Seminars and Colloquia by Series

Embeddings of manifolds and contact manifolds I

Series
Geometry Topology Working Seminar
Time
Friday, October 10, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech
This is the first of several talks disussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attentaion will be payed to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

Finite generation of symmetric toric ideals

Series
ACO Student Seminar
Time
Friday, October 10, 2014 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Krone – Georgia Tech –
Given a family of ideals which are symmetric under some group action on the variables, a natural question to ask is whether the generating set stabilizes up to symmetry as the number of variables tends to infinity. We answer this in the affirmative for a broad class of toric ideals, settling several open questions coming from algebraic statistics. Our approach involves factoring an equivariant monomial map into a part for which we have an explicit degree bound of the kernel, and a part for which we canprove that the source, a so-called matching monoid, is equivariantly Noetherian. The proof is mostly combinatorial, making use of the theory of well-partial orders and its relationship to Noetherianity of monoid rings. Joint work with Jan Draisma, Rob Eggermont, and Anton Leykin.

Nonlinear Dispersive Equations: A panoramic survey II

Series
PDE Working Seminar
Time
Thursday, October 9, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zaher Hani – GeorgiaTech
Nonlinear dispersive and wave equations constitute an area of PDE that has witnessed tremendous activity over the past thirty years. Such equations mostly orginate from physics; examples include nonlinear Schroedinger, wave, Klein-Gordon, and water wave equations, as well as Einstein's equations in general relativity. The rapid developments in this theory were, to a large extent, driven by several successful interactions with other areas of mathematics, mainly harmonic analysis, but also geometry, mathematical physics, probability, and even analytic number theory (we will touch on this in another talk). This led to many elegant tools and rather beautiful mathematical arguments. We will try to give a panoramic, yet very selective, survey of this rich topic focusing on intuition rather than technicalities. In this second talk, we continue discussing some aspects of nonlinear dispersive equations posed on Euclidean spaces.

The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-body Evolution

Series
PDE Seminar
Time
Tuesday, October 7, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xuwen Chen – Brown University
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to infinity. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

Natural Selection, Game Theory and Genetic Diversity

Series
Combinatorics Seminar
Time
Tuesday, October 7, 2014 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Georgios Piliouras – Cal Tech

Please Note: Bio: Georgios Piliouras is a postdoc at Caltech, Center for Mathematics and Computation. He received his PhD in Computer Science from Cornell University and has been a Georgia Tech postdoc at the EE department.

In a recent series of papers a strong connection has been established between standard models of sexual evolution in mathematical biology and Multiplicative Weights Updates Algorithm, a ubiquitous model of online learning and optimization. These papers show that mathematical models of biological evolution are tantamount to applying discrete replicator dynamics, a close variant of MWUA on coordination games. We show that in the case of coordination games, under minimal genericity assumptions, discrete replicator dynamics converge to pure Nash equilibria for all but a zero measure of initial conditions. This result holds despite the fact that mixed Nash equilibria can be exponentially (or even uncountably) many, completely dominating in number the set of pure Nash equilibria. Thus, in haploid organisms the long term preservation of genetic diversity needs to be safeguarded by other evolutionary mechanisms, such as mutation and speciation. This is joint work with Ruta Mehta and Ioannis Panageas.

Economics for tropical geometer

Series
Algebra Seminar
Time
Tuesday, October 7, 2014 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ngoc Mai Tran – UT Austin
This talk surveys the connection between economics and tropical geometry, as developed in the paper of Baldwin and Klemperer (Tropical Geometry to Analyse Demand). I will focus on translating concepts, theorems and questions in economics to tropical geometry terms.

Enumerating Polytropes

Series
Algebra Seminar
Time
Monday, October 6, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ngoc Mai Tran – UT Austin
Polytropes are both ordinary and tropical polytopes. Tropical types of polytropes in \R^n are in bijection with certain cones of a specific Gr\"obner fan in \R^{n^2-n}. Unfortunately, even for n = 5 the entire fan is too large to be computed by existing software. We show that the polytrope cones can be decomposed as the cones from the refinement of two fans, intersecting with a specific cone. This allows us to enumerate types of full-dimensional polytropes for $n = 4$, and maximal polytropes for $n = 5$ and $n = 6$. In this talk, I will prove the above result and describe the key difficulty in higher dimensions.

Some contact embeddings to the standard 5-sphere

Series
Geometry Topology Seminar
Time
Monday, October 6, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ryo Furukawa – University of Tokyo
In this talk we consider the contact embeddings of contact 3-manifolds to S^5 with the standard contact structure.Every closed 3-manifold can be embedded to S^5 smoothly by Wall's theorem. The only known necessary condition to a contact embedding to the standard S^5 is the triviality of the Euler class of the contact structure. On the other hand there are not so much examples of contact embeddings.I will explain the systematic construction of contact embeddings of some contact structures (containing non Stein fillable ones) on torus bundles and Lens spaces.If time permits I will explain relation between above construction and some polynomials on \mathbb C^3.

An Alternating Direction Approximate Newton Algorithm for Ill-conditioned inverse Problems with Application to Parallel MRI

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 6, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Maryam Yashtini – Georgia Tech Mathematics
An alternating direction approximate Newton method (ADAN) is developedfor solving inverse problems of the form$\min \{\phi(Bu) +1/2\norm{Au-f}_2^2\}$,where $\phi$ is a convex function, possibly nonsmooth,and $A$ and $B$ are matrices.Problems of this form arise in image reconstruction where$A$ is the matrix describing the imaging device, $f$ is themeasured data, $\phi$ is a regularization term, and $B$ is aderivative operator. The proposed algorithm is designed tohandle applications where $A$ is a large, dense ill conditionmatrix. The algorithm is based on the alternating directionmethod of multipliers (ADMM) and an approximation to Newton's method in which Newton's Hessian is replaced by a Barzilai-Borwein approximation. It is shown that ADAN converges to a solutionof the inverse problem; neither a line search nor an estimateof problem parameters, such as a Lipschitz constant, are required.Numerical results are provided using test problems fromparallel magnetic resonance imaging (PMRI).ADAN performed better than the other schemes that were tested.

On the growth of local intersection multiplicities in holomorphic dynamics

Series
CDSNS Colloquium
Time
Monday, October 6, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
William Gignac – School of Mathematics Georgia Inst. Technology
In this talk, we will discuss a question posed by Vladimir Arnold some twenty years ago, in a subject he called "dynamics of intersections." In the simplest setting, the question is the following: given a (discrete time) holomorphic dynamical system on a complex manifold X and two holomorphic curves C and D in X which pass through a fixed point P of the system, how quickly can the local intersection multiplicies at P of C with the iterates of D grow in time? Questions like this arise naturally, for instance, when trying to count the periodic points of a dynamical system. Arnold conjectured that this sequence of intersection multiplicities can grow at most exponentially fast, and in fact we can show this conjecture is true if the curves are chosen to be suitably generic. However, as we will see, for some (even very simple) dynamical systems one can choose curves so that the intersection multiplicities grow as fast as desired. We will see how to construct such counterexamples to Arnold's conjecture, using geometric ideas going back to work of Yoshikazu Yamagishi.

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