Seminars and Colloquia by Series

3-manifolds and lagrangian subspaces of character varieties

Series
Geometry Topology Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
A. SikoraSUNY Buffalo
W. Goldman proved that the SL(2)-character variety X(F) of a closed surface F is a holonomic symplectic manifold. He also showed that the Sl(2)-characters of every 3-manifold with boundary F form an isotropic subspace of X(F). In fact, for all 3-manifolds whose SL(2)-representations are easy to analyze, these representations form a Lagrangian space. In this talk, we are going to construct explicit examples of 3-manifolds M bounding surfaces of arbitrary genus, whose spaces of SL(2)-characters have dimension as small as possible. We discuss relevance of this problem to quantum and classical low-dimensional topology.

Conformal dimension of self-affine sets and modulus of a system of measures

Series
Analysis Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Hrant HakobyanUniversity of Toronto
A mapping F between metric spaces is called quasisymmetric (QS) if for every triple of points it distorts their relative distances in a controlled fashion. This is a natural generalization of conformality from the plane to metric spaces. In recent times much work has been devoted to the classification of metric spaces up to quasisymmetries. One of the main QS invariants of a space X is the conformal dimension, i.e the infimum of the Hausdorff dimensions of all spaces QS isomorphic to X. This invariant is hard to find and there are many classical fractals such as the standard Sierpinski carpet for which conformal dimension is not known. Tyson proved that if a metric space has sufficiently many curves then there is a lower bound for the conformal dimension. We will show that if there are sufficiently many thick Cantor sets in the space then there is a lower bound as well. "Sufficiently many" here is in terms of a modulus of a system of measures due to Fuglede, which is a generalization of the classical conformal modulus of Ahlfors and Beurling. As an application we obtain a new lower bound for the conformal dimension of self affine McMullen carpets.

The existence of three-dimensional generalized solitary waves

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 6, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shengfu DengSchool of Mathematics, Georgia Tech
We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.

On the inertia set of a graph

Series
Graph Theory Seminar
Time
Monday, October 6, 2008 - 11:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Hein van der HolstUniversity of Eindhoven
For an undirected graph G=(V,E) with V={1,...,n} let S(G) be the set of all symmetric n x n matrices A=[a_i,j] with a_i,j non-zero for distinct i,j if and only if ij is an edge. The inertia of a symmetric matrix is the triple (p_+,p_-,p_0), where p_+, p_-,p_0 are the number of positive, negative, and null eigenvalues respectively. The inverse inertia problem asks which inertias can be obtained by matrices in S(G). This problem has been studied intensively by Barrett, Hall, and Loewy. In this talk I will present new results on the inverse inertia problem, among them a Colin de Verdiere type invariant for the inertia set (this is the set of all possible inertias) of a graph, a formula for the inertia set of graphs with a 2-separation, and a formula for the inertia set of the join of a collection of graphs. The Colin de Verdiere type invariant for the inertia set is joint work with F. Barioli, S.M. Fallat, H.T. Hall, D. Hershkowitz, L. Hogben, and B. Shader, and the formula for the inertia set of the join of a collection of graphs is joint work with W. Barrett and H.T. Hall.

Maps and Branched Covers - Combinatorics, Geometry and Physics

Series
Combinatorics Seminar
Time
Friday, October 3, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ian GouldenUniversity of Waterloo
This is an expository account of recent work on the enumeration of maps (graphs embedded on a surface of arbitrary genus) and branched covers of the sphere.  These combinatorial and geometric objects can both be represented by permutation factorizations, in the which the subgroup generated by the factors acts transitively on the underlying symbols (these are called "transitive factorizations"). Various results and methods are discussed, including a number of methods from mathematical physics, such as matrix integrals and the KP hierarchy of integrable systems. A notable example of the results is a recent recurrence for triangulations of a surface of arbitrary genus obtained from the simplest partial differential equation in the KP hierarchy. The recurrence is very simple, but we do not know a combinatorial interpretation of it, yet it leads to precise asymptotics for the number of triangulations with n edges, of a surface of genus g.

Large N duality and integrable systems

Series
Geometry Topology Seminar
Time
Friday, October 3, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Tony PantevDept of Mathematics, University of Penn
I will describe a framework which relates large N duality to the geometry of degenerating Calabi-Yau spaces and the Hitchin integrable system. I will give a geometric interpretation of the Dijkgraaf-Vafa large N quantization procedure in this context.

Perfect simulation of Matern Type III point processes

Series
Stochastics Seminar
Time
Thursday, October 2, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mark HuberDepartments of Mathematics and Statistical Sciences, Duke University
Spatial data are often more dispersed than would be expected if the points were independently placed. Such data can be modeled with repulsive point processes, where the points appear as if they are repelling one another. Various models have been created to deal with this phenomenon. Matern created three algorithms that generate repulsive processes. Here, Matérn Type III processes are used to approximate the likelihood and posterior values for data. Perfect simulation methods are used to draw auxiliary variables for each spatial point that are part of the type III process.

Geometry and Topology in Fluid Mechanics

Series
School of Mathematics Colloquium
Time
Thursday, October 2, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John EtnyreSchool of Mathematics, Georgia Tech
Describe the trajectories of particles floating in a liquid. This is a surprisingly difficult problem and attempts to understand it have involved many diverse techniques. In the 60's Arold, Marsden, Ebin and others began to introduce topological techniques into the study of fluid flows. In this talk we will discuss some of these ideas and see how they naturally lead to the introduction of contact geometry into the study of fluid flows. We then consider some of the results one can obtain from this contact geometry perspective. For example we will show that for a sufficiently smooth steady ideal fluid flowing in the three sphere there is always some particle whose trajectory is a closed loop that bounds an embedded disk, and that (generically) certain steady Euler flows are (linearly) unstable.

Geometry and Topology in Fluid Mechanics

Series
School of Mathematics Colloquium
Time
Thursday, October 2, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John EtnyreSchool of Mathematics, Georgia Tech
Describe the trajectories of particles floating in a liquid. This is a surprisingly difficult problem and attempts to understand it have involved many diverse techniques. In the 60's Arold, Marsden, Ebin and others began to introduce topological techniques into the study of fluid flows. In this talk we will discuss some of these ideas and see how they naturally lead to the introduction of contact geometry into the study of fluid flows. We then consider some of the results one can obtain from this contact geometry perspective. For example we will show that for a sufficiently smooth steady ideal fluid flowing in the three sphere there is always some particle whose trajectory is a closed loop that bounds an embedded disk, and that (generically) certain steady Euler flows are (linearly) unstable.

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