# The 1st Workshop of JSPS-MAE Sakura Program "Geometry and Combinatorics of Hyperplane Arrangements and Related Problems"

1 (Mon) -- 5 (Fri) September, 2014.
Department of Mathematics, Hokkaido University (Room 4-501).

This is the 1st workshop of France-Japan Joint Project "Geometry and Combinatorics of Hyperplane Arrangements and Related Problems" (Sakura Program) spported by JSPS-MAE.

Invited Speakers:

Schedule (A talk added in Thu (Last update: 24th August))
1 (Mon) Sep.
10:45-11:45 Hiro-o Tokunaga (Tokyo Metropolitan), Zariski N-plets for arrangement of curves of low degrees and rational elliptic sufaces
14:00-15:00 Kyoji Saito (IPMU), Towards primitive forms of type $A_{\frac{1}{2}\infty}$ and $D_{\frac{1}{2}\infty}$
15:15-16:15 Akishi Ikeda (Tokyo), Space of stability conditions for the preprojective algebra and the regular orbit of the Weyl group
16:30-17:30 Daisuke Suyama (Hokkaido University), The freeness of the Ish arrangement

2 (Tue) Sep.
9:30-10:30 Toshiyuki Akita (Hokkaido), Vanishing theorems for p-local homology groups of Coxeter groups and their alternating subgroups
10:45-11:45 Benoit Guerville (Pau), Vector fields and invariant line arrangements
14:00-15:00 Jean Valles (Pau), Jumping lines of logarithmic bundles
15:15-16:15 Takuro Abe (Kyoto), Stability of line arrangements in the projective plane

3 (Wed) Sep.
9:30-10:30 Mutsumi Saito (Hokkaido), Limits of Jordan Lie subalgebras
10:45-11:45 Hiroaki Terao (Hokkaido), Multiple addition theorem on arrangements of hyperplanes and a proof of the Shapiro-Steinberg-Kostant-Macdonlald dual-partition formula
13:00- , free discussion.

4 (Thu) Sep.
9:30-10:30 Tadashi Ishibe (Tokyo), A generalization of the theory of Artin groups
10:45-11:45 Shuhei Tsujie (Hokkaido), Canonical systems of basic invariants for finite reflection groups
14:00-15:00 Juan Viu Sos (Pau), On periods of Kontsevich-Zagier
15:15-16:15 Michele Torielli (Hokkaido University), Resonant bands, Aomoto complex and real 4-nets.
16:30-17:30 Nguyen Viet Dung (Hanoi), On the higher topological complexity of hyperplane arrangements

5 (Fri) Sep.
9:30-10:30 Jacky Cresson (Pau), Transcendence of periods and complexity
10:45-11:45 Masahiko Yoshinaga (Hokkaido), Zero recognitions for periods and holonomic real numbers.
12:00-15:00, free discussion.

Abstracts:
1 (Mon) Sep.
10:45-11:45 Hiro-o Tokunaga (Tokyo Metropolitan)
Title: Zariski N-plets for arrangement of curves of low degrees and rational elliptic sufaces
Abstract: We consider geometry and (elementary) arithmetic for sections and bisections on rational elliptic surfaces. We apply our result in constructing Zariski N-plet for arrangement of curves of low degrees via dihedral covers.

14:00-15:00 Kyoji Saito (IPMU)
Title: Towards primitive forms of type $A_{\frac{1}{2}\infty}$ and $D_{\frac{1}{2}\infty}$
Abstract: We intend to study primitive forms for the transcendental functions of type $A_{\frac{1}{2}\infty}$ and $D_{\frac{1}{2}\infty}$. Since they have infinite dimensional lattices of vanishing cycles and, hence, have infinite dimensional deformation parameter spaces, we don't know yet what is a reasonable formulation towards them. On the other hand, classsical thoery of primitive forms asserts that the data of a primitive form are equivalent to the data of a good section, where the good section is defined on the relative de Rham cohomology group of un-deformed setting, which seems still accesible by classical tools. We ivestigate this direction, which seem to lead again a study of new type transcendental functions.

15:15-16:15 Akishi Ikeda (Tokyo)
Title: Space of stability conditions for the preprojective algebra and the regular orbit of the Weyl group
Abstract: In this talk, we show that the space of Bridgeland stability conditions for the preprojective algebra is described as the covering space of the regular orbit space of the Weyl group associated to the Kac -Moody Lie algebra. Since the Artin group is given by the fundamental group of the regular orbit space, the Artin group acts on the space of stability conditions as deck transformations. We construct this action categorically as Seidel-Thomas spherical twists.

16:30-17:30 Daisuke Suyama (Hokkaido University)
Title: The freeness of the Ish arrangement
Abstract: The Ish arrangement is an affine arrangement obtained by adding several parallel translations of hyperplanes in the Weyl arrangement of the type A to the Weyl arrangement. This arrangement was introduced by D. Armstrong to study diagonal harmonics. I will talk about the freeness of the Ish arrangement. This is a joint work with Shuhei Tsujie.

2 (Tue) Sep.
9:30-10:30 Toshiyuki Akita (Hokkaido),
Title: Vanishing theorems for p-local homology groups of Coxeter groups and their alternating subgroups
Abstract: Given an odd prime $p$, we estimate vanishing ranges of $p$-local homology of Coxeter groups and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka and Burichenko for symmetric groups and alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

10:45-11:45 Benoit Guerville (Pau),
Title: Vector fields and invariant line arrangements
Abstract: In collaboration with J. Cresson and J. Viu Sos Abstract: Hilbert's sixteenth problem is concerned with counting the maximal number of limit cycles for polynomial vector field. An easier problem is to replace limit cycles by algebraic limit cycles of algebraic curves. As a first step in this direction, we study the maximal number of lines for a given polynomial vector field. Our approach is then to focus on the relation between real line arrangements and polynomial vector fields. Thus, we try to understand the influence of the combinatorial structure of an arrangement on the minimal degree of the logarithmic derivatives of the arrangement (fixing only a finite number of lines). From here we investigate the bounds of the maximal number of lines invariant by a polynomial vector field of fixed degree.

14:00-15:00 Jean Valles (Pau),
Title: Jumping lines of logarithmic bundles
Abstract: PDF File

15:15-16:15 Takuro Abe (Kyoto),
Title: Stability of line arrangements in the projective plane
Abstract: There have been a lot of studies on freeness of line arrangements in the projective plane. Not only freeness, but also the (semi)stability is very important topic in vector bundle theory. However, there have been not so many studies on stability.
In this talk, we give a simple sufficient condition for the freeness and construct many stable line arrangements.

3 (Wed) Sep.
9:30-10:30 Mutsumi Saito (Hokkaido),
Title: Limits of Jordan Lie subalgebras
Abstract: Let g be a simple Lie algebra of rank n over C. We show that abelian ideals of dimension n of a Borel subalgebra of g are limits of Jordan Lie subalgebras. Combining this with a classical result by Kostant, we show that the g-module spanned by all abelian n-dimensional Lie subalgebras of g is actually spanned by the Jordan Lie subalgebras.

10:45-11:45 Hiroaki Terao (Hokkaido),
Title: Multiple addition theorem on arrangements of hyperplanes and a proof of the Shapiro-Steinberg-Kostant-Macdonlald dual-partition formula
Abstract: The addition-deletion theorem (1980) gave a criterion for a new hyperplane to be added to a given free arrangement without destroying the freeness. In this talk we will present a new theorem called the multiple addition theorem (MAT) which gives a sufficient condition for two or more hyperplanes to be added to a given free arrangement without destroying the freeness. The sufficient condition is related to the maximum exponent of the given free arrangement. We briefly outline our proof of MAT. As an application of MAT, we may prove that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko in 2006. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald.
(This is a joint work with Takuro Abe, Mohamed Barakat, Michael Cuntz and Torsten Hoge. The paper (arXiv:1304.8033) will appear in J. Eur. Math. Soc. )

4 (Thu) Sep.
Title: A generalization of the theory of Artin groups.
Abstract: For a finite reflection group $W$, we consider the fundamental group of regular orbit space of the finite reflection group $W$. The fundamental group admits a special presentation whose defining relations correspond to the finite Coxeter diagram of type $W$. The group (resp. monoid) defined by that presentation is called an Artin group (resp. Artin monoid). By showing a certain lemma for Artin monoid, we conclude that Artin monoid is a cancellative monoid and the LCM condition (i.e. for any elements $\alpha$, $\beta$, there exist left and right least common multiples of them) is satisfied. As a result, some decision problems in Artin groups can be solved. We may say that the above lemma is a key to success in the theory of Artin groups. In the talk, emphasizing the role of the lemma, we will consider a generalization of the theory of Artin groups.

10:45-11:45 Shuhei Tsujie (Hokkaido),
Title: Canonical systems of basic invariants for finite reflection groups
Abstract: A canonical system of basic invariants for a finite reflection group is a system of basic invariants which satisfies the orthogonality condition with respect to the pairing of differentiation. This notion was first introduced by Flatto and Wiener for finite Euclidean reflection groups to study the mean value property for polytopes. In this talk I will present the existence and the construction of canonical systems for unitary reflection groups.
These studies are joint-works with Norihiro Nakashima and Hiroaki Terao.

14:00-15:00 Juan Viu Sos (Pau),
Title: On periods of Kontsevich-Zagier.
Abstract: Introduced by M. Kontsevich and D. Zagier in their paper in 2001, periods are a class of numbers which contains most of the important constants in mathematics, as well as they are strongly related with transcendence theory in number theory, modern arithmetic and algebraic theory and also in differential equations theory.
In this talk, we present a semi-canonical geometrical representation form for periods in order to develop a graduation theory, with application to the study of transcendence and the two most important open problems of periods: the Kontsevich-Zagier conjecture and the Equality algorithm.

15:15-16:15 Michele Torielli (Hokkaido University),
Title: Resonant bands, Aomoto complex and real 4-nets.
Abstract: In this talk I will recall the notion of a k-net for a line arrangement, describe the connection between this notion, the cohomology of Aomoto complex and the resonant band algorithm. Finally, I will prove that real arrangements that support a 4-net structure do not exist. This is a joint work with M. Yoshinaga.

16:30-17:30 Nguyen Viet Dung (Hanoi),
Title: On the higher topological complexity of hyperplane arrangements
Abstract: (Joint work with Nguyen Van Ninh) The higher topological complexity is a generalization of the notion topological complexity, suggested by M. Faber, in relation to the problem of motion planning algorithm. In this talk we will compute the higher topological complexity for some spaces, including the complement of a generic arrangement. We then will discuss the combinatorial-determined property of the (higher) topological complexity.

5 (Fri) Sep.
9:30-10:30 Jacky Cresson (Pau),
Title: Transcendence of periods and complexity
Abstract: (joint work with J. Viu-Sos) Using the semi-canonical representation of periods obtained by J. Vius-Sos, we know that periods can be defined as volumes of compact rational semi-algebraic set. The complexity of a given period is then encoded by the minimal dimension the compact semi-algebraic set used in Viu-Sos representation (Wann's notion of degree for periods) and the classical complexity measure for semi-algebraic sets counting the number and the maximal degree of the defining equations. The degree distinguish between algebraic and transcendent periods and gives a natural classification of transcendent periods. We gives using a classical result of Van der Poorten a characterization of periods of degree 2 associated to rational functions. We also discuss several open problems.

10:45-11:45 Masahiko Yoshinaga (Hokkaido),
Title: Zero recognitions for periods and holonomic real numbers.
Abstract: We will introduce a class of real numbers so called "holonomic real numbers", which is conjecturally wider than periods of Kontsevich-Zagier. I will discuss zero recognition problems for periods and holonomic real numbers.

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Contact
Masahiko Yoshinaga(yoshinaga (at) math.sci.hokudai.ac.jp)