Thursday, January 10

Friday, January 11

Saturday, January 12

Sunday, January 13

Monday, January 14

Tuesday, January 15

 
Kazuhiko Aomoto
Nagoya University
Gauss-Manin Connections for Hypersphere Arrangements
(A Variant of Schlafli Formulae and Related Problems)

Eduardo Cattani
University of Massachusetts
Binomial residues
   A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A-hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A. This is joint work with Alicia Dickenstein (U. of Buenos Aires) and Bernd Sturmfels (U. C. Berkeley).

Michael Falk
Northern Arizona University
Geometry and combinatorics of resonant weights
   Let A denote the Orlik-Solomon algebra of an arrangement A of linear hyperplanes in C^l, with coefficients in a field k. Each x in A¹ determines a degree one differential A --> A by left multiplication. In this talk, we call x a resonant weight if the cohomology H¹(A,x) is nonzero. The collection of all resonant weights in A¹ = kⁿ forms the resonance variety R₁(A,k). It is filtered by dim_kH¹(A,x). When k=C, resonance varieties are related to the cohomology of rank-one local systems over the complement M of A. When k=Z_p, the resonance varieties give enumerations of classes of subgroups of the fundamental group of M.
   We describe a combinatorial characterization of resonant weights, involving partitions and bilinear forms, which is valid over any field k. To illustrate the method we enumerate some classes of index p normal subgroups of fundamental groups of graphic arrangements, including the pure braid group.
   When k=C and l=3, a resonant weight gives rise to a pencil of curves in CP², whose singular members include (at least three) products of lines. The union of these is the projective image of the original arrangement. When all singular elements are products of lines, it follows that the complement M is an Eilenberg-Mac Lane space. One infinite family and one sporadic example have been known for some time. We present a two new families of such fibered arrangements, discovered via resonant weights, and discuss the possible existence of second sporadic example.
   Also, in case k=C, the resonance varieties are known to be unions of linear subspaces. There are at least three proofs of the fact for k=C, none of which generalize to other fields. We conjecture that the same assertion holds over any algebraically closed field k. We present an example in characteristic two which exhibits features not found over C, and give some combinatorial explanations for the behavior. We formulate a description of resonance varieties which proves the conjecture at least in some cases, and illustrate with another unusual example in characteristic two.

Toshitake Kohno
University of Tokyo
Elliptic KZ system and related algebras
   I am going to describe the elliptic KZ system associated with elliptic solutions of the Yang-Baxter equation due to Belavin and Drinfel'd. A relation to the algebra of chord diagrams on surfaces and the homology of the loop space of orbit configuration spaces will be discussed.

Peter Orlik
University of Wisconsin
Gauss-Manin Connections for Hyperplane Arrangements
   In joint work with Dan Cohen, we construct a universal Gauss-Manin connection for the moduli space of an arrangement of hyperplanes in the cohomology of a complex rank one local system. We prove that the eigenvalues of this connection are integral linear combinations of the weights which define the local system.

Alexander Suciu
Northeastern University
Lower central series, free resolutions, and homotopy Lie algebras of arrangements   [slides]
   The two main objects in the topological study of hyperplane arrangements are the cohomology ring and the fundamental group of the complement. The ring admits a readily computable, combinatorial description, due to Orlik and Solomon. The (rational) graded Lie algebra associated to the lower central series of the group is also combinatorially determined, though no effective procedure for computing its ranks is known, except in the case when the OS-algebra is Koszul.
   In the first part of the talk, I will describe recent work with Hal Schenck, in which we determine the first four LCS ranks of the group from the Betti numbers of the free resolution of the OS-algebra over the exterior algebra. For several classes of arrangements, we make precise conjectures, expressing the LCS ranks in terms of known polynomials, and verify those conjectures in low ranks.
   To a hyperplane arrangement A, Dan Cohen, Fred Cohen and Miguel Xicoténcatl associate a sequence of ``redundant" subspace arrangements, A^k. For fiber-type arrangements, they show that the homotopy Lie algebra of A^k equals (up to rescaling) the graded Lie algebra associated to the fundamental group of A.
   In the second part of the talk, I will describe recent work with Stefan Papadima, in which we extend this "Rescaling Formula" for (rational) homotopy groups, to an arbitrary space with Koszul cohomology ring, and its corresponding sequence of homological rescalings. If the starting space is formal, we further upgrade this formula to the level of Malcev completions, and Milnor-Moore groups.

Hiroaki Terao
Tokyo Metropolitan University
The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements
   Let W be a finite irreducible orthogonal reflection group acting on an l-dimensionel Euclidean vector space V. The quotient space V/W has rich differential geometric structures: it has flat coordinates, the Hodge filtration (in the sense of K. Saito) and has the Frobenius manifold structure (in the sense of B. Dubrovin). In this talk we introduce the contact-order filtration of the derivation module D(A) along the corresponding Coxeter arrangement A and prove that the filtration is essentially equivalent to the above-mentioned differential-geometric structures.

Sergey Yuzvinsky
University of Oregon
Nets in CP² and cohomology jumping loci
   A k-net (k>2) in CP² is a pair of finite sets (A,X) of lines and points respectively such that A is partitioned in k classes with precisely one line from each class passes through any point of X and every two lines from different classes intersect at a point from X. This is a discrete analog of the notion of web popular in differential geometry. It turns out that k-nets are precisely those line arrangements whose complements have local system cohomology of dimension k-2 with minimal multiplicities of points of intersection.
   There are strong restrictions on nets in the complex plane. For instance, k can be equal to 3,4 or 5 only. There are more subtle restrictions on the combinatorial types of nets, especially for k=3. Every 3-net defines a binary operation and one can ask what finite groups can appear this way. We will discuss this question exhibiting old and new series of examples and restrictions.