Seminars and Colloquia by Series

Differential equations for colored triangulations

Series
Combinatorics Seminar
Time
Wednesday, October 29, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Olivier Bernardi – Brandeis University
We will present the solution to a statistical mechanics model on random lattices. More precisely, we consider the Potts model on the set of planar triangulations (embedded planar graph such that every face has degree 3). The partition function of this model is the generating function of vertex-colored triangulations counted according to the number of monochromatic edges and dichromatic edges. We characterize this partition function by a simple system of differential equations. Some special cases, such as properly 4-colored triangulations, lead to particularly simple equations waiting for a more direct combinatorial explanation. This is joint work with Mireille Bousquet-Melou.

The Colored Jones Polynomial and the Volume Conjecture

Series
Geometry Topology Student Seminar
Time
Wednesday, October 29, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Paprocki – Georgia Tech

Please Note: This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.

We will present an introduction to the notion of quantum invariants of knots and links, and in particular the colored Jones polynomial. We will also introduce the Volume Conjecture, which relates a certain limiting behavior of a quantum invariant (the colored Jones polynomial of a link) with a classical invariant (the hyperbolic volume of the hyperbolic part of a link complement in S^3) and has been proven in a number of cases.

Invariants of embeddings and immersions via contact geometry

Series
Research Horizons Seminar
Time
Wednesday, October 29, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. John Etnyre – Georgia Tech Math Department
There is a beautiful idea that one can study spaces by studying associated geometric objects. More specifically one can associate to a manifold (that is some space) a symplectic or contact manifold (that is the geometric object). The question is how useful is this idea. We will discuss this idea and related questions for subspaces (that is immersions and embeddings) with a focus on curves in the plane and knots in three space. If time permits we will discuss powerful new tools from contact geometry that allow one use this idea to construct invariants of knots and more generally embeddings and immersions in any space.

Regularity of Solutions of Hamilton-Jacobi Equation on a Domain

Series
PDE Seminar
Time
Tuesday, October 28, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert Fathi – École Normale Supérieure de Lyon, France
In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p. Let D be a compact smooth domain with boundary $\partial D$ contained in $T^N \times ]0,+\infty[$ . We show that if U is differentiable at each point of $\partial D$, then this is also the case on the interior of D. There are several variants of this result in different settings. To make the result accessible to the layman, we will explain the method on the function distance to a closed subset of an Euclidean space. This example contains all the ideas of the general case.

Intuitive Dyadic Calculus

Series
Analysis Working Seminar
Time
Monday, October 27, 2014 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Robert Rahm – School of Math
We discuss an approach to dyadic lattices (and their applications to harmonic analysis) presented by Lerner and Nazarov in their manuscript, Intutive Dyadic Calculus.

On complexity of 3-manifolds/On coordinates on virtual braid groups

Series
Geometry Topology Seminar
Time
Monday, October 27, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Evgeny Fominykh and Andrei Vesnin – Chelyabinsk State University –
These are two half an hour talks.Evgeny's abstract: The most useful approach to a classication of 3-manifolds is the complexity theory foundedby S. Matveev. Unfortunately, exact values of complexity are known for few infinite seriesof 3-manifold only. We present the results on complexity for two infinite series of hyperbolic3-manifolds with boundary.Andrei's abstract: We define coordinates on virtual braid groups. We prove that these coordinates are faithful invariants of virtual braids on two strings, and present evidence that they are also very powerful invariants for general virtual braids.The talk is based on the joint work with V.Bardakov and B.Wiest.

Southeast Geometry Seminar XXV

Series
Other Talks
Time
Sunday, October 26, 2014 - 08:30 for 8 hours (full day)
Location
University of Tennessee Knoxville
Speaker
Southeast Geometry Seminar – University of Tennessee Knoxville
The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: Emory University; Georgia Institute of Technology; University of Alabama at Birmingham; University of Tennessee Knoxville. The following five speakers will give presentations: Sigurd Angenent (University of Wisconsin-Madison); Omer Bobrowski (Duke University); Tom Ivey (College of Charleston); Ken Knox (University of Tennessee); Facundo Memoli (Ohio State University). Please email oliker@mathcs.emory.edu if you plan to attend and wish to request support.

Embeddings of manifolds and contact manifolds II

Series
Geometry Topology Working Seminar
Time
Friday, October 24, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John Etnyre – Georgia Tech
This is the second of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid 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.

Singularity formation in Compressible Euler equations

Series
PDE Working Seminar
Time
Thursday, October 23, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ronghua Pan – GeorgiaTech –
Compressible Euler equations describe the motion of compressible inviscid fluid. Physically, it states the basic conservation laws of mass, momentum, and energy. As one of the most important examples of nonlinear hyperbolic conservation laws, it is well-known that singularity will form in the solutions of Compressible Euler equations even with small smooth initial data. This talk will discuss some classical results in this direction, including some most recent results for the problem with large initial data.

Zeros of random polynomials

Series
School of Mathematics Colloquium
Time
Thursday, October 23, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Igor Pritsker – Oklahoma State University
The area was essentially originated by the general question: How many zeros of a random polynomials are real? Kac showed that the expected number of real zeros for a polynomial with i.i.d. Gaussian coefficients is logarithmic in terms of the degree. Later, it was found that most of zeros of random polynomials are asymptotically uniformly distributed near the unit circumference (with probability one) under mild assumptions on the coefficients. Thus two main directions of research are related to the almost sure limits of the zero counting measures, and to the quantitative results on the expected number of zeros in various sets. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by various bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.

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