Séminaires : Séminaire CAGe: Combinatorics, Arithmetic and Geometry

10:30-11:20:  Rakhi Pratihar (INRIA Saclay centre)

Title: Homological Invariants of q-Matroids

Abstract:

The matroid complex, the simplicial complex of independent sets of a finite matroid, carries many significant invariants of the matroid by associating topological (T0-Alexandroff space) and algebraic structure (Stanley–Reisner ring) to the matroid. A central theme is to understand the homological invariants, in particular, the simplicial homology groups, Cohen–Macaulayness, and the graded Betti numbers. Foundational work of Reisner (1978) and Stanley (1976) explains how Cohen–Macaulay properties of Stanley–Reisner rings are obtained by the homology of links, while Hochster gives a direct for-
mula relating graded Betti numbers to simplicial homology. On the topological side, Provan (1977) proved that matroid complexes are shellable, which implies that their reduced homology to be concentrated in top dimension. q-Matroids are a natural q-analogue of matroids where subsets are replaced by vector subspaces, and in the representable case, they correspond to rank metric codes. In this talk, I will present q-analogues of the results above, based on the research works [1, 2, 3, 4] that initiates a homological framework for q-matroids. If time permits, I will discuss an application of these results to determining important discrete invariants of linear error-correcting codes, namely the higher weight spectra, in both Hamming and rank metrics.

11:30-12:20:  Pavel Kalugin (University of Paris-Sud)

Title: Robust minimal matching rules for quasicrystals

Abstract: Tilings of $\mathbb{R}^d$ with finite local translational complexity can be lifted to form part of a periodic arrangement of polyhedra in a higher-dimensional space $\mathbb{R}^n$. Certain matching rules for tilings (such as those of Penrose tilings) fix an irrational slope of the lifted tiling. We propose to describe the matching rules of such quasiperiodic patterns using flat-branched semi-simplicial complexes. The lifting of a tiling is defined via an embedding of such a complex $B$ into the real $n$-dimensional torus $\mathbb{T}^n$, and the slope-fixing property of the matching rules is determined by the image of the induced homomorphism $H_d(B) \to H_d(\mathbb{T}^n)$.

14:30-15:20:  Alp Müyesser (University of Oxford)

Title: Graham's rearrangement conjecture

Abstract: Many problems in combinatorics are difficult for a similar reason: the constraints are too loose for explicit constructions to be easy, yet too rigid for purely probabilistic methods to apply. We will discuss how a combination of probabilistic and algebraic methods can be effective in this setting, via the following innocuous-looking conjecture of Graham (1971): for any prime p, any set of integers modulo p admits an ordering s_1,...,s_k such that all partial sums s_1+...+s_j are distinct modulo p. 

This conjecture has been resolved in the affirmative for all sufficiently large primes through a series of works using a wide range of techniques. We will outline some of the ideas behind these proofs.

15:30-16:20:  Fedor Selyanin (Skoltech and HSE, RUSSIA)

Title: Negligible and thin polytopes

Abstract: The h*-polynomial of a lattice polytope encodes the number of lattice points in its integer dilations. The local h*-polynomial (or ℓ*-polynomial) arises naturally in the Katz–Stapledon decomposition formulas for the h*-polynomial in case of polyhedral subdivisions. A polytope P is called thin if ℓ*(P; 1) = 0.

According to the global Kouchnirenko's theorem, an affine hypersurface {f = 0} ⊂ Cn with a convenient Newton polytope P ⊂ Rn≥0 and non-degenerate coefficients has the homotopy type of a bouquet of ν(P) spheres of dimension n − 1. Here, ν(P) is a certain alternating sum of volumes, known as the Newton number. A convenient polytope P is called negligible if ν(P) = 0.

 Following the paper arXiv:2507.03661, we will classify negligible polytopes as certain Cayley sums, called Bk-polytopes, using the Furukawa–Ito classification of dual-defective sets. By employing a generalization of the Katz–Stapledon decomposition formulas, we will show that for any convenient polytope P, the inequality ℓ*(P; 1) ≤ ν(P) holds. Consequently, negligible polytopes are thin.

14:00-15:00:  Anya Nordskova (University of Tokyo, Japan)

Title: Banach's problem in dimension 4 

Abstract: 90 years ago S. Banach asked the following question. Let V be a normed (real or complex) vector space and assume that all its subspaces of a fixed finite dimension k, where 1 < k < n = dim V, are isometric to each other. Is B necessarily Euclidean (that is, the norm is induced by an inner product)?  Translating the question into the language of convex sets: Let B be a convex centrally symmetric body in an n-dimensional normed vector space and assume that all its cross-sections by k-dimensional vector subspaces are linearly equivalent to each other. Is B necessarily an ellipsoid? 

In general, the question remains open, but affirmative answers were given in many special cases by Auerbach, Mazur and Ulam (1935), Dvoretzky (1959), Gromov (1967), Milman (1971), Bor, Hernandez-Lamoneda, Jimenez-Desantiago (2019). Almost all of these works are based on methods of algebraic topology.

Together with S. Ivanov and D. Mamaev (Invent. Math, 2023) we managed to solve Banach's problem in the smallest previously unknown case, namely, for k+1=n= 4. Due to the parallelizability of the three-dimensional sphere, topological arguments used in previous works do not provide any information in our case. Hence, we develop a different, differential geometric approach. 

This talk continues from our February 24th session, providing further details on the proof.

15:30-16:30:  Shai Haran (Technion, Israel)

Title: Non additive geometry 

Abstract: The usual dictionary between geometry and commutative algebra is not appropriate for Arithmetic geometry because addition is a singular operation at the "Real prime". We replace Rings, with addition and multiplication, by Props (=strict symmetric monoidal category generated by one object), or by Bioperad (=two closed symmetric operads acting on each other): to a ring we associate the prop of all matrices over it, with matrix multiplication and block direct sums as the basic operations, or the bioperad consisting of all raw and column vectors over it. We define the "commutative" props and bioperads, and using them we develop a generalized algebraic geometry, following Grothendieck footsteps closely. This new geometry is appropriate for Arithmetic (and potentially also for Physics).

This this continues from our February 24th session.

11:30-12:30:  Alex Fink (Queen Mary University of London)

Title: A couple ways matroids are like algebraic varieties

Abstract: Matroids are fundamental combinatorial objects. One way to think about a matroid is as recording the combinatorics of which coordinates can vanish together on a linear subspace. Not all matroids come from linear spaces, but in a surprisingly rich collection of ways they behave like they do: matroids have properties that come from an algebro-geometric construction when the linear space exists, but that still hold when it isn't. Time permitting I'll talk about a few of these, but my target will be new work in progress with Eur and Larson stating some cohomology vanishing theorems that give a second proof of Speyer's f-vector conjecture.

14:45-15:45: Oliver Lorscheid (University of Groningen, Netherlands)

Title: Tits's dream of F_1, combinatorial flag varietes and moduli spaces of matroids

Abstract: The first mathematicians that mentioned the desire for a field F_1 with one element, and of geometry over such an elusive object, was Jacques Tits who pursued this idea in the 1950s. His hope was to explain the analogy between geometries over finite fields F_q and certain incidence geometries that behave like the limit q -> 1. Much later, around 2000, Borovic, Gelfand and White expanded Tits's perspective towards combinatorial flag varieties, which are incidence geometries that stem from matroid theory.

In this talk, we introduce a formalism for algebraic geometry over F_1 that captures all these effects in terms of moduli spaces of flag matroids: finite field geometries emerge as F_q-rational points of these moduli spaces, combinatorial flag varieties arise as rational points with values in the so-called Krasner hyperfield and the Tits's incidence geometries resurface as the subsets of closed points of these moduli spaces. We conclude the talk with an explanation on how algebraic groups generalize to F_1 and in which sense SL(n) acts on the moduli space of flags.

16:00-17:00: Jana Rodriguez Hertz (Sustech, China)

Title: HOW FREQUENT IS THE BUTTERFLY EFFECТ?

Abstract: We pose the following conjecture: The fact that a small amount of disorder (positive Lyapunov exponents) leads to general stable disorder (stable ergodicity and even stable Bernouliness) is the most frequent situation in conservative dynamics. We survey some of the recent advances.

Schedule

11:30-12:30:  Veronica Fantini (LMO, Orsay)

Title:  Resurgent series from local Calabi-Yau threefolds and arithmetic 

Abstract:  Formal divergent power series appear in various contexts, and the theory of resurgence introduced by Écalle is a prominent tool to study them. In fact, it associates to a divergent power series a collection of exponentially small corrections with a set of complex numbers known as Stokes constants. In this talk, I will introduce some of the main ideas of resurgence and discuss the arithmetic structure of the Stokes constants of the divergent series associated with locally weighted projective spaces.

14:45-15:45: Eitan Bachmat (Ben-Gurion University)

Title:   Combinatorics, geometry and lenses

Abstract:  We will explore the connection between combinatorial notions such as the chain polytope and the entropy of a poset, Lorentzian geometry and super lenses in hyperbolic metamaterials.

Schedule

11:30-12:30:  Meral Tosun (Galatasaray University)

Title:  Geometric and combinatorial views on singularities

Abstract:  We focus on a special class of singularities that allow a polyhedral and

combinatorial description. We show how jet schemes capture fine local data

and lead to a resolution process in this class.

14:45-15:45: Dmitry Mineev ( Bar-Ilan University)

Title:  From tropical modifications of Bergman fans to correspondences and flag fans

Abstract:  Bergman fans are tropical counterparts of matroids. Strong maps between matroids admit a description as morphisms between Bergman fans. However, taking limits of even the simplest diagrams of matroids requires so-called weak maps, which do not translate to the tropical side. We suggest a fix, generalizing both weak maps of matroids and tropical morphisms, and construct a functor relating the two. We also introduce flag fans as a convenient tool for computations in this extended setting.

16:00-17:00: Christos Tatakis (University of Western Macedonia)

Title: The structure of complete intersection graphs and their planarity.

Abstract:  

Let G be a connected, undirected, finite and simple graph. We study the complete intersection property on the toric ideal $I_G$. In general, the toric ideal $I_G$ is complete intersection if and only if it can be generated by h binomials, where h=m-n+1 if G is a bipartite graph or h=m-n if G is not a bipartite graph, where by m we denote the number of the edges of G and by n the number of its vertices. The answer is known in the case of bipartite graphs, i.e. graphs with no odd cycles. In the last years, several useful partial results have been proved and they provide key properties of complete intersection toric ideals of graphs. 

We focus on the general case, where G is a random graph and we present a structural theorem which gives us necessary and sufficient conditions in which the toric ideal $I_G$ is complete intersection. Moreover, we characterize with sufficient and necessary conditions the complete intersection graphs which are planar. The talk is based on a joint work with Apostolos Thoma.

11:30-12:30:  Carsten Peterson (Sorbonne University)

Title:  A degenerate version of Brion's formula

Abstract: Brion's formula says that the continuous (resp. discrete) Fourier-Laplace transform of a polytope $P$ (resp. lattice points in a rational polytope) is equal to the sum of the continuous (resp. discrete) Fourier-Laplace transforms of the tangent cones of the vertices. However, whereas the former is an entire function, each latter function is merely meromorphic with singularities on the dual vectors $\xi$ which are constant on some positive-dimensional face of the polytope (resp. constant on the sublattice parallel to some positive-dimensional face). Because of this, one cannot ``plug into'' Brion's formula at such points.

We shall present a ``degenerate'' extension of Brion's formula for which one can still ``plug in'' at such troublesome points. Like Brion's formula it will be made up of terms each of which only depends on some local geometry of $P$. Our formula is particularly useful for understanding how the Fourier-Laplace transform varies over a family of polytopes with the same normal fan. In the generic case our formula reduces to the original Brion's formula, and in the maximally degenerate case ($\xi = 0$) it reduces to the volume of the polytope (resp. the Ehrhart quasi-polynomial).

14:45-15:45: Jacinta Torres (Jagiellonian University)
 

Title:  A new branching model in terms of flagged hives

Abstract: We prove a bijection between the branching models of Sundaram and Kwon. Along the way, we obtain a new branching model In terms of flagged hives polytopes. This is joint work with Sathish Kumar.

16:00-17:00: Evrydiki Nestoridi (Stony Brook University)

Title: Shuffling via transpositions

Abstract: In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of $n$ cards sufficiently well via random transpositions takes $1/2 n log n$ steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on two other shuffles, generalizing random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint works with A. Yan and S. Arfaee.

11:30-12:30: Danylo Radchenko

Title: Polylogarithms and the Steinberg module
Abstract: I will talk about a surprising connection between the Steinberg module of rationals and a certain space of multiple polylogarithms on tori. I will expain how this idea leads to several important implications for Goncharov's program on structure of multiple polylogarithms, and if time permits I will also discuss how it relates to the Church-Putman-Farb conjecture for the cohomology of GL_n(Z). The talk is based on a joint work in progress with Steven Charlton and Daniil Rudenko.





14:45-15:45: Olga Trapeznikova

Title: Intersection cohomology of moduli spaces of semistable bundles on curves

Abstract: The study of the intersection cohomology of moduli spaces of semistable bundles on Riemann surfaces began in the 80’s with the works of Frances Kirwan. Motivated by the work of Mozgovoy and Reineke, in joint work with Camilla Felisetti and Andras Szenes, we give a complete description of these structures via a detailed analysis of the Decomposition Theorem applied to a certain map. We also give a new formula for the intersection Betti numbers of these moduli spaces, which has a clear geometric meaning. In this talk, I will describe our results.




16:05-17:05: Claire Burrin

Title: Rational points on spheres

Abstract: A sequence of point-sets is considered optimally distributed with respect to covering it its covering exponent is 1. I will discuss some new results on the covering exponent for sequences of rational points on spheres. This is joint work with Matthias Gröbner.

10:00-11:00: Gavin Brown (University of Warwick)

 Title: Noncommutative singularity theory

Abstract:  I describe a noncommutative version of Arnold's classification of function germs and its application to the classification of simple 3-fold flops. The connection is the noncommutative deformation theory of a crepant rational curve on a 3-fold, which in turn exposes an ADE classification on noncommutative Jacobian algebras. This is joint work with Michael Wemyss (Glasgow).

11:30-12:30: Alexander Esterov (LIMS)

Title: Solvable systems of equations, Galois groups in enumerative geometry, and small lattice polytopes

Abstract: The general polynomial of a degree higher than 4 cannot be solved by radicals. This classical theorem has a  multidimensional version: solvable general systems of polynomial equations are in (almost) one-to-one correspondence with lattice polytopes of volume 4, and the latters admit a finite classification. In the narrow sense, I will talk about this xix-century-style result. In a broader sense, we shall look at the Galois groups of problems of enumerative geometry (such as Schubert calculus), and how their study leads to seemingly distant topics such as polyhedral geometry and braid groups.


l15:00-16:00: Joni Teräväinen (University of Turku)

Title: Uniformity of the primes in short intervals

Abstract: Gowers norms are a measure of the pseudorandomness properties of a set. Green, Tao and Ziegler showed in 2012 that the set of prime numbers is Gowers uniform, in the sense that a suitably normalised version of it has small Gowers norms. We show that the primes are Gowers uniform also when restricted to short intervals [x,x+x^c] for suitable c. Morover, the admissible value of c is smaller if we look at primes in almost all short intervals. I will also discuss an application of such results to an averaged version of the Hardy--Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.


16:30-17:30: Sean Eberhard (Queen's University)

Title: Diameter bounds for finite classical groups generated by transvections

Abstract: The diameter of a group G with respect to a symmetric generating set X is the smallest integer d such that every element of G is the product of at most d elements of X. A well-known conjecture of Babai predicts that every nonabelian finite simple group G has diameter (log |G|)^O(1) with respect to any generating set. This is known to be true for bounded-rank groups of Lie type (Helfgott; Pyber--Szabo; Breuillard--Green--Tao), but the conjecture is wide open for high-rank groups. There has bee a good deal of progress recently for generating sets containing either special elements or random elements. In this talk I will outline the proof that the conjecture holds for the classical groups SL_n(q), Sp_{2n}(q), SU_n(q) and any generating set containing a transvection. The proof is based essentially on (a) the positive resolution of Babai's conjecture in bounded rank and (b) a result of Kantor classifying finite irreducible linear groups generated by transvections.