## Seminars and Colloquia by Series

### Expanders via Random Spanning Trees

Series
Combinatorics Seminar
Time
Friday, December 5, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Luis RademacherSchool of Computer Science, Georgia Tech
Expanders via Random Spanning Trees Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph G_{n,p}, for p > c (log n)/n, two spanning trees give an expander. This is suggested by the case of the complete graph, where we prove that two random spanning trees give an expander. The construction of the splicer is elementary — each spanning tree can be produced independently using an algorithm by Aldous and Broder: a random walk in the graph with edges leading to previously unvisited vertices included in the tree. A second important application of splicers is to graph sparsification where the goal is to approximate every cut (and more generally the quadratic form of the Laplacian) using only a small subgraph of the original graph. Benczur-Karger as well as Spielman-Srivastava have shown sparsifiers with O(n log n/eps^2) edges that achieve approximation within factors 1+eps and 1-eps. Their methods, based on independent sampling of edges, need Omega(n log n) edges to get any approximation (else the subgraph could be disconnected) and leave open the question of linear-size sparsifiers. Splicers address this question for random graphs by providing sparsifiers of size O(n) that approximate every cut to within a factor of O(log n). This is joint work with Navin Goyal and Santosh Vempala.

### Transient (Electro)Chemical Imaging of Reacting Interfaces - Physical Concepts and Mathematical Challenges

Series
Mathematical Biology Seminar
Time
Wednesday, December 3, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Andrei FedorovSchool of Mechanical Engineering, Georgia Tech
In this presentation I will outline physical principles of two analytical techniques, the Scanning ElectroChemical Microscopy (SECM) and Scanning Mass Spectrometry (SMS), which can be used to obtain the spatially resolved images of (bio/electro)chemically active interfaces. The mathematical models need to be employed for image interpretation and mapping measured quantities (e.g., an electrode current in SECM) to biochemically relevant quantities (e.g., kinetics of exocytotic signaling events in cellular communications), and I will review the key ideas/assumptions used for the model formulation and the main results of analysis and simulations. In conclusion, an alternative approach to spatially-resolved imaging based on the multi-probe array will be introduced along with intriguing opportunities and challenges for mathematical interpretation of such images.

### Oblique derivative problems for elliptic equations

Series
PDE Seminar
Time
Tuesday, December 2, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Gary M. LiebermanIowa State University
The usual boundary condition adjoined to a second order elliptic equation is the Dirichlet problem, which prescribes the values of the solution on the boundary. In many applications, this is not the natural boundary condition. Instead, the value of some directional derivative is given at each point of the boundary. Such problems are usually considered a minor variation of the Dirichlet condition, but this talk will show that this problem has a life of its own. For example, if the direction changes continuously, then it is possible for the solution to be continuously differentiable up to a merely Lipschitz boundary. In addition, it's possible to get smooth solutions when the direction changes discontinuously as well.

### Broken Lefschetz fibrations and Floer theoretical invariants

Series
Geometry Topology Seminar
Time
Monday, December 1, 2008 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Yanki LekiliMIT
A broken fibration is a map from a smooth 4-manifold to S^2 with isolated Lefschetz singularities and isolated fold singularities along circles. These structures provide a new framework for studying the topology of 4-manifolds and a new way of studying Floer theoretical invariants of low dimensional manifolds. In this talk, we will first talk about topological constructions of broken Lefschetz fibrations. Then, we will describe Perutz's 4-manifold invariants associated with broken fibrations and a TQFT-like structure corresponding to these invariants. The main goal of this talk is to sketch a program for relating these invariants to Ozsváth-Szabó invariants.

### A Categorification of the Burau Representation via Contact Geometry

Series
Geometry Topology Seminar
Time
Monday, December 1, 2008 - 14:30 for 2 hours
Location
Skiles 269
Speaker
Sandra RitzUniversity of South Carolina
We will begin with an overview of the Burau representation of the braid group. This will be followed by an introduction to a contact category on 3-manifolds, with a brief discussion of its relation to the braid group.

### A general monotonicity concept and its applications in harmonic analysis and approximation theory

Series
Analysis Seminar
Time
Monday, December 1, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Sergey TikhonovICREA and CRM, Barcelona
In this talk we will discuss a generalization of monotone sequences/functions as well as of those of bounded variation. Some applications to various problems of analysis (the Lp-convergence of trigonometric series, the Boas-type problem for the Fourier transforms, the Jackson and Bernstein inequalities in approximation, etc.) will be considered.

### A note on Olsen inequality

Series
Analysis Seminar
Time
Wednesday, November 26, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yoshihiro SawanoGakushuin University, Japan

Let I_\alpha be the fractional integral operator. The Olsen inequality, useful in certain PDEs, concerns multiplication operators and fractional integrals in the L^p-norm, or more generally, the Morrey norm. We strenghten this inequality from the one given by Olsen.

### A Constructive Characterization of the Split Closure of a Mixed Integer Linear Program

Series
ACO Student Seminar
Time
Wednesday, November 26, 2008 - 13:30 for 2 hours
Location
ISyE Executive Classroom
Speaker
Juan Pablo VielmaISyE, Georgia Tech
Two independent proofs of the polyhedrality of the split closure of Mixed Integer Linear Program have been previously presented. Unfortunately neither of these proofs is constructive. In this paper, we present a constructive version of this proof. We also show that split cuts dominate a family of inequalities introduced by Koppe and Weismantel.

### High-order numerical methods for nonlinear PDEs

Series
PDE Seminar
Time
Tuesday, November 25, 2008 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Bojan PopovTexas A&amp;amp;M University

In this talk we will consider three different numerical methods for solving nonlinear PDEs:

1. A class of Godunov-type second order schemes for nonlinear conservation laws, starting from the Nessyahu-Tadmor scheme;
2. A class of L1 -based minimization methods for solving linear transport equations and stationary Hamilton- Jacobi equations;
3. Entropy-viscosity methods for nonlinear conservation laws.

All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.

### Dunkl processes, eigenvalues of random matrices and the Weyl-chamber

Series
Stochastics Seminar
Time
Tuesday, November 25, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Nizar DemniUniversity of Bielefeld
We will introduce the Dunkl derivative as well as the Dunkl process and some of its properties. We will treat its radial part called the radial Dunkl process and light the connection to the eigenvalues of some matrix valued processes and to the so called Brownian motions in Weyl chambers. Some open problems will be discussed at the end.