Mohamed Barakat
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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
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Junyan Chu
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Emanuele Delucchi
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Kenji Fukumizu
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Alice L.L.Gao
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Koji Imamura
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Satoru Iwata
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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
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Hal Schenck
Title: Lefschetz properties for Graphic Artinian Orlik-Terao algebras
Abstract: In 1994, Orlik and Terao introduced a commutative Artinian analog of the Orlik-Solomon algebra of a hyperplane arrangement to answer a question of Aomoto. A central topic of investigation in the study of commutative Artinian algebras is the Weak Lefschetz Property (WLP). We analyze WLP for the Artinian Orlik-Terao algebra of graphic 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, and apply this in the graphic case. Joint work with Nicholas 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
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Mehdi Tibouchi
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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
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