Oct 19

TBA

Departamental meeting


Oct 26

Integral geometry and valuation theory in pseudoRiemannian spaces

Dmitry Faifman (Tel Aviv University)

We will discuss the Blaschke branch of integral geometry and its manifestations in pseudoRiemannian space forms. First we will recall the fundamental notion of intrinsic volumes, known as quermassintegrals in convex geometry. Those notions were extended later to Riemannian manifolds by H. Weyl, who discovered a remarkable fact: given a manifold M embedded in Euclidean space, the volume of the epsilontube around it is an invariant of the Riemannian metric on M. We then discuss Alesker’s theory of smooth valuations, which provides a framework and a powerful toolset to study integral geometry, in particular in the presence of various symmetry groups.
Finally, we will use those ideas to explain some recent results in the integral geometry of pseudoRiemannian manifolds, in particular a collection of principal Crofton formulas in all space forms, and a ChernGaussBonnet formula for metrics of varying signature.
Partially based on joint works with S. Alesker, A. Bernig, G. Solanes.

Nov 2

Riemannian metrics on diffeomorphism groups — the good, the bad, and the unknown

Cy Maor (Hebrew University)

In finite dimensional Riemannian geometry, everything behaves nicely — the Riemannian metric induces a distance function, geodesics exist (at least for some time), and so on. In infinite dimensional Riemannian geometry, however, chaos reigns. In this talk I will focus on diffeomorphism groups, and on a particularly important hierarchy of Riemannian metrics on them: rightinvariant Sobolev metrics. These arise in many different contexts, from purely mathematical ones, to applications in hydrodynamics and imaging. I will give a brief introduction to these metrics, why we care about them, and what we know (and don’t know) about their properties.
Parts of the talk will be based on joint works with Bob Jerrard and Martin Bauer.

Nov 9

Order and disorder in multiscale substitution tilings

Yotam Smilansky (Rutgers University)

The study of aperiodic order and mathematical models of quasicrystals is concerned with ways in which disordered structures can nevertheless manifest aspects of order. In the talk I will describe examples such as the aperiodic Penrose and pinwheel tilings, together with several geometric, functional, dynamical and spectral properties that enable us to measure how far such constructions are from demonstrating latticelike behavior. A particular focus will be given to new results on multiscale substitution tilings, a class of tilings that was recently introduced jointly with Yaar Solomon.

Nov 16

Big Fiber Theorems and IdealValued Measures in Symplectic Topology

Yaniv Ganor (Technion)

In various areas of mathematics there exist “big fiber theorems”, these are theorems of the following type: “For any map in a certain class, there exists a ‘big’ fiber”, where the class of maps and the notion of size changes from case to case.
We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov’s notion of idealvalued measures.
We adapt the latter notion to the realm of symplectic topology, using an enhancement of a certain cohomology theory on symplectic manifolds introduced by Varolgunes, allowing us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.
Necessary preliminaries will be explained.
The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.

Nov 23

Randomness, genericity, and ubiquity of hyperbolic behavior in groups.

Ilya Gekhtman (Technion)

Consider an infinite group G acting by isometries on some metric space X.
How does a “typical” element act?
Consider a representation of G into some matrix group. What sort of matrix represents “typical” elements of G?
The answer depends on what we mean by the word “typical,” of which there are at least two reasonable notions. We may take a random walk on G and look where it lands after a large number of steps. We may also fix a generating set for G and look how large balls are distributed.
I will talk about how these two notions of genericity are related and how they differ, focusing on the setting of hyperbolic groups.
I will also explain that the following is true with respect to both notions:
For a group acting on a Gromov hyperbolic metric space typical elements act loxodromically, i.e. with northsouth dynamics.
For a representation of a large class of groups (including hyperbolic groups) into SL_n R, typical elements map to matrices whose eigenvalues are all simple and have distinct moduli.

Nov 30

Tiling the integers with translates of one tile: the CovenMeyerowitz tiling conditions for three prime factors

Itay Londner (UBC)

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered “standard” tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M.
In joint work with Izabella Laba (UBC), we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce some ingredients from the proof.
