Seminars and Colloquia by Series

Hydrodynamic limit of vortices in Ginzburg-Landau theory

Series
PDE Seminar
Time
Tuesday, September 30, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Spirn – University of Minnesota
Vortices arise in many problems in condensed matter physics, including superconductivity, superfluids, and Bose-Einstein condensates. I will discuss some results on the behavior of two of these systems when there are asymptotically large numbers of vortices. The methods involve suitable renormalization of the energies both at the vortex cores and at infinity, along with a renormalization of the vortex density function.

Goodness-of-fit testing in the Ising Model

Series
Algebra Seminar
Time
Monday, September 29, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Abraham Martin del Campo – IST Austria
Markov bases have been developed in algebraic statistics for exact goodness-of-fit testing. They connect all elements in a fiber (given by the sufficient statistics) and allow building a Markov chain to approximate the distribution of a test statistic by its posterior distribution. However, finding a Markov basis is often computationally intractable. In addition, the number of Markov steps required for converging to the stationary distribution depends on the connectivity of the sampling space.In this joint work with Caroline Uhler and Sarah Cepeda, we compare different test statistics and study the combinatorial structure of the finite lattice Ising model. We propose a new method for exact goodness-of-fit testing. Our technique avoids computing a Markov basis but builds a Markov chain consisting only of simple moves (i.e. swaps of two interior sites). These simple moves might not be sufficient to create a connected Markov chain. We prove that when a bounded change in the sufficient statistics is allowed, the resulting Markov chain is connected. The proposed algorithm not only overcomes the computational burden of finding a Markov basis, but it might also lead to a better connectivity of the sampling space and hence a faster convergence.

Approximating Real Stability Radii

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 29, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Manuela Manetta – Georgia Tech Mathematics
The distance of a nxn stable matrix to the set of unstable matrices, the so-called distance to instability, is a well-known measure of linear dynamical system stability. Existing techniques compute this quantity accurately but the cost is of the order of multiple SVDs of order n, which makes the method suitable to middle size problems. A new approach is presented, based on Newton's iteration applied to pseudospectral abscissa, whose implementation is obtained by discretization on differential equation for low-rank matrices, particularly suited for large sparse matrices.

Nearly integrable systems with orbits accumulating to KAM tori

Series
CDSNS Colloquium
Time
Monday, September 29, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marcel Guardia – Univ. Polit. Catalunya
The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian system of n degrees of freedom on a typical energy surface has a dense orbit. This question is wide open. In this talk I will explain a recent result by V. Kaloshin and myself which can be seen as a weak form of the quasi-ergodic hypothesis. We prove that a dense set of perturbations of integrable Hamiltonian systems of two and a half degrees of freedom possess orbits which accumulate in sets of positive measure. In particular, they accumulate in prescribed sets of KAM tori.

How quantum theory and statistical mechanics gave a polynomial of knots

Series
Stelson Lecture Series
Time
Thursday, September 25, 2014 - 16:35 for 1 hour (actually 50 minutes)
Location
Clary Theater, Student Success Center
Speaker
Vaughan Jones – University of Vanderbilt –
We will see how a result in von Neumann algebras (a theory developed by von Neumann to give themathematical framework for quantum physics) gave rise, rather serendipitously, to an elementary but very usefulinvariant in the theory of ordinary knots in threel space. Then we'll look at some subsequent developments of the theory, and talk about a thorny problem which remains open.

Second order free CLT

Series
Stochastics Seminar
Time
Thursday, September 25, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ionel Popescu – Georgia Tech
The CLT for free random variables was settled by Voiculescu very early in this work on free probability. He used this in turn to prove his main result on aymptotic freeness of independent random matrices. On the other hand, in random matrices, fluctuations can be understood as a second order phenomena. This notion of fluctuations has a conterpart in free probability which is called freenes of second order. I will explain what this is and how one can prove a free CLT result in this context. It is also interesting to point out that this is a nontrivial calculation which begs the same question in the classical context and I will comment on that.

Tropical K_4 curves

Series
Algebra Seminar
Time
Wednesday, September 24, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Melody Chan – Harvard University
This is joint work with Pakwut Jiradilok. Let X be a smooth, proper curve of genus 3 over a complete and algebraically closed nonarchimedean field. We say X is a K_4-curve if the nonarchimedean skeleton G of X is a metric K_4, i.e. a complete graph on 4 vertices.We prove that X is a K_4-curve if and only if X has an embedding in p^2 whose tropicalization has a strong deformation retract to a metric K_4. We then use such an embedding to show that the 28 odd theta characteristics of X are sent to the seven odd theta characteristics of g in seven groups of four. We give an example of the 28 bitangents of a honeycomb plane quartic, computed over the field C{{t}}, which shows that in general the 4 bitangents in a given group need not have the same tropicalizations.

The Gaussian Radon Transform for Infinite-Dimensional Banach Spaces

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, September 23, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Irina Holmes – School of Mathematics, Georgia Tech
In this talk we construct an infinite-dimensional, stochastic version of the Radon transform. We work within the framework of abstract Wiener spaces, introduced by Leonard Gross. We present some basic properties of this transform, as well as compute some specific examples on the classical Wiener space.

Artin fans in tropical geometry

Series
Algebra Seminar
Time
Monday, September 22, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Martin Ulirsch – Brown University
Recent work by J. and N. Giansiracusa, myself, and O. Lorscheid suggests that the tropical geometry of a toric variety $X$, or more generally of a logarithmic scheme $X$, can be formalized as a "Berkovich analytification" of a scheme over the field $\mathbb{F}_1$ with one element that is canonically associated to $X$.The goal of this talk is to introduce the theory of Artin fans, originally due to D. Abramovich and J. Wise, which can be used to lift rather unwieldy $\mathbb{F}_1$-geometric objects to the more familiar realm of algebraic stacks. Artin fans are \'etale locally isomorphic to quotient stacks of toric varieties by their big tori and their glueing data has a completely combinatorial description in terms of Kato fans.I am going to explain how to use the ideas surrounding the notion of Artin fans to study tropicalization maps associated to toric varieties and logarithmic schemes. Surprisingly these techniques allow us to give a reinterpretation of Tevelev's theory of tropical compactifications that can be generalized to compactifications of subvarieties in logarithmically smooth compactifcations of smooth varieties. For example, we can introduce definitions of tropical pairs and schoen varieties in terms of Artin fans that are equivalent to Tevelev's notions.

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