Abstracts

Dec. 15 (Mon)          Dec. 16 (Tue)          Dec. 17 (Wed)          Dec. 18 (Thu)          Dec. 19 (Fri)

Dec. 15 (Mon)

9:30 ~ 10:30

Sakumi Sugawara     (Hokkaido University)

Hyperplane arrangements and related 3-manifolds

In this talk, we will discuss two kinds of 3-manifolds related to hyperplane arrangements. The first is the boundary manifold of a line arrangement in $\mathbb{C}P^2$. Cohen--Suciu proved that the cohomology ring of the boundary manifold is isomorphic to the double of the Orlik--Solomon algebra of the arrangement. We will give a new proof of this formula and generalize it to arbitrary combinatorial line arrangements. The second is the Milnor fiber boundary of arrangements in $\mathbb{C}^3$. The homeomorphism type of the Milnor fiber boundary is combinatorially determined, but the explicit formula for the homology group is unknown. We will give an explicit formula for the first homology group for generic arrangements and several conjectures for general cases.

10:50 ~ 11:50

Brendon Rhoades     (University of California)

Hyperplanes and superspace

The rank n superspace ring is a tensor product of a rank n polynomial ring with a rank n exterior algebra. The superspace ring comes equipped with a diagonal action of the symmetric group. The  superspace coinvariant ring SR_n is the quotient by the ideal generated by S_n-invariants with vanishing constant term. We calculate the bigraded Frobenius image of SR_n. A key tool in our analysis are Solomon--Terao algebras associated to arrangements. Joint with Robert Angarone, Patricia Commins, Trevor Karn, Satoshi Murai, and Andy Wilson.

14:00 ~ 15:00

Shuhei Tsujie     (Hokkaido University of Education)

MAT-partitions of the braid arrangement and social choice theory

An MAT-partition is a set partition of an arrangement that guarantees the arrangement is MAT-free. MAT-partitions of the braid arrangement can be realized as edge-labeled complete graphs. Moreover, they correspond to regular vines, combinatorial objects that originate in the study of statistics. In this talk, we introduce a connection between MAT-partitions of the braid arrangement and maximal Arrow's single-peaked domains, which arise in social choice theory. This talk is based on joint work with Hung Manh Tran and Tan Nhat Tran.

Dec. 16 (Tue)

9:30 ~ 10:30

Saiei-Jaeyeong Matsubara-Heo     (Tohoku University)

The Reciprocal A-determinant

Given a configuration of lattice points, which we denote by A, Gelfand, Kapranov, and Zelevinsky introduced the principal A-determinant as a product of A-discriminants, which is a central notion in their theory of multidimensional discriminants. The theory is controlled by the combinatorial geometry of the toric variety associated to A. On the other hand, Proudfoot and Speyer studied another class of varieties whose geometry is controlled by the matroid of A: the reciprocal linear space. We propose a natural analogue of the principal A-determinant in this context. Keywords are the following: projective dual variety, Chow form, Horn--Kapranov parametrization, Euler discriminant, tropical linear space, characteristic variety, and holonomic D-modules. This is an ongoing joint work with Simon Telen (MPI MiS).

10:50 ~ 11:50

Clément Dupont     (University of Montpellier)

The homological linking invariant for line arrangements

I will revisit a topological invariant for line arrangements due to Artal, Florens, and Guerville--Ballé, analogous to the linking number in knot theory, and which has been used to find Zariski pairs, i.e., two line arrangements with the same combinatorics but different topologies. I will explain that this invariant is of a homological nature, and will explain its properties using the toolbox of the homological study of algebraic varieties. This is joint work with Benoît Guerville--Ballé (Sassari).

14:00 ~ 15:00

Ye Liu     (Xi'an Jiaotong-Liverpool University)

Magnitude of a real hyperplane arrangement

Leinster introduced the notion of magnitude of a metric space as a size-like invariant. In particular, the magnitude of a finite undirected simple graph is an integer power series whose coefficients admit a counting formula. On the other hand, for a real hyperplane arrangement, the tope graph (chamber adjacency graph) can recover the underlying oriented matroid (up to reorientation). In this talk, we consider the magnitude of the tope graph.

15:20 ~ 16:20

Shunya Adachi     (Utsunomiya University)

Transformations for linear Pfaffian systems with singularities along hyperplane arrangementsⅠ

Motivated by the work of Haraoka (2012), research on linear Pfaffian systems with singularities along hyperplane arrangements has advanced significantly.

The main tool is middle convolution, which is an operation that maps one linear Pfaffian system to another.

It was originally introduced by N. Katz for linear ordinary differential equations as a refinement of the Euler integral transform, and was extended by Haraoka to linear Pfaffian systems with logarithmic singularities along hyperplane arrangements. This provides an effective tool for constructing and analyzing linear Pfaffian systems. 

In particular, for equations of Knizhnik--Zamolodchikov (KZ) type, T. Oshima has obtained many interesting results by using middle convolution. These results will be explained in his talk.

Inspired by these developments, I have introduced an operation called the middle Laplace transform, which provides a refinement of the Laplace transform in the spirit of middle convolution and is expected to be useful for constructing and analyzing linear Pfaffian systems with irregular singularities.

In this talk, I will first give a brief review of the theory of middle convolution, with an emphasis on aspects related to hyperplane arrangements, and then present the formulation and properties of the middle Laplace transform.

Although the middle Laplace transform and the middle convolution are transformations of differential equations, they also induce transformations of the hyperplane arrangements describing the singular locus. Since the topology of the singular locus of a linear Pfaffian system strongly affects the structure of its solutions, it seems natural and interesting to investigate these transformations from the viewpoint of hyperplane arrangements. In this direction, there has been intriguing recent progress on middle convolution due to Oshima, which will also be explained in his talk.

16:40 ~ 17:40

Toshio Oshima     (University of Tokyo)

Transformations for linear Pfaffian systems with singularities along hyperplane arrangements Ⅱ

Rigid Fuchsian ordinary differential equations (ODEs) can be extended to KZ-type Pfaffian systems by treating their singular points as variables. For any hyperplane arrangement, one can construct a Pfaffian system whose singular locus is a hyperplane arrangement containing it. Such systems are obtained from the trivial equation u' = 0 via the middle convolution. These Pfaffian systems are defined by residue matrices along singular hyperplanes. I introduce a generalization of these matrices to those associated with lower-dimensional intersections and describe how the middle convolution transforms their conjugacy classes, which are determined by commutative families of these generalized residue matrices. The middle convolution may create new singular hyperplanes. I classify the arrangements for which this never occurs and call such arrangements stable. In the case of the KZ-type equation, whose singular locus is stable, I explain that conjugacy classes of the commutative families are connected to the resolution of singularities of the equation and that they provide an extension of the Riemann scheme for ODEs.

Dec. 17 (Wed)

9:30 ~ 10:30

Tsuyoshi Miezaki     (Waseda University)

Universal graph series, chromatic functions, and their index theory

In this talk, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs. 

10:50 ~ 11:50

Laurentiu Maxim     (University of Wisconsin-Madison)

Maximal twisted Betti numbers of complex hyperplane arrangement complements

I will discuss work in progress with Y. Liu and B. Wang, in which we show that the maximal Betti numbers of a local system on an essential complex hyperplane arrangement complement are achieved only when the local system is the constant sheaf. This answers a question recently posed by Liu--Yoshinaga.

12:10 ~ 13:10

Misha Feigin     (University of Glasgow)

Quasi-invariants and Arnold--Maxwell topological theorem

m-quasi-invariant polynomials can be associated to any Weyl group W and they form a free module of rank |W| over W-invariant polynomials for any integer multiplicity m. When m = 0 they coincide with all the polynomials. Representation of W on coinvariants in m-quasi-invariants is isomorphic to the regular representation of W, and the isotypic component of the reflection representation is equivalent to W-invariant logarithmic vector fields for the corresponding reflection arrangement with multiplicity 2m + 1. The origin of m-quasi-ivariants is in Calogero--Moser integrable systems.

It is tempting to try to find nice topological spaces whose cohomology rings would be naturally isomorphic to the rings of m-quasi-invariants, and which would reduce to complete flag varieties in the case when W is the symmetric group and m = 0. Recently a homotopical construction of such spaces was given by Berest and Ramadoss for the two-element group W = S_2, and another homotopical construction of spaces with slightly different cohomologies was given by Berest, Liu, and Ramadoss for general W. I am going to present a direct geometrical construction of the corresponding topological spaces in the case of W = S_2 which relies on Arnold's topological interpretation and generalisation of Maxwell's theorem about spherical harmonics. It allows to define an action of the unitary group U(2) on any even-dimensional sphere whose equivariant cohomology we compute. 

This is a joint work with K. Feldman.

Dec. 18 (Thu)

9:30 ~ 10:30

Paul Mücksch     (Leibniz University Hannover)

On the K(π,1)-problem for arrangements

The question of when complements of complex hyperplane arrangements form Eilenberg--MacLane K(π,1)-spaces is a central topic in the field. Over the decades, many remarkable results were obtained but still this property remains only partially understood, and notable conjectures persist. In this talk, I will survey the current state with particular attention to unresolved problems, and I will present new connections among existing criteria as well as further developments concerning necessary and sufficient conditions.

10:50 ~ 11:50

Akihiro Higashitani     (The University of Osaka)

Characteristic quasi-polynomials of deletions of Shi arrangements

Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers. In this talk, we compute the characteristic quasi-polynomials of the Shi arrangement of type B, C and D restricted by one given hyperplane. As a corollary, we determine when period collapse occurs for those characteristic quasi-polynomials.

14:00 ~ 15:00

Haru Negami     (Chiba University)

Unitarity of multiplicative middle convolution and its application to quantum computing

The multiplicative middle convolution of KZ-type equations is known to preserve the unitarity of representations. In this talk, we provide conditions under which a unitary representation has a finite image, i.e., when the associated monodromy group is finite. The fundamental group of the domain of a KZ-type equation is the pure braid group, and among unitary representations of braid groups, those with finite images are of particular interest due to quantum computing. We conclude the talk by discussing such applications to quantum error correction codes.

15:20 ~ 16:20

Hiroo Tokunaga     (Tokyo Metropolitan University)

Some conic-line arrangements of degree 7 and 8

We consider some examples for pairs of  conic-line arrangements $(B_1, B_2)$ of degree 7 and 8 such that (i) both of $B_i$ ($I = 1, 2$) have the same combinatorics, (ii) there exists no homeomorphism $h : (\mathbb{P}^2 \to \mathbb{P}^2$ with $h(B_1) = B_2$ and (iii) the jacobian ideals of  $B_1$ and $B_2$ have distinct minimum free resolution. These statement shows that examples for $(B_1, B_2)$  are Zariski pairs as well as strong Ziegler pairs. We also explain their construction based on  geometry and arithmetic of  certain rational elliptic surfaces.

16:40 ~ 17:40

Alexandru Suciu     (Northeastern University)

The effective Chen ranks conjecture

The Chen ranks conjecture asserts that the lower central series ranks of the maximal metabelian quotient of a hyperplane arrangement group are determined by the dimensions of the irreducible components of its first resonance variety. Equivalently, in sufficiently high degrees, the Hilbert series of the first homology of the associated Koszul complex is governed by the support of this module.

We prove an effective version of the conjecture: for any 1-formal group G whose resonance components are linear, isotropic, and separable, the Hilbert function of the Koszul module is explicitly described in all degrees $\ge b_1(G)-1$. These geometric hypotheses hold, for example, for complex line arrangements with at most triple points. 

Joint work with Marian Aprodu, Gavril Farkas, and Claudiu Raicu. 

Dec. 19 (Fri)

9:30 ~ 10:30

So Yamagata     (Fukuoka University)

A presentation of the pure braid group via braid monodromy

In this talk, I will introduce a method to derive an explicit presentation of the pure braid group as the fundamental group of the braid arrangement, using the technique of braid monodromy. The resulting presentation is shown to coincide with the modified Artin presentation given by Margalit and McCammond. I will also discuss an attempt to extend this framework to the Manin--Schechtman arrangements.

10:50 ~ 11:50

Roberto Pagaria     (Bologna University)

Generating Functions for the Chow Polynomials of Wonderful Models in Types A and B

We study the Chow ring of De Concini--Procesi wonderful models of types A_n and B_n with respect to the minimal building set. In the former case, the model coincides with the Deligne--Mumford compactification of M_{0,n}. Our focus is on the exponential generating function of the Poincaré polynomials, for which we provide several explicit formulas.  These results recover the classical formulas of Keel, Getzler, Manin, and Yuzvinsky, and yield new ones in type B.

This is joint work with L. Ferroni and L. Vecchi.

14:00 ~ 15:00

Yusuke Nakamura     (Nagoya University)

Ehrhart theory on periodic graphs

A periodic graph is defined as a graph on which the lattice Z^N acts freely, such that its quotient graph is a finite graph. Periodic graphs are objects of study in mathematical crystallography, and they also appear naturally in geometric group theory as Cayley graphs of virtually abelian groups. The growth sequence b(n) of a graph is defined as the number of vertices within a graph distance of n or less from a starting vertex. In this talk, we will first introduce the result (Nakamura--Sakamoto--Mase--Nakagawa) that the growth sequence of a periodic graph is of quasi-polynomial type (i.e., it is a quasi-polynomial for sufficiently large n). In the remaining time, we would also like to discuss phenomena analogous to Ehrhart theory.

15:20 ~ 16:20

Takuya Saito     (Hokkaido University)

Varieties associated with the combinatorics of discriminantal arrangements

Manin and Schechtman introduced discriminantal arrangements, which are generalizations of braid arrangements. They are constructed from given hyperplane arrangements, but the combinatorics of discriminantal arrangements are not uniquely determined by the combinatorics of original arrangements.In this talk, we consider subvarieties of the Grassmannian to classify the combinatorics of discriminantal arrangements. This talk includes the results of joint work with Das and Settepanella.

16:40 ~ 17:40

Graham Denham     (University of Western Ontario)

topology of Schubert arrangements

Just over 50 years ago, Hattori showed that the complement of a generic complex hyperplane arrangement is homotopy equivalent to a fat wedge of circles.  The Schubert arrangements (or lattice-path arrangements) generalize the generic arrangements, and they are of particular interest because they index a basis of the valuative group of matroids.  Using Yoshinaga's tools for Morse theory on complements of complexified real arrangements, we show that (complex) Schubert arrangement complements are homotopy equivalent to certain coordinate subcomplexes of tori. This generalizes Hattori's result and leads to a complete description of the cohomology of rank-1 local systems, in all degrees, for such arrangements. Joint work with Prajwal Udanshive.