Seminars and Colloquia by Series

The Extremal Nevanlinna-Pick problem for Riemann Surfaces

Series
Analysis Seminar
Time
Wednesday, October 28, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mrinal Ragupathi – Vanderbilt University
Given points $z_1,\ldots,z_n$ on a finite open Riemann surface $R$ and complex scalars $w_1,\ldots,w_n$, the Nevanlinna-Pick problem is to determine conditions for the existence of a holomorphic map $f:R\to \mathbb{D}$ such that $f(z_i) = w_i$. In this talk I will provide some background on the problem, and then discuss the extremal case. We will try to discuss how a method of McCullough can be used to provide more qualitative information about the solution. In particular, we will show that extremal cases are precisely the ones for which the solution is unique.

Soul Theorem and moduli spaces

Series
Research Horizons Seminar
Time
Wednesday, October 28, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Igor Belegradek – School of Mathematics, Georgia Tech –
The Soul Theorem, proved by Cheeger and Gromoll forty year ago, reveals a beautiful structure of noncompact complete manifolds of nonnegative curvature. In the talk I will sketch a proof of the Soul Theorem, and relate it to my current work on moduli spaces of nonnegatively curved metrics.

Why Decussate? Topological constraints on 3D wiring

Series
Mathematical Biology Seminar
Time
Wednesday, October 28, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Troy Shinbrot – Biomedical Engineering, Rutgers University
Many vertebrate motor and sensory systems "decussate," or cross the midline to the opposite side of the body. The successful crossing of millions of axons during development requires a complex of tightly controlled regulatory processes. Since these processes have evolved in many distinct systems and organisms, it seems reasonable to presume that decussation confers a significant functional advantage - yet if this is so, the nature of this advantage is not understood. In this talk, we examine constraints imposed by topology on the ways that a three dimensional processor and environment can be wired together in a continuous, somatotopic, way. We show that as the number of wiring connections grows, decussated arrangements become overwhelmingly more robust against wiring errors than seemingly simpler same-sided wiring schemes. These results provide a predictive approach for understanding how 3D networks must be wired if they are to be robust, and therefore have implications both regenerative strategies following spinal injury and for future large scale computational networks.

Joint ACO/DOS - Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Series
Other Talks
Time
Wednesday, October 28, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom
Speaker
Pushkar Tripathi – ACO, Computing Science and Systems, Georgia Tech

Please Note: Organizer: Daniel Dadush, ACO Student, ISyE

Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. We introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems. This talk is based on joint work with Gagan Goel, Chinmay Karande and Wang Lei. This is a joint ACO/DOS seminar, so please come a little early for pizza and refreshments sponsored by ACO.

Introduction to Bordered Heegaard-Floer homology II

Series
Geometry Topology Working Seminar
Time
Wednesday, October 28, 2009 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Shea Vela-Vick – Columbia University
Here we will introduce the basic definitions of bordered Floer homology. We will discuss bordered Heegaard diagrams as well as the algebraic objects, like A_\infinity algebras and modules, involved in the theory. We will also discuss the pairing theorem which states that if Y = Y_1 U_\phi Y_2 is obtained by identifying the (connected) boundaries of Y_1 and Y_2, then the closed Heegaard Floer theory of Y can be obtained as a suitable tensor product of the bordered theories of Y_1 and Y_2.Note the different time and place!This is a 1.5 hour talk.

Notes on the blow-up problem of the Euler equations

Series
PDE Seminar
Time
Tuesday, October 27, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Dongho Chae – Sungkyunkwan University, Korea and Universty of Chicago
We first discuss blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. In the second part of the talk we discuss some observations on the Euler equations with symmetries, which shows that the point-wise behavior of the pressure along the flows is closely related to the blow-up of of solutions.

Testing the stability of the functional autoregressive processwith an application to credit card transaction volume data

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, October 27, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Piotr Kokoszka – Utah State University
The functional autoregressive process has become a useful tool in the analysis of functional time series data. In this model, the observations and the errors are curves, and the role of the autoregressive coefficient is played by an integral operator. To ensure meaningful inference and prediction, it is important to verify that this operator does not change with time. We propose a method for testing its constancy which uses the functional principal component analysis. The test statistic is constructed to have a Kiefer type asymptotic distribution. The asymptotic justification of the procedure is very delicate and touches upon central notions of functional data analysis. The test is implemented using the R package fda. Its finite sample performance is illustrated by an application to credit card transaction data.

Eigenvalue Inequalities for Relativistic Hamiltonians and Fractional Laplacian

Series
Dissertation Defense
Time
Tuesday, October 27, 2009 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 269
Speaker
Selma Yildirim-Yolcu – School of Mathematics, Georgia Tech
In this thesis, some eigenvalue inequalities for Klein-Gordon operators and restricted to a bounded domain in Rd are proved. Such operators become very popular recently as they arise in many problems ranges from mathematical finance to crystal dislocations, especially the relativistic quantum mechanics and \alpha-stable stochastic processes. Many of the results obtained here concern finding bounds for some spectral functions of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogues results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the operator Hm, are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved.

Blow-up or no Blow-up? The Interplay Between Analysis and Computation in the Millennium Problem on Navier-Stokes Equations

Series
Stelson Lecture Series
Time
Tuesday, October 27, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Thomas Y. Hou – California Institute of Technology, Applied and Computational Mathematics

Please Note: This lecture will be more for the mathematical audience

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the seven Millennium Problems posted by the Clay Mathematical Institute. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex stretching. This is also confirmed by our large scale computations for some of the most well-known blow-up candidates. We also investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear stability and the global regularity of this class of solutions.

Multiscale Modeling and Computation - The Interplay Between Mathematics and Engineering Applications

Series
Stelson Lecture Series
Time
Monday, October 26, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
SST Room 2
Speaker
Thomas Y. Hou – California Institute of Technology, Applied and Computational Mathematics

Please Note: This lecture is more for the general audience.  Reception following lecture. Organizers: Chongchun Zeng and Hao Min Zhou

Many problems of fundamental and practical importance contain multiple scale solutions. Composite and nano materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Direct numerical simulations using a fine grid are very expensive. Developing effective multiscale methods that can capture accurately the large scale solution on a coarse grid has become essential in many engineering applications. In this talk, I will use two examples to illustrate how multiscale mathematics analysis can impact engineering applications. The first example is to develop multiscale computational methods to upscale multi-phase flows in strongly heterogeneous porous media. Multi-phase flows arise in many applications, ranging from petroleum engineering, contaminant transport, and fluid dynamics applications. Multiscale computational methods guided by multiscale analysis have already been adopted by the industry in their flow simulators. In the second example, we will show how to develop a systematic multiscale analysis for incompressible flows in three space dimensions. Deriving a reliable turbulent model has a significant impact in many engineering applications, including the aircraft design. This is known to be an extremely challenging problem. So far, most of the existing turbulent models are based on heuristic closure assumption and involve unknown parameters which need to be fitted by experimental data. We will show that how multiscale analysis can be used to develop a systematic multiscale method that does not involve any closure assumption and there are no adjustable parameters.

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