Sasha Barvinok: Wild convex bodies

"I would talk about the cone of non-negative multivariate forms

and other such families of convex sets whose precise structure "cannot

be known" because of the computational complexity reasons.

Depending on how much time I have, I could start with the cones

of positive semidefinite matrices and non-negative univariate

polynomials whose structure is known but maybe not as widely

as it deserves.

"

Schubert varieties are a class of algebraic projective

varieties associated to permutations. There follows that

all the geometric properties of these varieties (such as,

for example, dimension, inclusion, smoothness, singularities,

homology, cohomology, intersection (co)homology, etc...) depend

only on the starting permutation. In these lectures I will first

give an elementary (linear algebra) definition of Schubert

varieties, and then I will explain how various geometric

properties are encoded combinatorially in, and can be computed

from, the permutation. These encodings give rise to some beautiful,

and usually challenging, purely combinatorial questions,

many of which are open and have been the subject of intense

recent research. If time permits I will also mention, at the end,

other classes of algebraic varieties which are similarly encoded

by combinatorial objects. The prerequisites for these lectures are

linear algebra and elementary combinatorics.

Generalized associahedra and cluster algebras Cluster algebras are a class of commutative rings that we introduced three years ago in an attempt to design an algebraic framework for the dual canonical bases in quantum groups and their representations. The main result of the theory developed so far is the classification of cluster algebras of finite type, which, for reasons mysterious at the moment, turns out to be yet another instance of the famous Cartan-Killing classification. The underlying combinatorial structure of a cluster algebra of finite type is captured by its "cluster complex". We identify this simplicial complex as the dual complex of the generalized associahedron, a beautiful spherical cell complex (indeed, a convex polytope) associated with the corresponding root system. Time permitting, I will discuss connections with other topics, such as algebraic Y-systems, the Laurent phenomenon, realizations of cluster algebras as coordinate rings, Catalan numerology of arbitrary Lie type, and total positivity.

Abstract: I will discuss some fixed point theorems and techniques

for groups acting on acyclic complexes. This will include earlier works

of myself and Aschbacher-Segev, and I will focus on joint work with Bob

Oliver which should appear in the latest issue of Acta. Math. (see:

http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Segev/2dim).

The work with Oliver classifies those finite groups that can act without

fixed points on a 2-dimensional Z-acyclic (CW-) complex. See also the

presentation of this work in the Bourbaki seminar by A. Adem

(http://www.math.wisc.edu/~adem/aa-bourbaki.pdf).

Maria Chudnovsky: The Strong Perfect Graph Theorem Abstract: A graph is called perfect if for every induced subgraph the size of its largest clique equals the minimum number of colors needed to color its vertices. In 1960's Claude Berge made a conjecture that has become one of the most well-known open problems in graph theory: any graph that contains no induced odd cycles of length greater than three or their complements is perfect. This conjecture is known as the Strong Perfect Graph Conjecture. We call graphs containing no induced odd cycles of length greater than three or their complements Berge graphs. A stronger conjecture was made recently by Conforti, Cornuejols and Vuskovic that any Berge graph either belongs to one of a few well understood basic classes or has a decomposition that can not occur in a minimal counterexample to Berge's Conjecture. In joint work with Neil Robertson, Paul Seymour and Robin Thomas we were able to prove this conjecture and consequently the Strong Perfect Graph Theorem.

Emmanuel Farjoun: The Fundamental group of limits of simplicial complexes. Known result about group actions on trees and complexes can be derived from a old-new approach to the fundamental groupoid of limits. This gives also a combinatorial defintion for homotopy limits of complexes and groups.

Vitali Milman: How far is a convex body from an ellipsoid? We study convex sets in high dimensional spaces, and discuss various asymptotic phenomena when the dimension tends to infinity. Many of the problems of this type studied recently were motivated by applying a complexity-theoretic point of view which influenced and corrected our intuition in several directions. This influence is mostly on an "ideological" level and motivates the type of questions asked, and is not on a technical level, but I believe that such influence is, perhaps, the most important part of a successful cooperation between different fields. I will try to convey this feeling, as well as explain some concrete examples of results which support it. The talk is aimed for a general mathematical audience and should hopefully be comprehensible for all.

Shmuel Weinberger - Combinatorics and Homology, Relations between combinatorics and homology, and especially L-p Homology, connection between a conjecture of Hopf in differential geometry and combinatorics and related topics will be discussed.