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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 ,
ndexter@utk.edu ,
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.

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: Combinatorics Seminar

This is Lecture 3 of a series of 3 lectures. See the abstract on Tuesday's ACO colloquium of this week.(Please note that this lecture will be 80 minutes' long.)

Series: Math Physics Seminar

Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).Pipe, channel and plane flows, however, are flows on infinite spatial domains. We propose to recast the Navier-Stokes equations as a space-time theory, with the unstable invariant solutions now being the space-time tori (and not the 1-dimensional periodic orbits of the classical periodic orbit theory). The symbolic dynamics is likewise higher-dimensional (rather than a single temporal string of symbols). In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.We illustrate the strategy by solving a very simple classical field theory on a lattice modelling many-particle quantum chaos, adiscretized screened Poisson equation, or the ``spatiotemporal cat.'' No actual cats, graduate or undergraduate, have showninterest in, or were harmed during this research.

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.

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.

Friday, March 2, 2018 - 10:00 ,
Location: Skiles 254 ,
Marcel Celaya ,
Georgia Tech ,
mcelaya@gatech.edu ,
Organizer: Kisun Lee

In this talk we will discuss the paper of Adiprasito, Huh, and Katz titled "Hodge Theory for Combinatorial Geometries," which establishes the log-concavity of the characteristic polynomial of a matroid.

Series: Stelson Lecture Series

How is it possible to send encrypted information across an insecure channel (like the internet) so that only the intended recipient can decode it, without sharing the secret key in advance? In 1976, well before this question arose, a new mathematical theory of encryption (public-key cryptography) was invented by Diffie and Hellman, which made digital commerce and finance possible. The technology advances of the last twenty years bring new and urgent problems, including the need to compute on encrypted data in the cloud and to have cryptography that can withstand the speed-ups of quantum computers. In this lecture, we will discuss some of the history of cryptography, as well as some of the latest ideas in "lattice" cryptography which appear to be quantum resistant and efficient.