Hyperplane Arrangements and related topics

13 (Mon) -- 14 (Tue) February, 2017. Hokkaido University, Sapporo, Japan.

We are organizing a workshop on the occasion of Professor Hiroaki Terao's final lecture (14th Feb. 2017).

Invited Speakers:

13 (Mon) Feb. 2017 (Rm 4-501, Dept of Math, Hokkaido University)
9:30-10:30, Yoshinaga,
10:50-11:50, Settepanella,
13:30-14:30, Suyama,
14:45-15:45, Nakashima,
16:00-17:00, Wakefield,

Morning 14 (Tue) Feb. 2017 (Rm 4-501, Dept of Math, Hokkaido University)
9:30-10:30, Feichtner,
10:40-11:40, Abe,

Afternoon 14 (Tue) Feb. 2017 (Conference Hall, Hokkaido University)
Final Lectures by Prof. Yamaguchi and Prof. Tsuda and Prof. Terao
13:30-14:30, Prof. Keizo Yamaguchi.
15:00-16:00, Prof. Ichiro Tsuda.
16:30-17:30, Prof. Hiroaki Terao.
18:30-, Dinner (Need registration before 12 Jan. 2017. Please contact organizers for details.)


13 (Mon) Feb.
9:30-10:30 Yoshinaga
Title: Around the h-shift problem.
Abstract: The celebrated Terao's factorization theorem says that the freeness (= "algebraic property") of an arrangement implies factorization of the characteristic polynomial (= "combinatorial constraints"). Although the converse is not true, factorization of the characteristic polynomial is deduced from the freeness in many important cases.
It is natural to expect that other combinatorial phenomena of characteristic polynomials are also deduced from algebraic structures. In a survey paper "Freeness of hyperplane arrangements and related topics" (2014) I posed some problems in this direction. However, one of the problem has been disproved by Abe-Faenzi-Valles. We discuss possible modifications.

10:50-11:50 Settepanella
Abstract: Abstract: Compute the first homology group H1(F,C) of the Milnor fiber F of a line arrangement is a very well known and studied problem. Based on the minimal complex C*(S(A)) introdueced by Salvetti and Settepanella, we describe an algorithm which computes possible eigenvalues of the monodromy operator h1 of H1(F,C) in case of sharp line arrangement. We prove that, if a condition on some intersection points of lines in A is satisfied, then the only possible non trivial eigenvalues of h1 are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.

13:30-14:30 Suyama
Title: Freeness of certain subarrangements of Coxeter arrangements of type B.
Abstract: I will talk about partial results for characterization of freeness of subarrangements of Coxeter arrangements of the type B and a relationship between them and signed graphs. This is a joint work with M. Torielli and S. Tsujie.

14:45-15:45 Nakashima
Title: A module of high order differential operators on a hyperplane arrangement.
Abstract: A differential operator of order m on an arrangement is a polynomial coefficient linear combination of mth order partial derivatives by which the principal ideal generated by the defining polynomial of the arrangement is preserved. An arrangement is said to be m-free if the module of differential operator of order m on the arrangement is free over the polynomial ring. Holm asked two questions about m-freeness as follows: (1) Does m-free imply (m+1)-free for any arrangement? (2) Are all arrangements m-free for m large enough? In this talk, I present answers to these questions. This work is with T. Abe.

16:00-17:00 Wakefield
Title: Free multiplicities on the moduli of X_3
Abstract: The matroid of X_3 is the unique relaxation of the rank 3 braid matroid and is called the rank 3 whirl. It's characteristic polynomial does not factor hence all of it's moduli (a one dimensional family) are not free. However all the moduli support free multiplicities. In this talk we will discuss how to classify the free multiplicities using ideas originally from the work of Brandt and Terao.

14 (Tue) Feb.
9:30-10:30, Feichtner
Title: Leray models for Orlik-Solomon algebras
Abstract: Although hyperplane arrangement complements are rationally formal, we note that they have non-minimal rational (CDGA) models which are topologically and combinatorially significant, in view of recent work of Bibby and Dupont, as well as foundational results of De Concini and Procesi. We construct a family of CDGAs which interpolates between the Orlik-Solomon algebra and the cohomology algebras of arrangement compactifications.
Our construction is combinatorial and extends to all matroids, regardless of their (complex) realizability. The approach makes use of the notion of combinatorial blowups [F & Kozlov] as well as ideas in previous work with Yuzvinsky.
This is joint work with Christin Bibby and Graham Denham.

10:40-11:40, Abe
Title: Algebra and geometry of Solomon-Terao's formula
Abstract: Solomon-Terao's formula is one of the most important results in the theory of logarithmic derivation modules of hyperplane arrangements. To prove this formula, Solomon and Terao used so called the \eta-complex and its homology group. We study them from new points of view, define a new algebra from them, and give a geometric application of that algebra to Hessenberg varieties.
Takuro Abe(abe (at) imi.kyushu-u.ac.jp)
Toru Ohmoto(ohmoto (at) math.sci.hokudai.ac.jp)
Simona Settepanella(s.settepanella (at) math.sci.hokudai.ac.jp)
Michele Torielli(torielli (at) math.sci.hokudai.ac.jp)
Masahiko Yoshinaga(yoshinaga (at) math.sci.hokudai.ac.jp)