Program

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Monday, September 1
09:00-09:30		Registration
09:30-10:30	Alex Suciu	Topology of line arrangements
	I will discuss some recent advances in our understanding of the relationship between the topology, group theory, and combinatorics of a complex line arrangement.
10:30-11:00		Coffee Break
11:15-12:15	Michèle Vergne	Euler Mac Laurin formula for rational polytopes
	Let P be a polytope defined by rational inequalities. We consider a smooth function and we give an asymptotic formula for summing the function on points in P belonging to a rescaled lattice. This formula generalizes the local Euler-Mac Laurin formula defined by Baldoni-Berline-Vergne for polynomial functions. (Common work with Nicole Berline) Slides
12:30-14:00		Lunch
15:00-16:00	Dmitry Kozlov	Configuration spaces arising in distributed computing
	In this talk we shall describe a family of simplicial complexes, called protocol complexes, which arise naturally as some of the central objects in the field of theoretical distributed computing. These complexes give a description of the totality of all possible executions of distributed protocols in a fixed computational model. They are the natural analog of configuration spaces in this context. Part of the talk will be based in the recent book "Distributed Computing through Combinatorial Topology", joint with M. Herlihy and S. Rajsbaum.
16:00-16:30		Coffee Break
16:30-17:30	Michael Farber	Large random spaces and groups
	I will discuss several models producing random simplicial complexes. We are interested in geometric and topological properties of random simplicial complexes which are satisfied with probability tending to 1 as the number of vertices of the complex tends to infinity. In the talk I will explain why for random simplicial complexes the Whitehead and Eilenberg-Ganea Conjectures hold. I shall also describe torsion and the cohomological dimension of fundamental groups of random simplicial complexes. This is a joint work with Armindo Costa. Slides
Tuesday, September 2
09:30-10:30	Mike Davis	Complements of hyperplane arrangements as posets of spaces
	The complement of an affine hyperplane arrangement is a poset of spaces, where the indexing poset is the intersection poset and where the space corresponding to an intersection is the complement of the corresponding central arrangement. An application of this gives a spectral sequence for computing cohomology. Slides
10:30-11:00		Coffee Break
11:15-12:15	Jon McCammond	The structure of euclidean Artin groups
	The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several long-standing conjectures about these groups, proving for the first time that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. In my talk I will survey our results and the techniques we use to prove them. Slides
12:30-14:00		Lunch
15:00-16:00	Emanuele Delucchi	Recent developments in toric arrangements
	The study of toric arrangements is rooted in the literature in both its topological (since at least Looienga in 1993) and combinatorial aspects (e.g., Ehrenborg, Readdy and Slone 2009). Recent work of De Concini, Procesi and Vergne provided a fresh impulse towards a comprehensive study of this subject, viewed as a generalization of the successful theory of hyperplane (or subspace) arrangements in vector spaces. Out of this impulse grew many new results and techniques, concerning both topology and in combinatorics, which I will try to survey with an eye towards setting up a general combinatorial-topological framework which might lead to the treatment of even more general types of arrangements. Some of the results I will present have been obtained in joint works with Karim Adiprasito, Filippo Callegaro, Giacomo d'Antonio or Sonja Riedel. Slides
16:00-16:30		Coffee Break
16:30-17:30	Luca Moci	Matroids over a ring: motivations, examples, perspectives
	Several objects can be associated to a list of vectors with integer coordinates: a toric arrangement, a zonotope, a vector partition function. The linear algebra of the list is encoded by the notion of a matroid, but several properties of the objects above depend also on the arithmetics of the list: this is retained by the notion of a "matroid over Z". Similarly, applications to tropical geometry suggest to consider matroids over a discrete valuation ring. Motivated by these examples, we introduce the notion of a "matroid over a commutative ring R". When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids, and compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several known invariants. We will outline possible applications and open problems. (Based on joint work with A. Fink) Slides
17:45-19:15		Poster Session
	Haniya Azam	Cohomology of configuration spaces of Riemann surfaces
	Poster
	Christin Bibby	Cohomology of abelian arrangements
	Poster
	Elisabeth Leyton Chisholm	Braid Groups and Euclidean Simplices
	Poster
	Sonja Riedel	What is a toric pseudoarrangement?
	Poster
	Elia Saini	Phasing Classes of Matroids
	Poster
Wednesday, September 3
09:00-10:00	Hal Schenck	Chen ranks and resonance
	The Chen groups of a group G are the lower central series quotients of the maximal metabelian quotient of G. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of G, a jump locus for the cohomology of G. In the case where G is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu’s Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugating automorphisms of a finitely generated free group. (joint work with Dan Cohen, Lousiana State University)
10:00-10:30		Coffee Break
10:30-11:30	Alejandro Adem	An Infinite Loop Space Associated to Commuting Matrices
	Let G denote a Lie group. We show that a construction introduced by Adem-Cohen-Torres built out of the commuting elements in G plays the role of a classifying space for commutativity. We will discuss how this is reflected in properties of these spaces and show that for the unitary group U we obtain a new infinite loop space. This leads to the notion of commutative K-theory, with characteristic classes computed using multisymmetric polynomials. This is joint work with José Gómez, John Lind and Ulrike Tillmann.
11:45-12:45	Michael J. Falk	BGG resolutions and configuration spaces
	Let A be a projective arrangement and p ∶ X → ℙk the minimal resolution of A. Let D be a union of components of the normal-crossing divisor p−1(∪A), and let L be a rank-one local system on X−D. We present a complex A·(X,D;L), built out of Aomoto complexes of minors of A and residue maps between them, that computes the cohomology H*(X−D, L). When A=Ak,n is an sl2-discriminantal arrangement, and L is determined by a collection Lm1, …, Lmn of irreducible finite-dimensional sl2 modules, the skew-symmetric part of the complex A·(X,D;L) is isomorphic to the Bernstein-Gelfand-Gelfand resolution of Lm1 ⊗ ⋯ ⊗ Lmn. We illustrate with some examples. This is joint work with V. V. Schechtman and A. N. Varchenko.
13:00-14:00		Lunch
14:00-20:00		Free afternoon and guided tour of Cortona
20:00-.....		Social Dinner
Thursday, September 4
09:30-10:30	Eric Babson	Random triangulations of the two sphere
	We study the local geometry of random triangulations of the two sphere by constructing a foliated configuration space with a measure and a metric so that the metric balls in its leaves are those in the triangulations with the measure recording their frequency. This construction distinguishes a class of differential operators for which we study a density of states. Joint with: Nathan Hannon and Jerome Kaminker
10:30-11:00		Coffee Break
11:15-12:15	Anders Björner	Filtered geometric lattices and tropical Lefschetz theorem
	In the first part we prove a conjecture of Mikhalkin and Ziegler concerning positive sum systems in geometric lattices. This builds on methods from topological combinatorics. In the second part we establish analogues for smooth tropical varieties of the classical Lefschetz Section Theorem. For this the result from part one on geometric lattices provides a crucial index estimate for the stratified Morse data at critical points of the tropical variety, which we consider as a Whitney stratified space.
12:30-14:00		Lunch
15:00-16:00	Thomas Brady	Non-crossing partitions and monodromy of Milnor Fibres
	For arrangements associated to finite real reflection groups, we determine the action of the monodromy on simplicial models for the Milnor fibres. This is joint work with Mike Falk and Colum Watt. Slides
16:00-16:30		Coffee Break
16:30-17:30	Matthias Franz	Big polygon spaces
	Polygon spaces are configuration spaces of polygons with prescribed edge lengths. We present a related family of spaces, called big polygon spaces. They come with canonical torus actions, whose fixed point sets are polygon spaces. It turns out that big polygon spaces provide examples of torus actions on compact orientable manifolds whose existence has previously been an open question. For example, while the equivariant cohomology of a big polygon space is never free, it can be computed via the "GKM method", and the equivariant Poincaré pairing is perfect. More generally, big polygon spaces show that a certain bound obtained by Allday-Franz-Puppe and concerning syzygies in torus-equivariant cohomology is optimal. Slides
Friday, September 5
09:30-10:30	Graham Denham	Elliptic braid groups are duality groups
	The elliptic braid group is the fundamental group of a configuration space of n points in 2-dimensional torus. We show that such groups are duality groups, extending the known result for classical braid groups. The method is an instance of a more general cohomological vanishing construction which also has applications to torus arrangements, right-angled Artin groups, and hyperplane complements. This is joint work with Alex Suciu (Northeastern) and Sergey Yuzvinsky (Oregon).
10:30-11:00		Coffee Break
11:15-12:15	Sergey Yuzvinsky	Higher topological complexity of Eilenberg-MacLane spaces
	Topological complexity TC(X) of a topological space X was introduced by M. Farber about 10 years ago as a specialization of the Schwarz genus depending only on the homotopy type of X. About 5 years ago, Yu. Rudyak generalized Farber’s definition to obtain higher (s-th) topological complexity TCs(X) for s ≥ 2 that coincides with TC(X) for s = 2. For X being the complement of a complex arrangement of hyperplanes there were previous attempts to calculate TC(X). They were successful for particular classes of arrangements such as Coxeter infinite series (Farber and Y) and generic arrangements (Y, Cohen and Pruidze). In a recent paper by Gonzalez and Grant, TCs was computed (for an arbitrary s) for the Coxeter series of type A. In the talk, we will give a simple condition that allows us to calculate TCs for all complex reflection arrangements, i.e., for a class of K [π,1] spaces. If time allows we will talk about new upper and lower bounds for TCs for arrangement complements and for TCs for generic arrangements. Slides
12:30-14:00		Lunch
15:00-16:00	Gus Lehrer	Cohomology of the Milnor fibre of an arrangement
	I will discuss problems concerning the cohomology of the Milnor fibre of a hyperplane arrangement, both generally and in the case of the set of reflecting hyperplanes of a unitary reflection group. The basic problem is to determine the Hodge-Deligne polynomials, equivariantly, but this is currently a distant dream. I shall report on some results with Dimca, relating cohomology degree and Hodge structure to the order of the monodromy, as well as some more particular results. Slides
16:00-16:30		Coffee Break
16:30-17:30	Claudio Procesi	From the theorem of Amitsur-Levitzki to exterior algebras of simple Lie algebras
	The classical theorem of Amitsur-Levitzki tells us that the algebra on n x n matrices satisfies the standard polynomial in 2n variables, the essentially unique alternating multilinear non-commutative polynomial in degree 2n. We revisit this theorem in several ways, analyzing the structure of multilinear alternating and equivariant maps, first for matrices of various kinds and then in general for simple Lie algebras. The results obtained are quite precise and give the general framework for theorems of type similar to Amitsur-Levitzki. Slides