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
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.
Friday, March 2, 2018 - 15:05 , Location: Skiles 271 , Adrian P. Bustamante , Georgia Tech , Organizer:
Given a one-parameter family of maps of an interval to itself, one can observe period doubling bifurcations as the parameter is varied. The aspects of those bifurcations which are independent of the choice of a particular one-parameter family are called universal. In this talk we will introduce, heuristically, the so-called Feigenbaun universality and then we'll expose some rigorous results about it.
Series: Math Physics Seminar
Series: Combinatorics Seminar
This is Lecture 3 of a series of 3 lectures. See the abstract on Tuesday's ACO colloquium of this week.
Friday, March 2, 2018 - 14:00 , Location: Skiles 006 , Jen Hom , Georgia Tech , Organizer: Jennifer Hom
In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module. No prior knowledge of the Alexander module or Heegaard Floer homology will be assumed.
Friday, March 2, 2018 - 11:00 , Location: Skiles 006 , Jill Pipher , Brown University , Organizer: Mayya Zhilova
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.