Week of February 16, 2020

Tue Feb 18, 2020
11:00am - 340P - Combinatorics and Probability
Michael Krivelevich - (Tel Aviv University)
Embedding large minors in weak expanders and in sparse random graphs

A graph G on n vertices is called an alpha-expander if the external neighborhood of every vertex subset U of size |U|<=n/2 in G has size at least alpha*|U|.

Extending and improving the results of Plotkin, Rao and Smith, and of Kleinberg and Rubinfeld from the 90s, we prove that for every alpha>0, an alpha-expander G on n vertices contains every graph H with at most cn/log n vertices and edges as a minor, for c=c(alpha)>0.

Alternatively, every n-vertex graph G without sublinear separators contains all graphs with cn/logn vertices and edges as minors.

Consequently, a supercritical random graph G(n,(1+epsilon)/n) is typically minor-universal for the class of graphs with cn/log n vertices and edges.

The order of magnitude n/log n in the above results is optimal.

 

A joint work with Rajko Nenadov.

2:00pm to 2:50pm - RH 510R - Working Group in Information Theory
Amirhossein Taghvaei - (UC Irvine)
Local behavior of divergence

This week, we will discuss Section 4.2-4.3 of the lecture notes of Wu and Polyanski: 
http://people.lids.mit.edu/yp/homepage/papers.html

Working Group in Information Theory is a self-educational project in the department. Techniques based on information theory have become essential in high-dimensional probability, theoretical computer science and statistical learning theory. On the other hand, information theory is not taught systematically. The goal of this group is to close this gap.

3:00pm to 3:50pm - RH 306 - Analysis
Min Ru - (Houston University)
The geometric and arithmetic aspect of algebraic varieties

In this talk, I will first discuss some geometric properties of the algebraic varieties which 

are related to (and determine) their arithmetic properties (in particular the finiteness of the number of 

the rational points).  I will then state and discussion my recent result with Paul Vojta, as well as its application.

4:00pm - RH 306 - Differential Geometry
Daniel Stern - (University of Toronto)
Scalar curvature and harmonic functions

We'll discuss a new technique for relating scalar curvature bounds to the global structure of 3-dimensional manifolds, exploiting a relationship between the scalar curvature and the topology of level sets of harmonic functions. We will describe several geometric applications in both the compact and asymptotically flat settings, including a simple and effective new proof (joint with Bray, Kazaras, and Khuri) of the three-dimensional Riemannian positive mass theorem.

 

Wed Feb 19, 2020
3:00pm to 3:50pm - 340P - Harmonic Analysis
Ioan Bejenaru - (UCSD)
The multilinear restriction estimates for hypersurfaces with curvature

Abstract: I will present an overview of the multilinear restriction theory, with an emphasis on the case when the hypersurfaces have some curvature. I will discuss a new result: the case of n-1 hypersurfaces in n dimensions where a fairly general theory is developed. 

Thu Feb 20, 2020
2:00pm to 3:00pm - RH 340P - Differential Geometry
Zuoqin Wang - (USTC and MIT)
Semi-classical isotropic functions and applications

Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.

2:00pm to 3:00pm - 340P Rowland Hall - Differential Geometry
Zuoqin Wang - (USTC and MIT)
Semi-classical isotropic functions and applications

Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.
 

3:00pm - RH 306 - Algebra
Tomasz Prytuła - (Technical University of Denmark)
Commensurators of abelian subgroups in CAT(0) groups

This talk will introduce work in the area of Geometric Group Theory; no prior background in this area will be assumed. The commensurator of a subgroup H of a group G may be seen as a coarse approximation of the normalizer of H. We consider the situation where H is free abelian and G acts properly on a CAT(0) space, that is, a simply connected space of metric non-positive curvature. The structure of the normalizer of H and its action on the space are well understood in this context. However, the commensurator is more mysterious and it contains subtle information about the action which is not seen by the normalizer. For various classes of CAT(0) spaces we obtain structural results about the commensurator and its relation to the normalizer. In this talk, first I will give background on the commensurator and on CAT(0) spaces and groups, and then I will discuss various geometric tools and constructions used in our approach. This is joint work with Jingyin Huang.

3:00pm - RH 306 - Number Theory
Tomasz Prytuła - (Technical University of Denmark)
Commensurators of abelian subgroups in CAT(0) groups

This talk will introduce work in the area of Geometric Group Theory; no prior background in this area will be assumed. The commensurator of a subgroup H of a group G may be seen as a coarse approximation of the normalizer of H. We consider the situation where H is free abelian and G acts properly on a CAT(0) space, that is, a simply connected space of metric non-positive curvature. The structure of the normalizer of H and its action on the space are well understood in this context. However, the commensurator is more mysterious and it contains subtle information about the action which is not seen by the normalizer. For various classes of CAT(0) spaces we obtain structural results about the commensurator and its relation to the normalizer. In this talk, first I will give background on the commensurator and on CAT(0) spaces and groups, and then I will discuss various geometric tools and constructions used in our approach. This is joint work with Jingyin Huang.

4:00pm to 5:00pm - RH306 - Colloquium
Susan Friedlander - (USC)
Kolmogorov, Onsager and a stochastic model for turbulence

We will briefly review Kolmogorov’s (41) theory of homogeneous turbulence and Onsager’s ( 49)
conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of
vanishing viscosity. Although over the past 60 years there is a vast body of literature related to this subject, at present
there is no rigorous mathematical proof that the solutions to the Navier-Stokes equations yield
Kolmogorov’s laws. For this reason various models have been introduced that are more tractable but capture
some of the essential features of the Navier-Stokes equations themselves. We will discuss one such
stochastically driven dyadic model for turbulent energy cascades. We will describe how results for stochastic PDEs
can be used to prove that this dyadic model is consistent with Kolmogorov’s theory and Onsager’s conjecture.

This is joint work with Vlad Vicol and Nathan Glatt-Holtz.