Abstracts or informal descriptions
of lecture series and a few other lectures:
Eric Babson : On f-vectors
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.
" Francesco Brenti: Combinatorics of Schubert varieties
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.
Sergei Fomin: Generalized associahedra and cluster algebras
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.
Yoav Segev: Finite groups acting on acyclic complexes
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.