authors' main result gives a new family of quasi-Hermitian surfaces. Finally, the authors discuss in detail the previously known examples of quasi-Hermitian surfaces and explain how their construction fits into these previously known results.
Reviewer: Nathan Grieve (Wolfville)Classification of spreads of Tits quadrangles of order 64https://zbmath.org/1522.510082023-12-07T16:00:11.105023Z"Monzillo, Giusy"https://zbmath.org/authors/?q=ai:monzillo.giusy"Penttila, Tim"https://zbmath.org/authors/?q=ai:penttila.tim"Siciliano, Alessandro"https://zbmath.org/authors/?q=ai:siciliano.alessandroSpreads of Tits quadrangles \(T_2(\mathcal{O})\) where \(\mathcal{O}\) is an oval of \(\mathrm{PG}(2,q)\) with \(q\leq 32\) even were completely determined by \textit{M. R. Brown} et al. [Innov. Incidence Geom. 6--7, 111--126 (2008; Zbl 1175.51003)]. In the paper under review, the authors consider the next case of even order, \(q=64\). Using a computer and MAGMA, they show in the course of their classification that no new spreads arise, that is, all spreads of Tits quadrangles in order 64 were previously known.
The authors use the correspondences between spreads of Tits quadrangles and other geometric objects such as generalised fans of ovals, flocks of quadratic cones, flock quadrangles, 64-clans and herds of ovals in \(\mathrm{PG}(2, 64)\). They further utilise the classification of hyperovals in \(\mathrm{PG}(2, 64)\) by \textit{P. Vandendriessche} [Electron. J. Comb. 26, No. 2, Research Paper P2.35, 12 p. (2019; Zbl 1423.51008)]. Up to equivalence, there are exactly four such hyperovals, from which exactly 19 distinct ovals in \(\mathrm{PG}(2, 64)\) are obtained. Going through the list of these ovals, it is shown that the ovals in a generalized fan in \(\mathrm{PG}(2, 64)\) are all conics or all pointed conics.
The present authors [Finite Fields Appl. 81, Article ID 102035, 13 p (2022; Zbl 1487.51006)] previously obtained that there are, up to equivalence, exactly three flocks of the quadratic cone in \(\mathrm{PG}(3,64)\), the linear flock, the Subiaco flock and the Adelaide flock. These flocks are used to analyse how many inequivalent spreads of \(T_2(\mathcal{O})\) are obtained if \(\mathcal{O}\) is represented by an o-polynomial belonging to the set of o-polynomials defining the ovals in the herd associated with the 64-clans corresponding to the three flocks or the set of their inverses.
Reviewer: Günter F. Steinke (Christchurch)A note on small weight codewords of projective geometric codes and on the smallest sets of even typehttps://zbmath.org/1522.510092023-12-07T16:00:11.105023Z"Adriaensen, Sam"https://zbmath.org/authors/?q=ai:adriaensen.samSummary: In this paper, we study the codes \(\mathcal{C}_k(n,q)\) arising from the incidence of points and \(k\)-spaces in \(\mathrm{PG}(n,q)\) over the field \(\mathbb{F}_p\), with \(q=p^h\), \(p\) prime. We classify all codewords of minimum weight of the dual code \(\mathcal{C}_k(n,q)^\perp\) in case \(q\in\{4,8\}\). This is equivalent to classifying the smallest sets of even type in \(\mathrm{PG}(n,q)\) for \(q\in\{4,8\}\). We also provide shorter proofs for some already known results, namely, of the best known lower bound on the minimum weight of \(\mathcal{C}_k(n,q)^\perp\) for general values of \(q\), and of the classification of all codewords of \(\mathcal{C}_{n-1}(n,q)\) of weight up to \(2q^{n-1}\).Envy-free division using mapping degreehttps://zbmath.org/1522.510102023-12-07T16:00:11.105023Z"Avvakumov, Sergey"https://zbmath.org/authors/?q=ai:avvakumov.sergey-ya"Karasev, Roman"https://zbmath.org/authors/?q=ai:karasev.roman-nSummary: In this paper we study envy-free division problems. The classical approach to such problems, used by \textit{D. Gale} [Int. J. Game Theory 13, 61--64 (1984; Zbl 0531.90011)], reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (e.g., in the Knaster-Kuratowski-Mazurkiewicz and the Gale theorem). We follow \textit{E. Segal-Halevi} [``Fairly dividing a cake after some parts were burnt in the oven'', in: Proceedings of the 17th international conference on autonomous agents and multiagent systems, AAMAS 2018. New York, NY: Association for Computing Machinery (ACM). 1276--1284 (2018; \url{doi:10.5555/3237383.3237888})] and \textit{F. Meunier} and \textit{S. Zerbib} [Isr. J. Math. 234, No. 2, 907--925 (2019; Zbl 1432.91067)], and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when \(n\), the number of players that divide the segment, is a prime power, and we provide counterexamples for every \(n\) which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free division problems when \(n\) is odd and not a prime power.
{{\copyright} 2020 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence.}On the entropy of Hilbert geometries of low regularitieshttps://zbmath.org/1522.510112023-12-07T16:00:11.105023Z"Cristina, Jan"https://zbmath.org/authors/?q=ai:cristina.jan"Merlin, Louis"https://zbmath.org/authors/?q=ai:merlin.louisThe aim of this paper is to prove two main results on the volume entropy of Hilbert geometries. The first one states that if \(\Omega\) is a convex and relatively compact domain of \({\mathbb R}^2\) which is Ahlfors \(\alpha\)-regular, then \(h(\Omega) = \frac{2\alpha}{\alpha +1}\), where \(h(\Omega)\) stands for the volume growth entropy.
The second results strengthens the first main theorem of \textit{G. Berck} et al. [Pac. J. Math. 245, No. 2, 201--225 (2010; Zbl 1204.52003)] by weakening the assumption of \(C^{1,1}\)-regularity of the boundary of the convex set \(\Omega\).
Reviewer: Victor V. Pambuccian (Glendale)A note on the Hausdorff distance between norm balls and their linear mapshttps://zbmath.org/1522.510122023-12-07T16:00:11.105023Z"Haddad, Shadi"https://zbmath.org/authors/?q=ai:haddad.shadi"Halder, Abhishek"https://zbmath.org/authors/?q=ai:halder.abhishekSummary: We consider the problem of computing the (two-sided) Hausdorff distance between the unit \(\ell_{p_1}\) and \(\ell_{p_2}\) norm balls in finite dimensional Euclidean space for \(1 \leq p_1 < p_2 \leq \infty \), and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the \(k_1\) and \(k_2\) unit \(D\)-norm balls, which are certain polyhedral norm balls in \(d\) dimensions for \(1 \leq k_1 < k_2 \leq d\). When two different \(\ell_p\) norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different \(\ell_p\) unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.Spherical ellipses and equilateral spherical triangleshttps://zbmath.org/1522.510132023-12-07T16:00:11.105023Z"Maehara, Hiroshi"https://zbmath.org/authors/?q=ai:maehara.hiroshiSummary: For a spherical triangle \(ABC\) such that \(AC\ne BC\) on a sphere,
the circular arc \(\overset{\frown}{ACB}\) and the spherical ellipse on the sphere with foci \(A\), \(B\) that passes through \(C\) intersect in exactly two points, and the two curves cross each other at the intersection points. This fact is applied to show extremal properties for equilateral spherical triangles.Geometry of the ovoids: reptilian eggs and similar symmetric formshttps://zbmath.org/1522.510142023-12-07T16:00:11.105023Z"Mladenova, Clementina D."https://zbmath.org/authors/?q=ai:mladenova.clementina-d"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mIn analogy to the conic sections studied by Apollonius of Perga, the second century AD Greek geometer Perseus studied spiric sections, intersections of a torus with a plane that is parallel to its rotational symmetry axis. These were studied analytically by the second author in [J. Geom. Symmetry Phys. 58, 81--97 (2020; Zbl 1471.53004)], where their use as a geometrical model of the egg has been emphasized. Here, the authors derive explicit formulas for the volume, surface area and the curvatures of the egg-mimicking shapes (for avian or reptilian eggs, depending on the values of some parameters) and then compare them with experimental data.
For the entire collection see [Zbl 1516.53003].
Reviewer: Victor V. Pambuccian (Glendale)Yet another mathematical model of eggs: two-parametric Brandt's shapeshttps://zbmath.org/1522.510152023-12-07T16:00:11.105023Z"Mladenova, Clementina D."https://zbmath.org/authors/?q=ai:mladenova.clementina-d"Mladenov, Ivaïlo M."https://zbmath.org/authors/?q=ai:mladenov.ivailo-mThis is part of a ``series of short reviews in which the existing ``models'' [for eggs] are covered in some depth and where possible -- appropriately extended'' that the author have recently embarked upon (``an almost exhaustive list of such ``models'' can be found in \textit{W. Hortsch} [Alte und neue Eiformeln in der Geschichte der Mathematik. München: Selbstverlag Hortsch (1990)]''). The model analysed here was proposed in an obscure venue by \textit{G. Brandt} [The research of an equation of a shell formed by the two-focus curve, in: Sb. tr. VZPI: ``Stroitelstvo i arhitektura''. Moscow: VZBI. 76--86 (1973)]. It is a surface of revolution described by
\[
z^2 + y^2 = \frac{3x(2a - x)((x+a)^2-c^2)}{4(x+a)^2}\quad x\in[0, 2a]
\]
in which \(a > \)c are real positive parameters
Reviewer: Victor V. Pambuccian (Glendale)The proportion of non-degenerate complementary subspaces in classical spaceshttps://zbmath.org/1522.510162023-12-07T16:00:11.105023Z"Glasby, S. P."https://zbmath.org/authors/?q=ai:glasby.stephen-peter"Ihringer, Ferdinand"https://zbmath.org/authors/?q=ai:ihringer.ferdinand"Mattheus, Sam"https://zbmath.org/authors/?q=ai:mattheus.samSummary: Given positive integers \(e_1,e_2\), let \(X_i\) denote the set of \(e_i\)-dimensional subspaces of a fixed finite vector space \(V=({\mathbb{ F}}_q)^{e_1+e_2}\). Let \(Y_i\) be a non-empty subset of \(X_i\) and let \(\alpha_i = |Y_i|/|X_i|\). We give a positive lower bound, depending only on \(\alpha_1,\alpha_2,e_1,e_2,q\), for the proportion of pairs \((S_1,S_2)\in Y_1\times Y_2\) which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded \textit{S. P. Glasby} et al. [Finite Fields Appl. 82, Article ID 102055, 31 p. (2022; Zbl 1491.51017)].Analytical fuzzy space geometry. Ihttps://zbmath.org/1522.510172023-12-07T16:00:11.105023Z"Ghosh, Debdas"https://zbmath.org/authors/?q=ai:ghosh.debdas"Gupta, Diksha"https://zbmath.org/authors/?q=ai:gupta.diksha"Som, Tanmoy"https://zbmath.org/authors/?q=ai:som.tanmoySummary: In this paper, we introduce a few basic concepts of \textit{fuzzy space geometry} in the three-dimensional Euclidean space. The ideas that we study here are \textit{space fuzzy points}, \textit{distance between two space fuzzy points}, and \textit{space fuzzy line segments}. To represent a space fuzzy point, we introduce the idea of a reference function of three variables. Accordingly, we define an \(S\)-type space fuzzy point. The concepts of \textit{same points} and \textit{inverse points} with respect to two continuous space fuzzy points are studied to formulate the fuzzy space geometrical concepts. To formulate same and inverse points for space fuzzy points, we provide the concepts of a \textit{fuzzy number along a line} in the space. With the help of the introduced three-variable reference function and fuzzy number along a line, explicit general expressions of same and inverse points for space fuzzy points are provided. Employing the concept of inverse points, we define a fuzzy distance between two space fuzzy points. Using the idea of the same points, addition and convex combination of two space fuzzy points are defined. A fuzzy line segment is formulated by a convex combination of two space fuzzy points. In the sequel, a concept of coincidence of two space fuzzy points is also provided. All the provided ideas are supported with numerical examples and necessary pictorial illustrations. Importantly, we also provide algorithms to find
\begin{itemize}
\item the fuzzy distance between two space fuzzy points,
\item the membership value of a number in the fuzzy distance between two space fuzzy points and
\item the membership value of a point in the space fuzzy line segment.
\end{itemize}The icosidodecahedronhttps://zbmath.org/1522.520222023-12-07T16:00:11.105023Z"Baez, John C."https://zbmath.org/authors/?q=ai:baez.john-c(no abstract)On volumes of hyperbolic right-angled polyhedrahttps://zbmath.org/1522.520282023-12-07T16:00:11.105023Z"Alexandrov, Stepan A."https://zbmath.org/authors/?q=ai:alexandrov.stepan-a"Bogachev, Nikolay V."https://zbmath.org/authors/?q=ai:bogachev.nikolay-v"Vesnin, Andrei Yu."https://zbmath.org/authors/?q=ai:vesnin.andrei-yu"Egorov, Andrei A."https://zbmath.org/authors/?q=ai:egorov.andrei-aNew upper bounds for the volumes of right-angled polyhedra in the three-dimensional hyperbolic space are obtained in the following three cases: for ideal polyhedra with all vertices on the hyperbolic boundary; for compact polyhedra with only finite (=non-ideal) vertices; and for finite-volume polyhedra with vertices of both types.
Reviewer: Gaiane Panina (Sankt-Peterburg)Coprime Ehrhart theory and counting free segmentshttps://zbmath.org/1522.520312023-12-07T16:00:11.105023Z"Manecke, Sebastian"https://zbmath.org/authors/?q=ai:manecke.sebastian"Sanyal, Raman"https://zbmath.org/authors/?q=ai:sanyal.ramanIn this paper, the authors study lattice polytopes, that is, convex polytopes \(P\) with vertices in \(\mathbb{Z}^d\), for some \(d\). A lattice polytope \(P\) is called free (or empty) if \(P \cap \mathbb{Z}^d\) are precisely the vertices of \(P\). Free polytopes have been studied in relation to integer programming; see, for example, [\textit{H. W. Gould}, Fibonacci Q. 2, 241--260 (1964; Zbl 0129.02903); \textit{B. Reznick}, Discrete Math. 60, 219--242 (1986; Zbl 0593.52008); \textit{H. E. Scarf}, Math. Oper. Res. 10, 403--438 (1985; Zbl 0577.90054)]. Also, they are related to hollow polytopes, whose (relative) interiors do not contain lattice points. The interest of the authors in free polytopes comes from valuation theory and geometric combinatorics. For a set \(S \subset \mathbb{R}^d\), denote by \(|S| : \mathbb{R}^d \mapsto \{0,1\}\) its indicator function and let \(\alpha_{1}(S) := |S \cap \mathbb{Z}^d|\). In the context of valuation theory, \textit{D. A. Klain} proposed in [Adv. Math. 147, No. 1, 1--34 (1999; Zbl 0944.52007)], to investigate the functions \(\alpha_{i}(P,n)\) that count the number of free polytopes in \(n.P= \{np : p \in P\}\) with \(i\) vertices. For \( i = 1\), this is the well known Ehrhart polynomial [\textit{M. Beck} and \textit{R. Sanyal}, Combinatorial reciprocity theorems. An invitation to enumerative geometric combinatorics. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1411.05001)]; for \(i > 3\), the computation seems to be impossible. However, for \(i = 2, 3\) it is computationally challenging. In this paper, the authors develop a theory of coprime Ehrhart functions that count lattice points with relatively prime coordinates and use it to compute \(\alpha_{2}(P,n)\) for unimodular simplices. They show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and they give some applications to combinatorial counting. Among others, the authors of this article completely determine all functions \(\alpha_{i}(P,n)\) for unimodular triangles, and they propose to find an expression for \(\alpha_{3}(\Delta_{d},n)\), where \(\Delta_{d}\) is the standard unimodular simplex \(\subset \mathbb{R}^{d+1}\).
Reviewer: Boumédiène Et-Taoui (Mulhouse)Local geometry of polyhedra and Cauchy's rigidity theoremhttps://zbmath.org/1522.520412023-12-07T16:00:11.105023Z"Honvault, Pascal"https://zbmath.org/authors/?q=ai:honvault.pascalBased on the description of each star of a polyhedron of genus \(0\) in three-dimensional space by means of the quaternionic algebra introduced in [\textit{P. Honvault}, Forum Geom. 16, 313--316 (2016; Zbl 1352.51014)] and the Gram-Schmidt orthonormalisation process, the author derives a formula involving the angles of the faces (the ``internal angles''), and the angles between the faces (the ``external angles''), which leads to a new proof of Cauchy's rigidity theorem.
Reviewer: Victor V. Pambuccian (Glendale)Hyperbolic angles in Lorentzian length spaces and timelike curvature boundshttps://zbmath.org/1522.530252023-12-07T16:00:11.105023Z"Beran, Tobias"https://zbmath.org/authors/?q=ai:beran.tobias"Sämann, Clemens"https://zbmath.org/authors/?q=ai:samann.clemensInspired by \textit{ E. H. Kronheimer} and \textit{R. Penrose} [Proc. Camb. Philos. Soc. 63, 481--501 (1967; Zbl 0148.46502)], a framework for Lorentzian geometry in the synthetic geometric spirit of Alexandrov spaces and CAT(\(k\))-spaces was developed in [\textit{M. Kunzinger} and \textit{C. Sämann}, Ann. Global Anal. Geom. 54, No. 3, 399--447 (2018; Zbl 1501.53057)] under the name of Lorentzian (pre-)length spaces. A notion of hyperbolic angle, of an angle between time-like curves, of a time-like tangent cone, as well as of an exponential map were introduced to provide a language in which to ask questions similar to those asked in synthetic differential geometry. The exploration leads to results such as: a triangle inequality for (upper) angles, the characterization of time-like curvature bounds (defined via triangle comparison) with an angle monotonicity condition, an improvement of a geodesic non-branching result for spaces with time-like curvature bounded below.
Reviewer: Victor V. Pambuccian (Glendale)Stability of non-degenerate Ricci-type Palatini theorieshttps://zbmath.org/1522.830392023-12-07T16:00:11.105023Z"Annala, Jaakko"https://zbmath.org/authors/?q=ai:annala.jaakko"Räsänen, Syksy"https://zbmath.org/authors/?q=ai:rasanen.syksy(no abstract)Constraining Weil-Petersson volumes by universal random matrix correlations in low-dimensional quantum gravityhttps://zbmath.org/1522.830822023-12-07T16:00:11.105023Z"Weber, Torsten"https://zbmath.org/authors/?q=ai:weber.torsten"Haneder, Fabian"https://zbmath.org/authors/?q=ai:haneder.fabian"Richter, Klaus"https://zbmath.org/authors/?q=ai:richter.klaus-jurgen"Urbina, Juan Diego"https://zbmath.org/authors/?q=ai:urbina.juan-diegoSummary: Based on the discovery of the duality between Jackiw-Teitelboim quantum gravity and a double-scaled matrix ensemble by \textit{P. Saad} et al. in [``JT gravity as a matrix integral'', Preprint, \url{arXiv:1903.11115}], we show how consistency between the two theories in the universal random matrix theory (RMT) limit imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil-Petersson (WP) volumes, solving a celebrated nonlinear recursion formula that is notoriously difficult to analyse. Since our results imply \textit{linear} relations between the coefficients of the WP volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction. In this way, we propose a long-term program to improve the understanding of mathematically hard aspects concerning moduli spaces of hyperbolic manifolds by using universal RMT results as input.Stable big bang formation for Einstein's equations: the complete sub-critical regimehttps://zbmath.org/1522.832572023-12-07T16:00:11.105023Z"Fournodavlos, Grigorios"https://zbmath.org/authors/?q=ai:fournodavlos.grigorios"Rodnianski, Igor"https://zbmath.org/authors/?q=ai:rodnianski.igor"Speck, Jared"https://zbmath.org/authors/?q=ai:speck.jaredSummary: For \((t,x) \in (0,\infty )\times \mathbb{T}^{\mathfrak{D}} \), the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as \(t \downarrow 0\), i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents \(\widetilde{q}_1,\cdots ,\widetilde{q}_{\mathfrak{D}} \in \mathbb{R} \), which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at \(\lbrace t = 1 \rbrace \), as long as the exponents are ``sub-critical'' in the following sense: \( \underset{\substack{I,J,B=1,\cdots , \mathfrak{D}\\ I <J}}{\max } \{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1\). Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with \(\mathfrak{D}= 3\) and \(\widetilde{q}_1 \approx \widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3\), which corresponds to the stability of the Friedmann-Lemaître-Robertson-Walker solution's Big Bang or (2) the Einstein-vacuum equations for \(\mathfrak{D}\geq 38\) with \(\underset{I=1,\cdots ,\mathfrak{D}}{\max } |\widetilde{q}_I| < 1/6\). In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the \(1+\mathfrak{D} \)-dimensional Einstein-scalar field system for all \(\mathfrak{D}\geq 3\) and the \(1+\mathfrak{D} \)-dimensional Einstein-vacuum equations for \(\mathfrak{D}\geq 10\); both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in \(1+3\) dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized \(U(1)\)-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized \(U(1)\)-symmetric solutions.
Our proof relies on a new formulation of Einstein's equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi-Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to \(t\), and to handle this difficulty, we use \(t\)-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the \(t\)-weights and interpolating between the singularity-strength of the solution's low order and high order derivatives. Finally, we note that our formulation of Einstein's equations highlights the quantities that might generate instabilities outside of the sub-critical regime.A canonical complex structure and the bosonic signature operator for scalar fields in globally hyperbolic spacetimeshttps://zbmath.org/1522.834242023-12-07T16:00:11.105023Z"Finster, Felix"https://zbmath.org/authors/?q=ai:finster.felix"Much, Albert"https://zbmath.org/authors/?q=ai:much.albertSummary: The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.Quantum cosmology of pure connection general relativityhttps://zbmath.org/1522.834272023-12-07T16:00:11.105023Z"Gielen, Steffen"https://zbmath.org/authors/?q=ai:gielen.steffen"Nash, Elliot"https://zbmath.org/authors/?q=ai:nash.elliotSummary: We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebański formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.The polyhedral geometry of pivot rules and monotone pathshttps://zbmath.org/1522.900252023-12-07T16:00:11.105023Z"Black, Alexander E."https://zbmath.org/authors/?q=ai:black.alexander-e"De Loera, Jesús A."https://zbmath.org/authors/?q=ai:de-loera.jesus-a"Lütjeharms, Niklas"https://zbmath.org/authors/?q=ai:lutjeharms.niklas"Sanyal, Raman"https://zbmath.org/authors/?q=ai:sanyal.ramanSummary: Motivated by the analysis of the performance of the simplex method, we study the behavior of families of pivot rules of linear programs. We introduce \textit{normalized-weight pivot rules} which are fundamental for the following reasons: First, they are \textit{memory-less,} in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. The latter leads to the existence of two polytopes, the \textit{pivot rule polytopes} and the \textit{neighbotopes,} that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences and compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. We introduce a normalized-weight pivot rule, called the \textit{max-slope pivot rule}, which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine the \textit{monotone path polytopes} of Billera and Sturmfels. Our constructions are important beyond optimization and provide new perspectives on classical geometric combinatorics. Special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest-improvement pivot rules we draw connections to sweep polytopes and polymatroids.