Avraham Aizenbud, Weizmann
Title: What happens when you push measures using polynomial maps – a dictionary between algebraic geometry and analysis.
Abstract:
We will discuss the following question: given a polynomial map between two affine spaces V and W, what can be said about the collection of measures on W obtained as the pushforward of smooth, compactly supported measures?
We will discuss two perspectives on this question:
(1) what happens to individual measures, and
(2) what happens to the collection.
The talk is based on several joint works with Nir Avni and Shachar Carmeli.
Joseph Bernstein, Tel Aviv University
Title: Regular Densities and Differential Operators on Singular Varieties.
Abstract:
Let X be an irreducible complex algebraic variety.
I claim that there is a natural (canonical) local construction that associates to X a coherent OX-module ωX that I call the sheaf of Regular Densities on X.
On the open subset Xsm of smooth points of X this sheaf is just the sheaf of densities = volume forms = top degree differential forms. So my claim
is that this familiar sheaf of densities on the subset Xsm has a canonical extension to the whole variety X.
This construction is non-trivial and is based on the theory of D-modules.
In my talk I will describe this construction and discuss some very nice
properties of the sheaf ωX. I will also discuss some generalizations of this construction.
In particular, I will discuss its application to the following general
question: What is a ”correct” notion of a differential operator on a
singular variety X.
Michael Bialy, Tel Aviv University
Title: Rigidity of Mather $beta$-function for convex billiards.
Abstract:
I shall discuss an isoperimetric type inequality for Mather $beta$-function for convex billiards. It was recently found in a joint work with Stefano Baranzini and Alfonso Sorrentino. In particular we reprove the inequalities Length and Area of extremal polygons found by R. Schneider and E. Sas in the previous century.
Leonid Polterovich, Tel Aviv University
Title: The mathematics of egg-beaters.
Abstract:
The study of area-preserving maps on surfaces sits at the crossroads of dynamics, algebra, and geometry. We'll explore the group of such maps from a geometric viewpoint and discuss the contrast between maps with simple, predictable behavior and chaotic ones—'egg-beaters'. Joint with Lev Buhovsky, Ben Feuerstein, and Egor Shelukhin.
Lev Radzivilovsky, Weizmann
Title: Some three-dimensional problems presented with animations.
Abstract:
I plan to discuss two elementary geometric problems using three-dimensional GeoGebra animations.
The first problem is one of the Boris's favourite problems about the largest section of a tetrahedron.
The second problem is the problem about parabolas embracing circles (by Omri Solan and Yoav Krauz), which was the "green T-shirt problem" of Israeli math Olympiad 11 years ago. A parabola embraces a circle, if they have two tangent points. Given three circles, and three parabolas embracing different pairs of circles, the claim is that there is a line which is tangent to all three parabolas. The proof of that claim is 3-dimensional. The theorem was further generalized by Yaron Brodsky, Omri Peer and Omri Solan (one of generalizations has a 3-dimensional proof, another has a 4-dimensional proof).