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).
13 (Mon) Feb. 2017
(Rm 4-501, Dept of Math, Hokkaido University)
Morning 14 (Tue) Feb. 2017
(Rm 4-501, Dept of Math, Hokkaido University)
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.
Title: Around the h-shift problem.
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
Title: HOMOLOGY GRAPH OF REAL ARRANGEMENTS AND MONODROMY OF MILNOR FIBER
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.
Title: Freeness of certain subarrangements of Coxeter arrangements of type B.
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.
Title: A module of high order differential operators on a hyperplane arrangement.
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.
Title: Free multiplicities on the moduli of X_3
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.
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
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.
Title: Algebra and geometry of Solomon-Terao's formula
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)