The seminar meets on Tuesdays, 14:30-15:30, in Math -101

This Week

Chloé Perin (Hebrew University of Jerusalem)

First-order logic on the free group and geometry

We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions such as forking independence, etc. It turns out that techniques from geometric group theory are very useful to tackle such problems. Some of these questions have been answered, others are still open - our aim is to give a feel for the techniques and directions of this field.
We will assume no special knowledge of model theory.

We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions such as forking independence, etc. It turns out that techniques from geometric group theory are very useful to tackle such problems. Some of these questions have been answered, others are still open - our aim is to give a feel for the techniques and directions of this field.
We will assume no special knowledge of model theory.

Adam Sheffer (California Institute of Technology (Caltech))

While the topic of geometric incidences has existed for
several decades, in recent years it has been experiencing a
renaissance due to the introduction of new polynomial methods. This
progress involves a variety of new results and techniques, and also
interactions with fields such as algebraic geometry and harmonic
analysis.

A simple example of an incidences problem: Given a set of n points and
set of n lines, both in R^2, what is the maximum number of point-line
pairs such that the point is on the line. Studying incidence problems
often involves the uncovering of hidden structure and symmetries.

In this talk we introduce and survey the topic of geometric
incidences, focusing on the recent polynomial techniques and results
(some by the speaker). We will see how various algebraic and analysis
tools can be used to solve such combinatorial problems.

I will discuss how Hodge theory, and positivity phenomena from algebraic geometry in general, can be used to resolve fundamental conjectures in combinatorics, including Rotas conjecture for log-concavity of Whitney numbers and beyond. I will also discuss how combinatorics can in turn be used to explain and prove such phenomena, such as the Hodge-Riemann relations for matroids.

I will describe the theory of hyperbolic flows on three manifolds, and then describe a new approach to chaotic flows using knot theory, allowing for topological analysis of singular flows. I’ll use this to show that, surprisingly, the famous Lorenz flow on R^3 can be related to the geodesic flow on the modular surface. When changing the parameters, we also find a new type of topological phases in the Lorenz system.
This will be an introductory talk.

This is joint work with Michael Magee.
Since the 1970’s, Physicists and Mathematicians who study random matrices in the standard models of GUE or GOE,
are aware of intriguing connections between integrals of such random matrices and the enumeration of graphs on surfaces.
We establish a new aspect of this theory: for random matrices sampled from the group U(n) of Unitary matrices. The group structure of these matrices allows us to go further and find surprising algebraic quantities hidden in the values of these integrals.
The talk will be aimed at graduate students, and all notions will be explained.