Seminars and Colloquia by Series

Random Matrices and Subsequences

Series
Stochastics Seminar
Time
Thursday, October 30, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Hua Xu School of Mathematics, Georgia Tech
In this presentation, interactions between spectra of classical Gaussian ensembles and subsequence problems are studied with the help of the powerful machinery of Young tableaux. For the random word problem, from an ordered finite alphabet, the shape of the associated Young tableaux is shown to converge to the spectrum of the (generalized) traceless GUE. Various properties of the (generalized) traceless GUE are established, such as a law of large number for the extreme eigenvalues and the convergence of the spectral measure towards the semicircle law. The limiting shape of the whole tableau is also obtained as a Brownian functional. The Poissonized word problem is finally talked, and, with it, the convergence of the whole Poissonized tableaux is derived.

Cubic graphs and universality

Series
Graph Theory Seminar
Time
Thursday, October 30, 2008 - 11:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech

Please Note: PLEASE NOTE UNUSUAL TIME

We will consider the problem of counting the number T(n,g) of cubic graphs with n edges on a surface of of genus g, and review was is known in the combinatorial community in the past 30 years, what was conjectured in physics 20 years ago, and what was proven last month in joint work with Thang Le and Marcos Marino, using the Riemann-Hilbert analysis of the Painleve equation. No knowledge of physics or analysis is required.

Semiparametric Estimation of ARCH(∞) Model

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, October 29, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Lily WangDepartment of Statistics, University of Georgia
We analyze a class of semiparametric ARCH models that nests the simple GARCH(1,1) model but has flexible news impact function. A simple estimation method is proposed based on profiled polynomial spline smoothing. Under regular conditions, the proposed estimator of the dynamic coeffcient is shown to be root-n consistent and asymptotically normal. A fast and efficient algorithm based on fast fourier transform (FFT) has been developed to analyze volatility functions with infinitely many lagged variables within seconds. We compare the performance of our method with the commonly used GARCH(1, 1) model, the GJR model and the method in Linton and Mammen (2005) through simulated data and various interesting time series. For the S&P 500 index returns, we find further statistical evidence of the nonlinear and asymmetric news impact functions.

Random Words, Increasing Subsequences and Random Matrices

Series
Research Horizons Seminar
Time
Wednesday, October 29, 2008 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Christian HoudréSchool of Mathematics, Georgia Tech
This talk is not an appetizer to pizza, but rather an appetizer to the main course: Hua Xu's and Trevis Litherland's thesis defenses which will respectively take place on Thursday the 30th of October and November the 6th, in Skiles 269, at 3pm. I will present the history and origins of the problems they have been tackling ("Ulam's problems"). Various interactions with other fields such as Analysis, Algebra (Young Tableaux) or Bioinformatics (Sequence Comparison) will be touched upon. Then, some elementary but rather useful probabilistic techniques will also be introduced and shown how to be applied.

Bilinear Forms on the Dirichlet Space

Series
Analysis Seminar
Time
Monday, October 27, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Brett WickUniversity of South Carolina
The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is based on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer

The four-vertex-property and topology of surfaces with constant curvature

Series
Geometry Topology Seminar
Time
Monday, October 27, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad GhomiSchool of Mathematics, Georgia Tech
We prove that every metric of constant curvature on a compact 2-manifold M with boundary bdM induces (at least) four vertices, i.e., local extrema of geodesic curvature, on bdM, if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on M which induce only two vertices on bdM. Furthermore, we characterize the sphere as the only closed orientable Riemannian 2-manifold M which has the four-vertex-property, i.e., the boundary of every compact surface immersed in M has 4 vertices.

An efficient numerical method for vesicle simulations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 27, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
George BirosCSE, Georgia Tech
Fluid membranes are area-preserving interfaces that resist bending. They are models of cell membranes, intracellular organelles, and viral particles. We are interested in developing simulation tools for dilute suspensions of deformable vesicles. These tools should be computationally efficient, that is, they should scale well as the number of vesicles increases. For very low Reynolds numbers, as it is often the case in mesoscopic length scales, the Stokes approximation can be used for the background fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the nonlocal hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme. Joint work with: Shravan K. Veerapaneni, Denis Gueyffier, and Denis Zorin.

The triangle-free process

Series
Combinatorics Seminar
Time
Friday, October 24, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tom BohmanCMU
Consider the following random graph process. We begin with the empty graph on n vertices and add edges chosen at random one at a time. Each edge is chosen uniformly at random from the collection of pairs of vertices that do not form triangles when added as edges to the existing graph. In this talk I discuss an analysis of the triangle-free process using the so-called differential equations method for random graph processes. It turns out that with high probability the triangle-free process produces a Ramsey R(3,t) graph, a triangle-free graph whose independence number is within a multiplicative constant factor of the smallest possible.

A new topological bound for energy of fluid flows

Series
Geometry Topology Seminar
Time
Friday, October 24, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Rafal KomendarczykUniversity of Pennsylvania
In many physical situations we are interested in topological lower bounds for L^2-energy of volume preserving vector fields. Such situations include for instance evolution of a magnetic field in ideal magnetohydrodynamics. Classical energy bounds involve topological invariants like helicity which measure linkage of orbits in the flow. In this talk I will present a new lower bound in terms of the third order helicity, which is an invariant measuring a third order linkage of orbits. I will also discuss how the third order helicity can be derived from the Milnor's \mu-bar invariant for 3-component links.

The differential equations method for random graph processes

Series
Graph Theory Seminar
Time
Thursday, October 23, 2008 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tom BohmanCMU
In this lecture I will introduce the method and sketch some recent applications. The main idea is to exploit a natural connection between the evolution of discrete random processes and continuous functions on the real numbers. Roughly speaking, the method is as follows: Given a discrete random process, we calculate the expected change in the random variable (or variables) of interest in one step of the process, write a differential equation suggested by the expected change, and show that the evolution of the random variable over the course of the process is sharply concentrated around the trajectory given by the solution of the differential equation. This allows us to translate simple facts (often combinatorial in nature) about expected changes in one step of the process into strong statements about sharp concentration of the random variable over the entire course of the process.

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