Mohamed Barakat
Title: The interplay between classical and constructive mathematics
Abstract: This talk will address the historical controversy between classical and constructive mathematics and how it has evolved into a fascinating interplay between these two forms of mathematical reasoning. I will argue that classical mathematicians have good reasons to care more deeply about constructive mathematics.
Thomas Britz
Title: Wei duality, Fomin-Greene duality and demimatroids
Abstract: The Fomin-Greene Duality Theorem for finite posets and Wei's Duality Theorem for linear codes, and demimatroids more generally, are celebrated theorems that relate sets of extremal invariants via seemingly similar dualities. This talk will give an overview of these theorems and will describe how their two dualities relate.
In particular, it will be shown that the chain and antichain numbers each satisfy a Wei-type duality with antichain- and chain-deletion numbers, respectively, and that, for each poset with at least two elements, its chain and antichain demimatroids are not related by any composition of standard demimatroid duality operations. Their upper Wei-number numbers do however determine one another, via Fomin--Greene duality. The two dualities thereby coincide for Wei numbers but not at the level of the underlying rank functions.
Carlos Cid
Title: Cryptography from LPN: design, classical security and quantum attacks
Abstract: The Learning Parity with Noise problem is one of the most prominent hard learning problems with application in cryptography. Its search version can be viewed as the average-case problem of decoding random linear codes in the presence of noise. Although not as versatile as its generalisation over large prime fields, the Learning with Errors (LWE) problem, LPN and its variants has been used to define symmetric-key encryption and authentication protocols, collision-resistant hash functions, pseudo-random generators, public-key encryption, identification and commitment schemes, and zero-knowledge proofs. Its appeal includes the potential for lightweight implementations and security reductions to a well-studied hardness assumption. In this talk, we introduce LPN, survey selected cryptographic constructions based on it, and discuss their security against both classical and quantum adversaries.
Junyan Chu
Title: A Saito-type criterion for generators of logarithmic derivation modules
Abstract: For a hyperplane arrangement in an $\ell$-dimensional vector space, Saito’s criterion characterizes when a given set of $\ell$ derivations generates the entire logarithmic derivation module in terms of the determinant of their coefficient matrix. In this case, the arrangement is free. We extend this criterion to a non-free arrangement with larger families of derivations. Specifically, we study the maximal minors of the $p \times \ell$ coefficient matrix ($p \ge \ell$) formed by $p$ logarithmic derivations. In particular, when $p = \ell + 1$, we obtain an explicit characterisation of strictly plus-one generated (SPOG) arrangements.
Emanuele Delucchi
Title: Combinatorics and topology of arrangements: an introduction
Abstract: The theory of arrangements lies at the crossroads of discrete mathematics, topology and geometry. The classical case is that of arrangements of hyperplanes: here the interplay between matroid theory and the topology of the arrangement's complement has given rise to a sprawling research field over the last 60 years. Recently, the theory has broadened to include the study of arrangements of hypersurfaces, especially in the complex torus and in abelian Lie groups.
In this introductory talk I will review some basics of the classical theory and present an overview of current results in the case of hypersurface arrangements, with an eye to potential connections to the topics of the conference. A particular focus will be on the corresponding combinatorial framework, including some recent generalizations of matroid theory that arose in this context.
Kenji Fukumizu
Title: Flow Matching from Viewpoint of Proximal Operators
Abstract: We discuss the dynamics of Optimal Transport Conditional Flow Matching (OTCFM) under the manifold hypothesis, where the support of the target distribution is a low dimensional manifold. We first derive an exact proximal expression of the vector field, using an extended Brenier potential for the OT; the vector field can be expressed by a denoiser, which is presented as the proximal operator associated with the potential function. Based on this explicit proximal form of the vector field, we analyze the manifold stability for perturbations to initial points, using the Lyapunov exponents of the dynamics. By analyzing the local behavior with the tools in convex analysis, we show that the dynamics contract linearly in the perturbation of the normal directions to the manifold, while perturbations in the tangential directions are preserved; this implies the stability of the manifold structure to the perturbations.
Alice L.L.Gao
Title: Equivariant inverse Z-polynomials of matroids
Abstract: In this talk, I will discuss the equivariant inverse Z-polynomial of a matroid equipped with an action of a finite group. We prove that the coefficients of the equivariant inverse Z-polynomials are honest representations and that these polynomials are palindromic. Explicit formulas are obtained for uniform matroids equipped with the action of the symmetric group. The corresponding formulas for q-niform matroids are derived using the Comparison Theorem for unipotent representations. We show that these polynomials are equivariantly unimodal and strongly inductively log-concave for both uniform and q-niform matroids. For arbitrary equivariant paving matroids, explicit expressions are obtained by relating the polynomials of a matroid to those of its relaxation.
Koji Imamura
Title: On the quasi-polynomial structure of weight enumerators
Abstract: The weight enumerator of a linear code records the number of codewords of each Hamming weight. In this talk, we consider codes over the ring of integers modulo q obtained from a fixed integer generator matrix and study their weight enumerators as q varies. We describe their quasi-polynomial structure and its connection with characteristic quasi-polynomials of hyperplane arrangements. We then explain how this structure reduces the determination of minimum weights for the entire family to computations at finitely many values of q. We also discuss a Greene-type identity relating weight enumerators to the Tutte quasi-polynomial. This is joint work with Norihiro Nakashima and Takuya Saito.
Satoru Iwata
Title: Linear Matroid Parity: Algorithms and Applications
Abstract: Matching and matroid intersection are two primary frameworks of combinatorial optimization problems that admit polynomial-time algorithms. As a common generalization of these two frameworks, the matroid parity (or matroid matching) problem was introduced in the 70s. This framework is so general that it requires in the worst case an exponential number of independence oracle calls. Nevertheless, Lovász (1980) showed that this problem admits a min-max formula and a polynomial-time algorithm for linearly represented matroids. Since then, efficient algorithms have been developed for the linear matroid parity problem, which finds applications in various fields including electric circuit analysis, combinatorial rigidity theory, and topological graph theory. This talk provides an overview on linear matroid parity, putting emphasis on its applications and recent algorithmic developments.
Shota Maehara
Title: Combinatorial formality for certain free arrangements via lc-bases
Abstract: One of the central open problems in the theory of hyperplane arrangements is Terao’s conjecture. It asserts that the freeness of the logarithmic derivation module, which is defined by the defining polynomial of a hyperplane arrangement, depends only on its combinatorial structure. In the theory of hyperplane arrangements, there are several variations of the question of whether a certain property is determined by the combinatorial structure. For example, formality, which is a property strictly weaker than freeness, is known to be not combinatorial in general. In this talk, as an approach to Terao’s conjecture, we consider the following weaker conjecture.
Conj. 1: Every free arrangement is combinatorially formal.
In other words, Conj. 1 asserts that every arrangement having the same combinatorial type as a free arrangement is, even if we do not know whether it is free, at least formal. According to the theory of lc-bases introduced by M. Falk for matroids, Conj. 1 immediately follows if we assume the following assertion.
Conj. 2: Freeness implies the existence of an lc-basis of the associated linear matroid.
We take this as the main conjecture in this talk. As our main result, we verify the conjecture for certain classes of real central arrangements of rank three. In our proof, we reduce the problem to an argument based on counting the chambers of affine line arrangements in \mathbb{R}^2. This is joint work with Shizuo Kaji and Hiroki Sasaki.
Satoshi Murai
Title: Graphic arrangements and Stanley-Reisner theory
Abstract: Modules of logarithmic derivations and logarithmic differential 1-forms are central objects in the algebraic study of hyperplane arrangements. In this talk, I will show that these modules are related to face rings of simplicial posets and cover ideals of graphs that appear s in combinatorial commutative algebra. These connections enable us to express some basic algebraic invariants of these modules, such as their Hilbert series and local cohomology modules, in terms of combinatorial and topological data associated with clique complexes and simplicial posets arising from graphs. The talk is based on a joint work with Takuro Abe.
Haru Negami
Title: Integrable transformation of Knizhnik-Zamolodchikov-type equation and its application to quantum computing
Abstract: In topological quantum computation, quantum gates are realized as unitary representations of the braid group. In this talk, I will present a method for constructing unitary representations of the braid group as monodromy representations of KZ-type equations, by means of an integral transformation called middle convolution. I will also discuss applications to quantum computing.
Taihei Oki
Title: Exchangeability of Matroid Bases
Abstract: The symmetric basis exchange property states that for any bases $A$ and $B$ of a matroid and any $x \in A\setminus B$, there exists $y \in B\setminus A$ such that both $A-x+y$ and $B+x-y$ are bases. For representable matroids, this property can be derived from the Grassmann--Plücker identity. In this talk, we survey classical and recent exchange properties of matroid bases that go beyond the basic exchange axiom but can nevertheless be derived from it. We also present a general framework for deriving extensions of the Grassmann--Plücker identity. We also discuss applications of these exchange properties in theoretical computer science.
Hal Schenck
Title: Subarrangements of type A: the weak Lefschetz property of the Artinian Orlik-Terao algebra
Abstract: In 1994, Orlik and Terao introduced a commutative Artinian analog S/I(A) of the Orlik-Solomon algebra of a hyperplane arrangement A to answer a question of Aomoto. A central topic of investigation in the study of Artinian algebras is the Weak Lefschetz Property (WLP). We analyze WLP for the Artinian Orlik-Terao algebra of graphc arrangements. Even for chordal graphs (which give rise to Koszul algebras) WLP sometimes fails; conversely an analysis of the state polytope shows WLP can hold even when WLP fails for all possible initial ideals. More generally, for any algebra with a tensor product decomposition, we construct canonical elements in the kernel of the multiplication map, refining previous results in the literature. Joint work with N. Gaubatz
Akimichi Takemura
Title: Some applications of theory of hyperplane arrangements to statistics
Abstract: In this talk, I survey some of my works with Kamiya and Terao concerning applications of hyperplane arrangements to the study of ranking patterns of unfolding models in statistics.
Hopein Christofen Tang
Title: Column multiplicity approach to linear codes
Abstract: Hyperplane arrangements are closely related to the column compositions of generator matrices of linear codes. In this talk, we explore the advantages of considering column compositions, or column multiplicities, of generator matrices of linear codes over finite chain rings. We focus on the connections between column multiplicities and various weight functions in coding theory, as well as on the construction of optimal and few-weight codes using the column multiplicity approach.
Keiji Tatsumi
Title: Approximate Multi-Objective Multi-Class Support Vector Machines Based on Geometric Margins
Abstract: Despite recent advances in artificial intelligence, particularly in deep learning, support vector machines (SVMs) remain useful classification methods due to their relative simplicity and interpretability, which have become increasingly important in the context of explainable AI (XAI). In this talk, we review a series of studies on approximate multi-objective multi-class support vector machines (AMMSVMs), in which the geometric margins between different classes are treated as multiple objectives to be maximized simultaneously. To obtain computationally tractable formulations, the resulting multi-objective optimization problems are approximately reformulated as convex optimization problems that can be solved using second-order cone programming (SOCP). We first introduce the basic framework of AMMSVMs and then discuss its extensions, including a soft-margin formulation that allows violations of classification boundaries while considering the trade-offs between such violations and geometric margins. Finally, we review the main results of these studies and discuss the effectiveness and limitations of the proposed approaches.
Mehdi Tibouchi
Title: Indifferentiable hashing to elliptic curves
Abstract: As an illustration of the interplay between various mathematical techniques and cryptographic research, I would like to talk about the problem of hashing arbitrary values to points on an elliptic curve. This is a required step in many cryptographic constructions, and a number of algorithms have been proposed to do so over the years. Some of them are now even documented in an IETF RFC. I plan to give an overview about the problem and its history, and then discuss a number of interesting recent results, how they have improved our understanding, and what pieces of the puzzle we are still missing.
Shuhei Tsujie
Title: MAT-partitions of braid arrangements, regular vines, and Arrow's single-peaked domains
Abstract: MAT-partitions of braid arrangements arise in the theory of free hyperplane arrangements. Regular vines originate in probability theory, and Arrow's single-peaked domains arise in social choice theory. These three concepts were developed independently. Nevertheless, they share the same underlying mathematical structure. In this talk, we describe this common structure in the language of combinatorial species. This talk is based on joint work with Hung Manh Tran and Tan Nhat Tran.
Max Wakefield
Title: Applications of Chain Characteristic Polynomials
Abstract:The classic characteristic polynomials are ubiquitous in matroid and arrangement theory. They arose as chromatic polynomials of graphs and, via duality, as flow polynomials. Recently, studies in valuative matroid invariants has led to the development of finer universal polynomial invariants like Derken's G-invariant and the chain Tutte polynomials. Using these invariants we can define a generalization of characteristic polynomials which we call chain characteristic polynomials. Then we can study applications of these polynomials. We define a generalized proper vertex coloring on a graph, which we call coupled multicoloring. We also define a generalized nowhere zero flow, which we call coupled multicommodity flow. Then we show that these chain characteristic polynomials enumerate the number of coupled multicolorings and coupled multicommodity flows on a graph. We conclude by listing multiple problems on enumerative properties of chain characteristic polynomials.