Wednesday, May 23 
09:3010:30 
Mario Salvetti 
Discrete methods for the topological study of some Configuration Spaces


We apply some methods from "discrete topology" to the study of some "Configuration Spaces". In particular, we are interested in
(co)homological computations for configuration spaces associated with Artin and Coxeter groups.

10:3011:00   Coffee Break 
11:0012:00 
Giovanni Gaiffi 
Maximal models of root arrangements 

Given a subspace arrangement, there are several De ConciniProcesi models associated to it, depending on distinct sets of initial combinatorial data (building sets). We will describe this family of models in the case of root arrangements, focussing in particular on maximal models.

12:0014:00   Lunch 
14:0015:00 
Tom Brady 
Milnor fibers and noncrossing partitions 

We describe work in progress with Mike Falk and Colum Watt in which we construct a combinatorial chain complex which computes
the cohomology of the Milnor fiber of the discriminant associated with a finitetype Coxeter group, whose terms are cohomology groups of truncations of the associated noncrossing partition lattice.

15:0015:30   Coffee Break 
15:3016:30 
Filippo Callegaro 
Homology of complex braid groups 

I'll present some results of a recent work in collaboration with Ivan Marin (Paris). We studied the braid groups associated to the
reflection arrangements for complex reflection groups. For irreducible Coxeter groups there is a 1to1 correspondence with Artin
groups. With complex reflection groups it happens that different reflection groups can give the same braid group and it is not
always clear whether two complex braid groups are isomorphic. Homology can help to distinguish some of them and we can prove some
isomorphisms too, but still open problems remains. A direct limit of complex reflection groups can be defined in a natural way and
it turns out that its homology is related to the homology of some loop spaces.


Thursday, May 24 
09:3010:30 
Graham Denham 
Duality properties for abelian covers


In parallel with a classical definition due to Bieri and Eckmann, say
an FP group G is an abelian duality group if H^p(G,Z[G^{ab}]) is zero
except for a single integer p=n, in which case the cohomology group is
torsionfree. We make an analogous definition for spaces. In contrast
to the classical notion, the abelian duality property imposes some
obvious constraints on the Betti numbers of abelian covers.
While related, the two notions are inequivalent: for example, surface
groups of genus at least 2 are (Poincare) duality groups, yet they are
not abelian duality groups. On the other hand, using a result of Brady
and Meier, we find that rightangled Artin groups are abelian duality
groups if and only if they are duality groups: both properties are
equivalent to the CohenMacaulay property for the presentation
graph. Building on work of Davis, Januszkiewicz, Leary and Okun,
hyperplane arrangement complements are both duality and abelian
duality spaces. These results follow from a more general cohomological
vanishing theorem, part of work in progress with Alex Suciu and Sergey
Yuzvinsky.

10:3011:00   Coffee Break 
11:0012:00 
Michael Falk 
Resolutions of rankone local systems on hyperplane complements 

We describe a spectral sequence that computes the cohomology of a
rankone local system on the complement of a union of exceptional
divisors in the minimal resolution of a projective arrangement, in
terms of Aomoto complexes and residues, and do some examples. This is
a preliminary report on work in progress with A. N. Varchenko.

12:0014:00   Lunch 
14:0015:00 
Luca Moci 
Arithmetic matroids 

We introduce the notion of an arithmetic matroid, axiomatizing the linear algebra and the arithmetics of a list of elements
of a finitely generated abelian group. To an arithmetic matroid we associate an arithmetic Tutte polynomial, with applications
to toric arrangements, partition functions, zonotopes, and labeled graphs. Then we provide a combinatorial interpretation of
its coefficients, generalizing Crapo's formula for the classical Tutte polynomial.

15:0015:30   Coffee Break 
15:3016:30 
Emanuele Delucchi 
Minimality of toric arrangements 

A toric arrangement is given by a family A of level sets of characters of a complex torus T.
The focus of this talk will be on the topology of the complement M:=T \ A, and in particular on the extent to which it is determined
by the combinatorial data of the arrangement A.
After an introduction to toric arrangements, I will present some recent joint work with Giacomo d'Antonio, proving that M is a
minimal space (and thus homologically torsionfree).


Friday, May 25 
09:3010:30 
Alexandru Suciu 
The rational cohomology of smooth, real toric varieties


I will discuss the cohomology with coefficients in
rank one local systems for various polyhedral products,
including real DavisJanuszkiewicz spaces and toric complexes.
As an application, I will show how to determine the Betti
numbers and the cup products of real toric manifolds.
This is joint work with Alvise Trevisan.

10:3011:00   Coffee Break 
11:0012:00 
Simona Settepanella 
Vanishing results for the cohomology of complex toric hyperplane complements 

Suppose ℛ is the complement of an essential arrangement of toric hyperlanes in the complex torus (ℂ^*)^n
and π=π_1(ℛ).
Then H^*(ℛ;A) vanishes except in the top degree n when A is one of the following systems of local coefficients:
(a) a system of nonresonant coefficients in a complex line bundle,
(b) the von Neumann algebra ℕ π, or
(c) the group ring Z π.
In case (a) the dimension of H^n is e(ℛ) where e(ℛ) denotes the Euler characteristic, and in case (b) the
nth ℓ^2 Betti number is also e(ℛ).

12:0014:00   Lunch 
14:0015:00 
David Mond 
Partial normalisations and weak normalisations of free divisors. 

If D is a free divisor then its Jacobian ideal defines a codimension 2 CohenMacaulay scheme in the ambient space.
If the radical of the Jacobian ideal also defines a CohenMacaulay scheme then the endomorphisms of this ideal form a
ring whose spec, \tilde D, is a partial normalisation of the divisor. This procedure is purely local, so one can say
something about \tilde D by looking at generic points on D_sing. But a general description of \tilde D seems hard to find.
The talk discusses some examples. This is work in progress with Michele Torielli.
