Program: 

21st November

• 12:30–12:50: Yosuke Sato (Waseda University)

Graphical design and equitable partition 

A graphical design is a graph-theoretic analogue of a spherical design. In this talk, I will discuss how "extremal" graphical designs can be characterized by means of equitable partitions of graphs.

• 12:50–13:10: Yuki Terada (Waseda University)

Graphical designs of Paley graphs 

We give an algebraic proof that Paley graphs do not admit any nontrivial graphical designs.

• 13:10–13:30: Ryosuke Yamaguchi (Waseda University)

Approximate counting of graph colorings 

In this talk, I will introduce the use of Bubley and Dyer’s path coupling technique for approximate counting of graph k-colorings, and discuss how the method may be extended to the more general setting of H-colorings.

• 13:30–14:00: Jurre van Schie (Waseda University)

A Linear-Time Algorithm for Computing the Minimum Number of Trees to Cover the Edges of Phylogenetic Networks 

Phylogenetic networks capture complex evolutionary histories beyond what trees can represent. Simpson asked for the minimum number of embedded subtrees needed to cover all edges of a binary phylogenetic network, studying limited network classes. We address this problem for any directed phylogenetic network, binary or non-binary. We provide two explicit theorems showing that the minimum number k of subtrees required is determined by the maximum out-degree among tree vertices or maximum in-degree among reticulations. Combining these characterizations yields the first linear-time O(∣V∣+∣E∣) algorithm for the MINIMUM TREE-COVER PROBLEM.

• 14:00–14:30: Kevin Limanta (The University of New South Wales) 

Super Catalan Numbers and Polynomial Summation over Unit Circles over Finite Fields 

In this talk I will discuss polynomial summation over unit circles over finite fields of odd characteristic. There are nonetheless strong parallels to classical integration theory over a circle, and we show that the super Catalan numbers and closely related rational numbers lie at the heart of both theories, giving a uniform analytic meaning. In the end, I will also give some explorative implication of this result to finite field spherical design.

Program: 

12th September 

• 9:30–10:10: Ryo Uchiumi (The University of Osaka)

The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups 

In this talk, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod q hyperplane arrangements. We will show that its character is a quasi-polynomial in q and can be expressed as a sum of the induced characters of an equivariant version of the Ehrhart quasi-polynomials. If time permits, we will discuss the results for the Coxeter arranegements with Weyl group actions.

• 10:20–11:00: Masato Konoike (The University of Osaka)

Characteristic quasi-polynomials of restrictions of Shi arrangements of type D 

Characteristic quasi-polynomials enumerate the number of points in the complement of hyperplane arrangements modulo positive integers. In 2024, Higashitani and Nakashima computed the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type B by one given hyperplane. In this talk, we extend their work to type D: we compute the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type D by one given hyperplane. Furthermore, we determine whether period collapse occurs in the characteristic quasi-polynomials of the deletions of the Shi arrangements of type D.

• 11:20–12:00: Takuro Abe (Rikkyo University)

Double poins of projective line arrangements and algebra 

The famous Sylvester-Gallai theorem asserts that every non-pencil real projevtive line arrangements have at least one double points, but this is no more true when the base field is the complex number field. Anzis and Tohaneanu conjectured that several complex version of this for a supersolvable arrangements, which was solved by the speaker. We give an overview of them and introduce recent developments around this double point problem.

• 14:00–14:40: Sarbaini (Kanazawa University)

Centralizer Algebras of Two Permutation Groups of Order 1344

There are two permutation groups that they share the same character table of order $1344$. We take up natural representations on 8 and 14 letters respectively. The purpose of this paper is to examine the semi-simple structure of centralizing algebras in the tensor representation.

• 14:50–15:30: A.K.M. Selim Reza (Kanazawa University)

Computation of Vector-Valued Invariants for a Finite Complex Reflection Group 

We consider the complex reflection group \( H_1 \), identified as No. 8 in the Shephard-Todd classification. In this paper, we present computations of the vector-valued invariants associated with various representations of \( H_1 \). Additionally, we investigate the structure of the corresponding invariant rings.

• 15:50–16:30: Tongyu Nian (The University of Osaka)

q-deformation of graphic arrangements 

We first observed a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field Fq . These two arrangements are defined by the determinants of the Vandermonde and the Moore matrix, respectively. These two matrices are transformed to each other by replacing a natural number n with q^n (q-deformation). In the talk, I will introduce the notion of “q-deformation of graphical arrangements”  and extend it to simplicial complexes, followed by a main theorem concerning the q-analog. This new class of arrangements enjoy the similarities to graphical arrangements. We show that many invariants of the “q-deformation” behave as the ones of the graphical arrangements. This is a joint work with Shuhei Tsujie, Ryo Uchiumi and Masahiko Yoshinaga.

• 16:40–17:20: Himadri Shekhar Chakraborty（Shahjalal University of Science and Technology） 

Higher weight generalization of Jacobi polynomials for codes 

In this talk, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between the resulting Jacobi polynomials. Moreover, we present the Jacobi analogue of MacWilliams identity for both higher and extended weight enumerators. We also present that the higher Jacobi polynomials for linear codes whose subcode supports form $t$-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique. Finally, we demonstrate how higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials. This is a joint work with Tsuyoshi Miezaki.