On an extension of the Ito-Nisio theorem with applications to the continuity of Ito map and Levy processes
Series: Stochastics Seminar
We obtain an extension of the Ito-Nisio theorem to certain non separable Banach spaces and apply it to the continuity of the Ito map and Levy processes. The Ito map assigns a rough path input of an ODE to its solution (output). Continuity of this map usually requires strong, non separable, Banach space norms on the path space. We consider as an input to this map a series expansion a Levy process and study the mode of convergence of the corresponding series of outputs. The key to this approach is the validity of Ito-Nisio theorem in non separable Wiener spaces of certain functions of bounded p-variation. This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.
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: School of Mathematics Colloquium
The KLS conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) and present recent progress resulting in the current best bound, as well as a tight bound for the log-Sobolev constant (both with Yin Tat Lee). The conjecture has led to several techniques of general interest.
Wednesday, March 7, 2018 - 14:00 , Location: Atlanta , Agniva Roy , GaTech , Organizer: Anubhav Mukherjee
Three dimensional lens spaces L(p,q) are well known as the first examples of 3-manifolds that were not known by their homology or fundamental group alone. The complete classification of L(p,q), upto homeomorphism, was an important result, the first proof of which was given by Reidemeister in the 1930s. In the 1980s, a more topological proof was given by Bonahon and Hodgson. This talk will discuss two equivalent definitions of Lens spaces, some of their well known properties, and then sketch the idea of Bonahon and Hodgson's proof. Time permitting, we shall also see Bonahon's result about the mapping class group of L(p,q).
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.
Tuesday, March 6, 2018 - 15:00 , Location: skiles 005 , Marcel Guardia , Universitat Politècnica de Catalunya , Organizer: Livia Corsi
The restricted three body problem models the motion of a body of zero mass under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe Keplerian orbits. In 1922, Chazy conjectured that this model had oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Its existence was not proven until 1960 by Sitnikov in a extremely symmetric and carefully chosen configuration. In 1973, Moser related oscillatory motions to the existence of chaotic orbits given by a horseshoe and thus associated to certain transversal homoclinic points. Since then, there has been many atempts to generalize their result to more general settings in the restricted three body problem.In 1980, J. Llibre and C. Sim\'o, using Moser ideas, proved the existence of oscillatory motions for the restricted planar circular three body problem provided that the ratio between the masses of the two primaries was arbitrarily small. In this talk I will explain how to generalize their result to any value of the mass ratio. I will also explain how to generalize the result to the restricted planar elliptic three body problem. This is based on joint works with P. Martin, T. M. Seara. and L. Sabbagh.
Series: Combinatorics Seminar
How many triangles are needed to make the new graphs not look like random graphs? I am trying to answer this question. (The talk will be during 12:05-1:15pm; please note the room is *Skiles 256*)
Monday, March 5, 2018 - 13:55 , Location: Skiles 005 , Nick Dexter , University of Tennessee , email@example.com , Organizer: Wenjing Liao
We present and analyze a novel sparse polynomial approximation method for the solution of PDEs with stochastic and parametric inputs. Our approach treats the parameterized problem as a problem of joint-sparse signal reconstruction, i.e., the simultaneous reconstruction of a set of signals sharing a common sparsity pattern from a countable, possibly infinite, set of measurements. Combined with the standard measurement scheme developed for compressed sensing-based polynomial approximation, this approach allows for global approximations of the solution over both physical and parametric domains. In addition, we are able to show that, with minimal sample complexity, error estimates comparable to the best s-term approximation, in energy norms, are achievable, while requiring only a priori bounds on polynomial truncation error. We perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.
Monday, March 5, 2018 - 11:15 , Location: Skiles 005 , Prof. Evelyn Sander , George Mason University , Organizer: Molei Tao
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.