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Series: Analysis Seminar

An overarching problem in matrix weighted theory is the so-called A2 conjecture, namely the question of whether the norm of a Calderón-Zygmund operator acting on a matrix weighted L2 space depends linearly on the A2 characteristic of the weight. In this talk, I will discuss the history of this problem and provide a survey of recent results with an emphasis on the challenges that arise within the setup.

Series: Job Candidate Talk

I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the "method of interlacing polynomials") and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.

Series: CDSNS Colloquium

A trajectory is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus for which F has the form F(t) = t + rho (mod 1) for all points in the torus, and for some rho in the torus. There is an extensive literature on determining the coordinates of the vector rho, called the rotation numbers of F. However, even in the one-dimensional case there has been no general method for computing the vector rho given only the trajectory (u_n), though there are plenty of special cases. I will present a computational method called the Embedding Continuation Method for computing some components of r from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. There is however a caveat; the coordinates of the rotation vector depend on the choice of coordinates of the torus. I will give a statement of the various sets of possible rotation numbers that rho can yield. I will illustrate these ideas with one- and two-dimensional examples.

Series: Combinatorics Seminar

This is Lecture 3 of a series of 3 lectures. See the abstract on Tuesday's ACO colloquium of this week.

Series: Math Physics Seminar

TBA

Friday, March 2, 2018 - 14:00 ,
Location: Skiles 006 ,
Jen Hom ,
Georgia Tech ,
Organizer: Jennifer Hom

Series: School of Mathematics Colloquium

The regularity properties of solutions to linear partial differential equations in domains depend on the structure of the equation, the degree of smoothness of the coefficients of the equation, and of the boundary of the domain. Quantifying this dependence is a classical problem, and modern techniques can answer some of these questions with remarkable precision. For both physical and theoretical reasons, it is important to consider partial differential equations with non-smooth coefficients. We’ll discuss how some classical tools in harmonic and complex analysis have played a central role in answering questions in this subject at the interface of harmonic analysis and PDE.

Series: Graph Theory Seminar

This is Lecture 2 of a series of 3 lectures by the speaker. See the abstract on Tuesday's ACO colloquium of this week.

Series: Graph Theory Seminar

For a graph G, a set of subtrees of G are edge-independent with
root r ∈ V(G) if, for every vertex v ∈ V(G), the paths between v and r
in each tree are edge-disjoint. A set of k such trees represent a set of
redundant
broadcasts from r which can withstand k-1 edge failures. It is easy to
see that k-edge-connectivity is a necessary condition for the existence
of a set of k edge-independent spanning trees for all possible roots.
Itai and Rodeh have conjectured that this condition
is also sufficient. This had previously been proven for k=2, 3. We
prove the case k=4 using a decomposition of the graph similar to an ear
decomposition. Joint work with Robin Thomas.

Series: Analysis Seminar

Joint with Guth and Li, recently we showed that the solution to the free Schroedinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space H^s(R^2) with s>1/3. This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schroedinger maximal functions, which have some similar flavor as the Fourier restriction estimates. In this talk, I'll first show how to reduce the original problem in three dimensions to an essentially two dimensional one, via polynomial partitioning method. Then we'll see that the reduced problem asks how to control the size of the solution on a sparse and spread-out set, and it can be solved by refined Strichartz estimates derived from l^2 decoupling theorem and induction on scales.