Abstract: We will discuss connections between Gromov's work on isoperimetry of waists and Milman's work on the M-ellipsoid of a convex body. It is proven that any convex body K in an n-dimensional Euclidean space has a linear image K_1 of volume one satisfying the following waist inequality: Any continuous map f from K_1 to R^d has a fiber f^{-1}(t) whose (n-d)-dimensional volume is at least c^{n-d}, where c > 0 is a universal constant. Already in the case where f is linear, this constitutes a slight improvement over known results.
Manchester Building (Hall 2), Hebrew University Jerusalem
Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From a computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc.?
*** Please note the LOCATION ***
We shall give a simple generalization of commutative rings. The
category GR of such generalized rings contains ordinary commutative
rings (fully, faithfully), but also the "integers" and the "residue
field" at a real or complex place of a number field ; the "field with
one element" F1 (the initial object of GR) ; the "Arithmetical
Surface" (the categorical sum of the integers Z with them self). We
shall show this geometry sees the real and complex places of a number
Title: Spectral approach to the chromatic number of a simplicial complex
Abstract: In this talk, we'll summarize results obtained in recent years in a pursuit for spectral bounds for the chromatic number of a simplicial complex. As the principal application, we'll show that Ramanujan complexes serve as family of explicitly constructed complexes with large girth and large chromatic number. We'll also present other results, such as a bound on the expansion and a bound on the mixing of a complex, and refer to open questions.
Manchester Building (Hall 2), Hebrew University Jerusalem
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes are Riemann, Newton, Poincare, von Neumann, Kato, Loewner, Krein and Noether. This talk is in two parts. The second part will be given from 4:00 to 5:00 (not 5:30) in the Basic Notions seminar.
Manchester Building (Hall 2), Hebrew University Jerusalem
A finite set has an interesting numerical invariant - its cardinality. There are two natural generalizations of "cardinality" to a (homotopy) invariant for (suitably finite) spaces. One is the classical Euler characteristic. The other is the Baez-Dolan "homotopy cardinality". These two invariants, both natural from a certain perspective, seem to be very different from each other yet mysteriously connected. The question of the precise relation between them was popularized by John Baez as one of the "mysteries of counting".
Title: Rapid expansion in finite simple groups
Abstract: We show that small normal subsets $A$ of finite simple groups expand
very rapidly -- namely, $|A^2| \ge |A|^{2-\epsilon}$, where $\epsilon >0$ is
arbitrarily small.
Joint work with M. W. Liebeck and A. Shalev
We show that the generation problem in Thompson group F is decidable, i.e., there is an algorithm which decides if a finite set of elements of F generates the whole F. The algorithm makes use of the Stallings 2-core of subgroups of F, which can be defined in an analogue way to the Stallings core of subgroups of a free group. An application of the algorithm shows that F is a cyclic extension of a group K which has a maximal elementary amenable subgroup B. The group B is a copy of a subgroup of F constructed by Brin.
