Séminaires : Séminaire CAGe: Combinatorics, Arithmetic and Geometry

Les sujets sont ceux décrits par le titre :). Ils doivent être compris dans un sens large.

Notre objectif est de nous réunir avec une périodicité mensuelle.

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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.