Problems Around Hyperplane Arrangements

18 (Thu) June, 2015. Hokkaido University
10:00--15:00 (?? Depending on the number of talks)
Place: 3-205 Dapartment of Mathematics

This one-day workshop aims to share recent developments and problems around hyperplane arrangements. Everyone is welcome to present problems.

JSPS-MAE Sakura Program "Geometry and combinatorics of hyperplane arrangements and related topics".

Rules:

Each talk is either 10 minutes or 20 minutes.
The talk by experienced researcher should contain at least one open problem.
Students are encouraged to give a talk what he/she is studying or is interested in.
Welcome everyone.

Tentative Schedule: Thursday 18 June 2015
10:00-10:20, Wakefield 1
10:20-10:40, Wakefield 2
10:40-11:00, Wakefield 3
11:00-11:20, break
11:20-11:40, Terao
11:40-12:00, Hasebe
12:00-13:20, lunch
13:20-13:40, Faenzi
13:40-14:00, Bailet
14:00-14:20, Settepanella (1)
14:20-14:40, break
14:40-15:00, Settepanella (2)?
15:00-15:20, Yoshinaga
15:00-?? , some others?

Tentative speakers: Max Wakefield (2 or 3 talks), Daniele Faenzi, Simona Settepanella, Pauline Bailet, Hiroaki Terao, Takahiro Hasebe, Masahiko Yoshinaga and others.

Tentative titles and abstracts:

Max Wakefield (Naval Academy):
(1) Ghosts of Jacobian ideals
(Abstract) We define a ghost of an ideal as an associated prime of high codimension in a certain Ext module. The Ext modules of the Jacobian ideal give a kind of stratification of the non-freeness of an arrangement. There is a very nice classification of non-ghost associated primes of Ext modules as exactly the non-free flats of the arrangement. The main focus in this discussion is how to detect the ghosts in an arrangement. Only a few examples are known. We can construct a family of graphic arrangements using a kind of duality where we can compute that the first few members have ghosts. A good starting place would be the prove that this family always has ghosts. Then develop this duality in general and relate the the freeness of the arrangements which are dual to each other.

(2) Kazhdan-Lusztig polynomials for braid arrangements
(Abstract) The Kahzdan-Lusztig polynomials for arrangements exhibit many deep connections between combinatorics and geometry. These polynomials can be defined by a simple combinatorial recursion. However, these polynomials are still very difficult to compute. For the case of the braid arrangements we have a conjecture for the top coefficient but do not even know a conjectural formula for all the coefficients.

(3) Title: Degree sequences for graphic arrangements
(Abstract) The degrees of a minimal set of generators of the module of derivations of a general graphic arrangement are obviously combinatorial. We present a bound on the top degree. But determining exactly this degree sequence seems difficult. We propose a more comprehensible study of these degree sequences.

Daniele Faenzi (Bourgogne):
Remarks and conjectures on deformed Weyl arrangements beyond the free range.
(Abstract) Deformed Weyl arrangements are obtained adding integral translates in a given range of the (positive) reflecting hyperplanes of a root system Phi. The resulting arrangement is free if the range symmetric around 0, or off from symmetry by 1. For more unbalanced ranges, the arrangement is not free, so how to describe it? I will report on ongoing work with Abe and Valles, providing an answer when Phi=A_2 and some conjectures for more general Phi, modifying former conjectures by Yoshinaga.

Simona Settepanella (Hokkaido): STRATA OF DISCRIMINANTAL ARRANGMENTS

Pauline Bailet (Hokkaido): TBA

Hiroaki Terao (Hokkaido): TBA

Takahiro Hasebe (Hokkaido): Combinatorics of set partitions and ordered set partitions
(Abstract) I will talk about some classes of set partitions and ordered set partitions arising from noncommutative probability and permutation groups.

Masahiko Yoshinaga (Hokkaido):
(1) Brion-type theorem for Borel-Moore homology of hyperplane arrangements.
(Abstract) A famous theorem of Brion asserts that for a given rational polytope, the sum of formal power series defined by the cone of each vertex becomes a rational function. We observe that a similar phenomenon happens for Borel-Moore homology groups of arrangements (at least one-dimensional case).

(2) What should be the chromatic polynomial for infinite graphs?

Notice (28 May):

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