Summer Conference on Hyperplane Arrangements(SCHA) in Sapporo

Contents

Talks

Takuro Abe (Kyushu Univ.)
Title: Recent topics on free arrangements of hyperplanes
Abstract: Freeness has been one of the central topics among the theory of hyperplane arrangements. There are several important results like Terao's addition-deletion and factorization therems, the moduli theoretic approach by Yusvinsky and so on. In particular, Yoshinaga's criterion on freeness by using mutliarrangemens gave a breakthrough in this study area, and there have been a lot of new approaches on freeness problem.
In this talk, we discuss several recent results including freeness criterion, multiple addition theorems and divisional freeness. Also, we discuss the new algebraic class of line arramgenents and plane curves in the projective plane called near freeness by Dimca and Sticlaru. Moreover, we pose some problems which appeared in this ten years.
Kazuhiko Aomoto (Nagoya Univ.)
Title: Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants
Abstract: The $n$ dimensional hypergeometric integrals associated with a hypersphere arrangement S are formulated by the pairing of $n$ dimensional twisted cohomology $H^{n}_{\nabla}(X,\Omega)(\ast S))$ and its dual. Under the condition of general position there are stated some results and conjectures which concern a representation of the standard form of the twisted cohomology, the variational formula of the corresponding integral in terms of a special basis of invariant (under isometry) 1-forms using Cayley-Menger minor determinants , a connection relation of the unique twisted $n$-cycle identi ed with the unbounded chamber to a special basis of twisted $n$-cycles identi ed bounded chambers. As an application the variational formula of the volume of a domain bounded by hypersphere arrangements will be presented. This seems to be an extension of the well-known Schläfli formula for an n dimensional geodesic simplex.
Nero Budur (Univ. of Leuven)
Title: Simple D-modules
Abstract: We give a necessary and sufficient criterion for the simplicity as perverse sheaves, or as regular holonomic D-modules, of the direct images of rank-one local systems under an open embedding. For complements of hyperplane arrangements, this criterion is combinatorial. This problem is related with the still-conjectural combinatorial nature of the Betti numbers of the Milnor fibers of hyperplane arrangements. Joint work with Y. Liu, L. Saumell, B. Wang.
Graham Denham (Western Univ.)
Title: Tangent cones, Alexander invariants and rational models
Abstract: The Tangent Cone Theorem relates the cohomology of local systems on a space X to algebraic information extracted from the cohomology ring of X or, more generally, a rational (CDGA) model for X. I will describe some recent work with Alex Suciu that attempts to simplify the existing literature and that leads to a new Tangent Cone Theorem for Alexander invariants. Hyperplane arrangements and their Milnor fibres provide our motivating examples.
Michael Falk (Northern Arizona Univ.)
Title: On the cohomology of the Milnor fiber
Abstract: Let A be a central arrangement in Cn with (reduced) defining equation Q(x1, ..., xn). The Milnor fiber F of A is the affine variety defined by Q(x1, ..., xn)=1. We prove that the first betti number of F is determined by the intersection lattice L of A, resolving a long-standing conjecture. It is enough to prove the statement in case n=3. In this case we construct an explicit compactification S of F, with D=S-F a normal crossing divisor. By known results, it is enough to show the first betti number of S is determined by L. Using the residue double complex of [FSV15], the first cohomology of S is determined by residue maps in the natural stratification of S. The surface S constructed so that the pair (U,D), U a regular neighborhood of D in S, is determined up to analytic isomorphism by the intersection lattice. This implies the result.
Eva Maria Feichtner (Univ. Bremen)
Title: Matroids and rational models
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.
June Huh (Princeton Univ.)
Title: Hard Lefschetz theorem for matroids and other things
Abstract: I will explain the proof and implications of the Hard Lefschetz theorem and the Hodge-Riemann relations for matroids and related tropical varieties. Joint work with Karim Adiprasito and Eric Katz.
Toshitake Kohno (Univ. of Tokyo)
Title: Higher category extensions of holonomy maps for hyperplane arrangements
Abstract: We explain a method to construct higher category extensions of holonomy representations of homotopy path groupoids my means of Chen's formal homology connections. Applying this general method, we describe an explicit form of higher holonomy for homotopy path groupoids in the case of the complement of hyperplane arrangements. In particular, by means of a 2-functor from the path 2-groupoid of the configuration space, we construct representations of the 2-category of braid cobordisms.
Gustav Lehrer (Univ. of Sydney)
Title: Cohomology of arrangements and representations of reductive groups.
Abstract: I will discuss problems in arrangements and in root systems which arise from the representation theory of reductive groups over finite fields.
Anatoly Libgober (Univ. of Illinois at Chicago)
Title: Strata of Discriminantal arrangements
Abstract: In 1987 Manin-Schechtman proposed a generalization of braid arrangements such that the fundamental groups of the complements to these discriminantal arrangements provide a natural generalizations of pure braid groups. It turns out that such fundamental groups depend on "hidden" parameters controlling the structure of codimension 2 strata of Manin-Schechtman arrangements. We shall describe these possible variations in the structure of codimension 2 strata of discriminantal arrangements and respective variations in the fundamental groups. (This is report on join work with Simona Settepanella).
Eduard Looijenga (Univ. of Utrecht and YMSC Tsinghua Beijing)
Title: Configuration spaces of curves and conformal blocks
Abstract: WZW theory produces flat vector bundles over the moduli space of pointed curves, or rather over a C*-bundle over such spaces. Physicists assert that these flat bundles come with a flat inner product and this has been shown by mathematicians in a rather roundabout way in some cases. It is therefore fair to say that the theory is still incomplete, certainly from the point of view of an algebraic geometer. With this in mind we propose a construction that links these bundles to the Hodge theory of configuration spaces of curves. We believe the ingredients of this construction have an interest independent this envisaged application.
Brendon Rhoades (Univ. of California, San Diego)
Title: The Parking Conjecture
Abstract: Let W be an irreducible real reflection group. We will describe two models of "parking functions" attached to W, one combinatorial and one algebraic, and conjecture an isomorphism between these models. Our conjecture, proven in various cases, will uniformly imply a variety of facts in Coxeter-Catalan theory which are at present only understood in a case-by-case fashion. This is partially joint with Drew Armstrong, Vic Reiner, and Michelle Bodnar.
Mathias Schulze (Tech. Univ. Kaiserslautern)
Title: Duality on value semigroups
Abstract: The semigroup of values is a classical combinatorial invariant associated to a curve singularity. It is defined by taking all regular elements of the corresponding ring to the integral closure and taking a multivaluation. For a curve singularity with $r$ branches the semigroup of values is a submonoid of $\NN^r$. In the irreducible case it is a numerical semigroup, otherwise it is not even finitely generated. Due to Lejeune-Jalabert and Zariski, this value semigroup determines the topological type of plane complex curves.
As observed by Kunz in the irreducible case, the Gorenstein property of a curve singularity is equivalent to a symmetry of gaps and non-gaps in the (numerical) value semigroup. Delgado generalized this result to the reducible case introducing a non-obvious notion of symmetry of a semigroup. A canonical (fractional) ideal on a curve singularity defines a duality on fractional ideals. On the other hand taking multivaluations as above associates to any fractional ideal a value semigroup ideal. Such value semigroup ideals satisfy certain natural axioms defining the class of so-called good semigroup ideals. Barucci, D'Anna and Fr旦berg gave an example of a good semigroup that does not come from a ring. Extending Delgado's symmetry result, D'Anna described the value semigroup ideals of canonical ideals. In the Gorenstein case, Delphine Pol described the value semigroup ideal of duals. Unifying the work of D'Anna and Pol we establish a purely combinatorial duality on good semigroup ideals that mirrors the duality on fractional ideals. The talk is based on joint work with Philipp Korell and Laura Tozzo.
William Slofstra (Univ. of Waterloo)
Title: Freeness and local formality of inversion hyperplane arrangements
Abstract: Inversion hyperplane arrangements are subarrangements of root system arrangements indexed by elements of the corresponding finite Weyl group. Freeness of inversion arrangements is connected to the combinatorics of the Weyl group and to the geometry of Schubert varieties, and it seems reasonable to conjecture that this connection might extend to other properties of the inversion arrangement, especially those related to freeness. In this talk, I will review the connection between freeness of inversion arrangements and properties of the Weyl group, and explain how local formality can also be captured by the Weyl group.
This is joint work with Travis Scrimshaw.
Eric Sommers (Univ. of Massachusetts)
Title: From hyperplane arrangements to families of Weyl group representations
Abstract: We will discuss a family of representations of a Weyl group W that first arose in the setting of the hyperplane arrangement associated to W. This family also shows up in affine Springer theory, rational Cherednik algebras, and double harmonics. The focus of the talk will be on decomposing a graded version of these representations into induced pieces and using this to obtain q-analogues of the values of hyperplane characteristic polynomials (evaluated at certain positive integers). This yields q-analogues of well-known combinatorial quantities (namely, the Kreweras and Narayana numbers) that satsify the cyclic sieving property with respect to various subposets of the W-noncrossing partitions (joint work with Vic Reiner). If time permits, we will discuss implications for the double-graded version of this story, which relates to work of Haiman, Hikita, and others.
Alexandru Suciu (Northeastern Univ.)
Title: On the topology of line arrangements
Abstract: I will discuss some recent advances in our understanding of the diverse connections between the combinatorics of an arrangement of complex lines and the topology of the various spaces built out of it.
Uli Walther (Purdue Univ.)
Title: What kills an arrangement?
Abstract: I will discuss the current state of affairs on the questions "What are the annihilators inside the Weyl algebra of the D-module generated by f^s and the D-module generated by 1/f?"
Masahiko Yoshinaga (Hokkaido Univ.)
Title: Hyperplane arrangements and the Eulerian polynomial.
Abstract: This talk focuses on a relation between Eulerian polynomials and Linial arrangements. Eulerian polynomials were introduced by Euler in his study of special values of zeta functions at negative integers. The Linial arrangement is an affine hyperplane arrangement associated to root systems, which has intriguing relations with enumerative problems. Recently, it is found that the characteristic (quasi)polynomial of the Linial arrangement can be expressed in terms of the root system generalization of Eulerian polynomial introduced by Lam and Postnikov. A conjectured property "functional equation of the characteristic polynomial" is now settled by using the expression and the symmetry of the Eulerian polynomial. We will also discuss congruences satisfied by Eulerian polynomials.

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