A walk between hyperplane arrangements, computer algebra and algorithms

WEDNESDAY 31

11:00-11:15 Registration

11:15-12:15 Numata

12:15-14:15 Lunch break

14:15-15:15 Tokunaga

15:15-15:45 Coffee break

15:45-16:45 Ida

17:00-18:00 Pol

Social dinner

THURSDAY 1

9:30-10:30 Yoshinaga

10:30-11:00 Coffee break

11:00-12:00 Bigatti

12:00-13:00 Palezzato

13:00-14:30 Lunch break

Free afternoon

FRIDAY 2

10:00-11:00 Abe

11:00-11:30 Coffee break

11:30-12:30 Minato

12:30-14:00 Lunch break

14:00-15:00 Tsujie

15:00-15:30 Coffee break

15:30-16:30 Free discussion

Titles and abstracts:

- Takuro Abe (Kyushu University)

Title: Poincare polynomials and free arrangements

Abstract: The most useful method to construct freeness is Terao's addition-deletion theorem, and as its improvement, the division theorem enables us to check freeness based on the division of Poincare polynomials, or equivalently, some equation between second Betti numbers. We show how the first and second Betti numbers play the key role in the freeness framework of Terao's addition-deletion, proving that the deletion of free arrangement is combinatorial, and some related topics.

- Anna Maria Bigatti (University of Genova)

Title:CoCoA: Computations in Commutative Algebra.

Abstract: CoCoA-5 is an interactive Computer Algebra System for Computations in Commutative Algebra, particularly Groebner bases. It offers a dedicated, mathematicians-friendly programming language, with many built-in functions. Its mathematical core in CoCoALib, an open-source C++ library, designed to facilitate integration with other software. We give an overview of the latest developments of the library and of the system, in particular relating to the upcoming function package on hyperplane arrangements. (http://cocoa.dima.unige.it)

- Tetsuo Ida (Tsukuba University)

Title:Models of Origami for Automated Construction, Verification and Visualization

Abstract: Origami as an art of paper folding. It can also be view as the machinery of geometric construction in a similar sense to Turing machine, and the lambda calculus are computing machinery. Origami transforms geometrical objects by folds. As in computing, we have a variety of approaches to look into the process of the transformations of origami. We give the overview of the approaches that we have taken, having in mind that our goal is automating the construction, proving and visualization by a computer. Using illustrative origami examples, we discuss the logic of Huzita-Justin folds, the programming language for origami construction and proving, the method of origami theorem proving, and the method of visualization. Behind them, we developed abstract structures that are amenable to computing. We call the abstract structures collectively "models" of origami. We further explain those models.

- Shin-ichi Minato (Hokkaido University)

Title: BDD/ZDD-Based Algorithms and Algebras for Discrete Structure Manipulation

Abstract: BDD (Binary Decision Diagram) is a classical date structure foe representing a Boolean function. BDD-based algorithms were developed mainly for VLSI logic design in early 1990s. ZDD (Zero-suppressed BDD) is a variant of BDD, customized for representing a set of combinatorics, often appear in solving combinatorial problems. BDDs and ZDDs have become more widely known since D. Knuth intensively discussed them in his famous book series in 2009. Although a quarter of a century has passed since the original idea of BDD-based operations by R. Bryant, there are still many interesting research topics ongoing. In this talk, I will give an overview of the basic algorithms and algebras for BDDs/ZDDs. We will also look over some recent research topics related to real-life applications.

- Yasuhide Numata (Shinshu University)

Title: Spanning trees in a lattice and automata

Abstract: Let $n$ be a positive integer. We consider the set of spanning trees in $A_n\times A_m$, for some $m$. In this talk, we give a regular language whose words correspond to spanning trees.

- Elisa Palezzato (Hokkaido University)

Title: New characterizations of freeness for hyperplane arrangements.

Abstract: I will present two new characterizations of freeness for hyperplane arrangements. The first one, through the computation of the generic initial ideal of the Jacobean ideal of an arrangement. The second characterization, via the sectional matrix of the Jacobean ideal. This is part of my joint work with A.M. Bigatti and M. Torielli.

- Delphine Pol (Hokkaido University)

Title: Logarithmic forms along equidimensional subspaces

Abstract: The purpose of this talk is to discuss about logarithmic forms along equidimensional subspaces, which appear as a generalization of the logarithmic forms along hyper surfaces introduced by K. Saito. We will first consider the case of subspaces which are complete intersections. We will also give the definitions in the case of equidimensional reduced subspaces, which is closely related to Kersken's characterization of regular meromorphic forms. We will then study the case of 1-dimensional spaces, which includes in particular line arrangements.

- Hiroo Tokunaga (Tokyo Metropolitan University)

Title: Topology of arrangement of plane curves of low degree and "arithmetic" of double cover of $\mathbb{P}^2$

Abstract: Let $B$ be a reduced plane curve with irreducible components of low degree. $\sum_{i\in l}B_i$ denote its irreducible decomposition. Put $l=l_1\cup l_2$ be a partition of the index set $l$. We construct a double cover of $\mathbb{P}^2$ branched along $B_{l_1}:=\sum_{i\in l_1}B_i$ and denote its minimal resolution by $S_{B_{l_1}}$ and the covering morphism by ${\tilde f}_{B_{l_1}}:S_{B_{l_1}}\to{\mathbb P}^2$. We first study combinatorics or geometry of ${\tilde f}_{B_{l_1}}^{-1}{\sum_{i\in l_2}B_i}$ and apply results to consider topology of $({\mathbb P}^2,B)$.

- Shuhei Tsujie (Hokkaido University)

Title: Freeness of two families of hyperplane arrangements determined by gain graphs

Abstract: A gain graph is a graph whose edges are labeled by elements of a group. When the group is equal to $\mathbb{F}_2$ or $\mathbb{Z}$, we can construct two types of hyperplane arrangements from the gain graph. We will talk about correspondence about freeness of these arrangements. Moreover, we will characterize freeness of these arrangements when the group equals $\mathbb{F}_2$. This study is based on a joint work with Daisuke Suyama and Michele Torielli.

- Masahiko Yoshinaga (Hokkaido University)

Title: How many homomorphisms are expected? A motivation for G-Tutte polynomials.

Abstract: We discuss the expectation of the number of homomorphisms from a certain random finitely generated abelian groups to a fixed finite abelian group G, which is expressed by using G-Tutte polynomial introduced by Ye Liu, Tan Nhat Tran and the speaker last year.

Organizers: Toru Ohmoto (Hokkaido University), Michele Torielli (Hokkaido University), Masahiko Yoshinaga (Hokkaido University).

Contact: Michele Torielli (torielli [at] math.sci.hokudai.ac.jp)

This program is supported by: the "Budget Support for the Leader Developing Program".