Configuration Spaces: Geometry, Combinatorics and Topology

In collaboration with INdAM

2 May 2010 - 30 June 2010

Aims

NOTICE For all information about the activities, please select PLANNED ACTIVITIES.

AIMS AND RESEARCH DIRECTIONS

The importance of configuration spaces has been growing steadily in central areas of Mathematics often related to Theoretical Physics, hyperplane arrangements, and combinatorics.

Despite their simple definition, configuration spaces admit important, broad applications with deep classical ties to knot theory, homotopical algebra, the theory of operads as well as conformal field theory. The theory of configuration spaces also has links to the study of low dimensional topology, and combinatorics.

For example, these spaces have been central in Kontsevich's proof of deformation quantization.

Via the study of hyperplane arrangements and that of partition functions new interactions between splines, combinatorics and topology have developed with unexpected connections to equivariant K-theory.

The current time is a crucial moment to take advantage of the interaction between these communities.

Our goal is to bring together some of the leading experts to Pisa in these different, but significantly overlapping areas. Our intention is to inform on the current state of the art and to develop interaction among experts in order to foster new developments. With much recent progress, the time is ripe to exploit these deep connections and methods.

The subjects treated will include

I) Study of local systems on the complements of a hyperplane arrangements. Characteristic and resonance varieties.

II) Qualitative and quantitative problems related to the study of partition functions with applications to approximation theory.

III) Invariants of Braids and knots. Topological quantum field theory. Applications to low dimensional topology.

IV) Homotopy theory aspects of the study of configuration spaces.

V) Subspace arrangements. Their topology and combinatorics.

VI) Combinatorial aspects of the theory of Coxeter and Artin groups.

VII) Geometric group theory for certain choices of groups.

In view of these developments, we plan to have some mini-courses and seminars running over the two-month period, as well as two workshops.

I) Among minicourses, two at least will be on the following subjects:

(a) arrangements, and

(b) relations with the theory of splines and approximation theory.

II) Two weekly seminars.

III) Two workshops.

IV) In addition to senior people, we plan to provide support for a number of younger participants. This participation will expose younger mathematicians to connections outside of their main subjects, an opportunity rich in new ideas as well as unusual in these rapidly developing areas.