This school consists of lecturers (22-25 July) and workshop (25-26 July). The school is a part of Hokkaido Summer Institute 2019

Lecturers: Graham Denham (Western, Canada), Luca Moci (Bologna, Italy), Masahiko Yoshinaga (Sapporo)

Workshop Speakers: Ahmed Umer Ashraf (Western Ontario/Hokkaido), Shinzo Bannai (Ibaraki) , Takahiro Hasebe (Hokkaido), Anatol Kirillov (Kyoto RIMS), Ye Liu (Suzhou), Tsuyoshi Miezaki (Ryukyus), Takahiro Nagaoka (Kyoto), Yasuhito Nakajima (Tokyo), Delphine Pol (Kaiserslautern), Michele Torielli, Tan Nhat Tran (Hokkaido), Akiyoshi Tsuchiya, Akiko Yazawa (Shinshu)

Titles and Abstracts for Workshop (PDF) (updated 24 July 2019 22:30)

Room: 4-501, Department of Mathematics, Hokkaido University.

Schedules:

22 (Mon)

• 9:00-12:00, (Yoshinaga 1, 2) Introduction to matroids

23 (Tue)

• 9:00-12:00, (Yoshinaga 3, 4) Introduction to matroids

24 (Wed)

• 9:00-10:30 (Denham 1), Arrangement Compactifications
• 10:30-12:00 (Moci 1), Toric arrangements and beyond

25 (Thu)

• 9:00-10:30 (Denham 2), Chow rings of Matroids
• 10:30-12:00 (Moci 2), Arithmetic Tutte and G-Tutte polynomials
• 13:30-14:20 Kirillov: Ninth variation to study generalized cohomology of parabolic Flag and regular nilpotent Hessenberg varieties.
• 14:30-15:20 Miezaki: A generalization of the Tutte polynomials
• 15:35-15:50 Nakajima, Minimal tropical basis for Bergman fan of matroid .
• 15:50-16:05 Yazawa, The eigenvalues of the Hessian matrices of the generating functions for some truncated graphic matroids.
• 16:05-16:20 Torielli, Combinatorially equivalent hyperplane arrangements.

26 (Fri)

• 10:00-10:50 Hasebe, Non-commutative probability theory, set partitions and Tutte polynomials.
• 11:00-11:50 Nagaoka, The strong Lefschetz property and Hodge--Riemann relation for Maeno--Numata algbera at degree one.
• 13:30-13:45 Bannai, The matroid structure of vectors of the Mordell-Weil lattice and the embedded topology of certain plane curve arrangements.
• 13:45-14:00 Ashraf, Cyclic flat approach to matroid polytopes volume
• 14:00-14:15 Tsuchiya, Ehrhart theory and interior polynomials.
• 14:15-14:30 Tran, Positivity of the coefficients of $G$-Tutte polynomials.
• 14:50-15:40 Pol, Representable matroids and configuration polynomials
• 15:50-16:40 Liu, Topology of the icosidodecahedral arrangement.
• 17:00-18:00 Discussion

Title and Abstracts (Lectures):

Denham,

I: Arrangement Compactifications,

Abstracg: We continue some of the ideas from Lecture 3 and see how the combinatorics of matroids are reflected in invariants of constructions involving hyperplane arrangements. We consider examples such as the Solomon-Terao algebra, the De Concini-Procesi compactification, and the critical set variety. We introduce the Bergman fan as a basic object.

II: Chow rings of Matroids,

We further consider the combinatorics of the Bergman fan, which leads to the remarkable positivity properties of the Chow ring of a matroid. This leads to a new construction, the conormal fan of a matroid. We consider its applications.

Moci,

(1) Toric arrangements and beyond.

We will introduce the basic notions on toric arrangements, and show (by looking at some examples) how their combinatorics and topology differ from those of hyperplane arrangements. This will motivate the combinatorial constructions presented in the second lecture. If time allows we will also discuss elliptic and abelian arrangements.

(2) Arithmetic Tutte and G-Tutte polynomials.

We will briefly introduce arithmetic matroids and discuss some recent works of Pagaria and his coauthors, which shed light on the relation of arithmetic matroids with the topology of toric and elliptic arrangements. Then we will define arithmetic Tutte polynomials and G-Tutte polynomials: the latter were recently introduced by Liu, Tran and Yoshinaga.

Yoshinaga, (1) Basics (Definitions of matroids and Tutte polynomial). (2) Hyperplane arrangements and combinatorics, (3) Hyperplane arrangements and topology, (4) Applications of Tutte polynomial to enumerative problems.

Notes:

• Unfortunately, budgets is strictly limited and we can not provide financial support.
• We will have slots for "short communications" during the workshop. If you are interested in giving a short talk related to the theme of the workshop, please contact Masahiko Yoshinaga (Email: yoshinaga(at)math.sci.hokudai.ac.jp). Deadline: 20 June.