## Seminars and Colloquia by Series

### Elliptic hypergeometric integrals

Series
Analysis Seminar
Time
Monday, February 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Eric RainsCaltech
Euler's beta (and gamma) integral and the associated orthogonal polynomials lie at the core of much of the theory of special functions, and many generalizations have been studied, including multivariate analogues (the Selberg integral; also work of Dixon and Varchenko), q-analogues (Askey-Wilson, Nasrallah-Rahman), and both (work of Milne-Lilly and Gustafson; Macdonald and Koornwinder for orthgonal polynomials). (Among these are the more tractable sums arising in random matrices/tilings/etc.) In 2000, van Diejen and Spiridonov conjectured a further generalization of the Selberg integral, going beyond $q$ to the elliptic level (replacing q by a point on an elliptic curve). I'll discuss two proofs of their conjecture, and the corresponding elliptic analogue of the Macdonald and Koornwinder orthogonal polynomials. In addition, I'll discuss a further generalization of the elliptic Selberg integral with a (partial) symmetry under the exceptional Weyl group E_8, and its relation to Sakai's elliptic Painlev equation.

### The mathematical understanding of tau-leaping algorithm

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tiejun LiPeking University
The tau-leaping algorithm is proposed by D.T. Gillespie in 2001 for accelerating the simulation for chemical reaction systems. It is faster than the traditional stochastic simulation algorithm (SSA), which is an exact simulation algorithm. In this lecture, I will overview some recent mathematical results on tau-leaping done by our group, which include the rigorous analysis, construction of the new algorithm, and the systematic analysis of the error.

### Cubic graphs and number fields

Series
Geometry Topology Seminar
Time
Monday, February 23, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech
A cubic graph is a graph with all vertices of valency 3. We will show how to assign two numerical invariants to a cubic graph: its spectral radius, and a number field. These invariants appear in asymptotics of classical spin networks, and are notoriously hard to compute. They are known for the Theta graph, the Tetrahedron, but already unknown for the Cube and the K_{3,3} graph. This is joint work with Roland van der Veen: arXiv:0902.3113.

### Introduction to metric and comparison geometry

Series
Other Talks
Time
Friday, February 20, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Igor BelegradekSchool of Mathematics, Georgia Tech
Comparison geometry studies Riemannian manifolds with a given curvature bound. This minicourse is an introduction to volume comparison (as developed by Bishop and Gromov), which is fundamental in understanding manifolds with a lower bound on Ricci curvature. Prerequisites are very modest: we only need basics of Riemannian geometry, and fluency with fundamental groups and metric spaces. The second (2 hour) lecture is about Gromov-Hausdorff convergence, which provides a natural framework to studying degenerations of Riemannian metrics.

### Introduction to metric and comparison geometry

Series
Geometry Topology Working Seminar
Time
Friday, February 20, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Comparison geometry studies Riemannian manifolds with a given curvature bound. This minicourse is an introduction to volume comparison (as developed by Bishop and Gromov), which is fundamental in understanding manifolds with a lower bound on Ricci curvature. Prerequisites are very modest: we only need basics of Riemannian geometry, and fluency with fundamental groups and metric spaces. The second (2 hour) lecture is about Gromov-Hausdorff convergence, which provides a natural framework to studying degenerations of Riemannian metrics.

### Sums and products in C[x]

Series
Combinatorics Seminar
Time
Friday, February 20, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ernie CrootSchool of Mathematics, Georgia Tech
In this work (joint with Derrick Hart), we show that there exists a constant c > 0 such that the following holds for all n sufficiently large: if S is a set of n monic polynomials over C[x], and the product set S.S = {fg : f,g in S}; has size at most n^(1+c), then the sumset S+S = {f+g : f,g in S}; has size \Omega(n^2). There is a related result due to Mei-Chu Chang, which says that if S is a set of n complex numbers, and |S.S| < n^(1+c), then |S+S| > n^(2-f(c)), where f(c) -> 0 as c -> 0; but, there currently is no result (other than the one due to myself and Hart) giving a lower bound of the quality >> n^2 for |S+S| for a fixed value of c. Our proof combines combinatorial and algebraic methods.

### Coupling in ergodic problems for Stochastic Navier-Stokes

Series
Probability Working Seminar
Time
Friday, February 20, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 268
Speaker
Sergio AlmadaSchool of Mathematics, Georgia Tech
The talk is based on a paper by Kuksin, Pyatnickiy, and Shirikyan. In this paper, the convergence to a stationary distribution is established by partial coupling. Here, only finitely many coordinates in the (infinite-dimensional) phase space participate in the coupling while the dynamics takes care of the other coordinates.

### Introduction to basic governing equations of fluid dynamics

Series
SIAM Student Seminar
Time
Friday, February 20, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Ke YinSchool of Mathematics, Georgia Tech
In this introductory talk, I am going to derive the basic governing equations of fluid dynamics. Our assumption are the three physical principles: the conservation of mass, Newton's second law, and the conservation of energy. The main object is to present Euler equations (which characterize inviscid flow) and Navier-Stokes equations (which characterize viscid flow).

### Optimal alignments and sceneries

Series
Stochastics Seminar
Time
Thursday, February 19, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Heinrich MatzingerSchool of Mathematics, Georgai Tech
We explore the connection between Scenery Reconstruction and Optimal Alignments. We present some new algorithms which work in practise and not just in theory, to solve the Scenery Reconstruction problem

### Tiling R^n by unit cubes

Series
Graph Theory Seminar
Time
Thursday, February 19, 2009 - 12:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Peter HorakUniversity of Washington, Tacoma
Tiling problems belong to the oldest problems in whole mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1908. In this lecture we will discuss the conjecture, and other closely related problems.