## Seminars and Colloquia by Series

Thursday, November 5, 2009 - 11:00 , Location: Skiles 269 , Lyonia Bunimovich , Georgia Tech , Organizer: Guillermo Goldsztein
Real life networks are usually large and have a very complicated structure. It is tempting therefore to simplify or reduce the associated graph of interactions in a network while maintaining its basic structure as well as some characteristic(s) of the original graph. A key question is which characteristic(s) to conserve while reducing a graph. Studies of dynamical networks reveal that an important characteristic of a network's structure is a spectrum of its adjacency matrix. In this talk we present an approach which allows for the reduction of a general weighted graph in such a way that the spectrum of the graph's (weighted) adjacency matrix is maintained up to some finite set that is known in advance. (Here, the possible weights belong to the set of complex rational functions, i.e. to a very general class of weights). A graph can be isospectrally reduced to a graph on any subset of its nodes, which could be an important property for various applications. It is also possible to introduce a new equivalence relation in the set of all networks. Namely, two networks are spectrally equivalent if each of them can be isospectrally reduced onto one and the same (smaller) graph. This result should also be useful for analysis of real networks. As the first application of the isospectral graph reduction we considered a problem of estimation of spectra of matrices. It happens that our procedure allows for improvements of the estimates obtained by all three classical methods given by Gershgorin, Brauer and Brualdi. (Joint work with B.Webb) A talk will be readily accessible to undergraduates familiar with matrices and complex functions.
Thursday, October 22, 2009 - 11:00 , Location: Skiles 269 , Jerry Bona , University of Illinois at Chicago , Organizer: Guillermo Goldsztein
After a brief account of some of the history of this classical subject, we indicate how such models are derived. Rigorous theory justifying the models will be discussed and the conversation will then turn to some applications. These will be drawn from questions arising in geophysics and coastal engineering, as time permits.
Thursday, October 8, 2009 - 11:00 , Location: Skiles 269 , Laura DeMarco , Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago , Organizer: Guillermo Goldsztein
A classification of the dynamics of polynomials in one complex variable has remained elusive, even when considering only the simpler "structurally stable" polynomials. In this talk, I will describe the basics of polynomial iteration, leading up to recent results in the direction of a complete classification. In particular, I will describe a (singular) metric on the complex plane induced by the iteration of a polynomial. I will explain how this geometric structure relates to topological conjugacy classes within the moduli space of polynomials.
Thursday, September 24, 2009 - 11:05 , Location: Skiles 269 , Distinguished Professor Craig Tracy , University of California, Davis , Organizer: Guillermo Goldsztein
The asymmetric simple exclusion process (ASEP) is a continuous time Markov process of interacting particles on a lattice \Gamma. ASEP is defined by two rules: (1) A particle at x \in \Gamma waits an exponential time with parameter one, and then chooses y \in \Gamma with probability p(x, y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x. The main interest lies in infinite particle systems. In this lecture we consider the ASEP on the integer lattice {\mathbb Z} with nearest neighbor jump rule: p(x, x+1) = p, p(x, x-1) = 1-p and p \ne 1/2. The integrable structure is that of Bethe Ansatz. We discuss various limit theorems which in certain cases establishes KPZ universality.
Wednesday, August 26, 2009 - 15:00 , Location: Chemistry and Biochemistry Boggs Building, Room B-6A , , Department of Physics and Institute for Gravitational Physics and Geometry, Pennsylvania State University , , Organizer:

Pre-reception at 2:30 in Room N201.  If you would like to meet with Prof. Ashtekar while he is on campus (at the Center for Relativistic Astrophysics - Boggs building), please contact <A class="moz-txt-link-abbreviated" href="mailto:lori.federico@physics.gatech.edu">lori.federico@physics.gatech.edu</a>.

General relativity is based on a deep interplay between physics and mathematics: Gravity is encoded in geometry. It has had spectacular observational success and has also pushed forward the frontier of geometric analysis. But the theory is incomplete because it ignores quantum physics. It predicts that the space-time ends at singularities such as the big-bang. Physics then comes to a halt. Recent developments in loop quantum gravity show that these predictions arise because the theory has been pushed beyond the domain of its validity. With new inputs from mathematics, one can extend cosmology beyond the big-bang. The talk will provide an overview of this new and rich interplay between physics and mathematics.
Thursday, April 16, 2009 - 16:30 , Location: Skiles 269 , John McCuan , School of Mathematics, Georgia Tech , Organizer: Guillermo Goldsztein
Archimedes principle may be used to predict if and how certain solid objects float in a liquid bath. The principle, however, neglects to consider capillary forces which can sometimes play an important role. We describe a recent generalization of the principle and how the standard textbook presentation of Archimedes' work may have played a role in delaying the discovery of such generalizations to this late date.
Thursday, April 2, 2009 - 11:00 , Location: Skiles 269 , Marshall Slemrod , Department of Mathematics, University of Wisconsin , Organizer: Guillermo Goldsztein
In this talk I will outline recent results of G-Q Chen, Dehua Wang, and me on the problem of isometric embedding a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Remarkably there is very pretty duality between this problem and the equations of steady 2-D gas dynamics. Compensated compactness (L.Tartar and F.Murat) yields proof of existence of solutions to an initial value problem when the prescribed metric is the one associated with the catenoid.
Thursday, March 26, 2009 - 11:00 , Location: Skiles 269 , Walter Craig , McMaster University , Organizer: Guillermo Goldsztein
A new estimate on weak solutions of the Navier-Stokes equations in three dimensions gives some information about the partial regularity of solutions. In particular, if energy concentration takes place, the dimension of the microlocal singular set cannot be too small. This estimate has a dynamical systems proof. These results are joint work with M. Arnold and A. Biryuk.
Thursday, March 5, 2009 - 11:00 , Location: Skiles 269 , Rick Kenyon , Mathematics Department, Brown University , Organizer: Guillermo Goldsztein
This is joint work with Andrei Okounkov. The honeycomb dimer model'' is a natural model of discrete random surfaces in R^3. It is possible to write down a Law of Large Numbers" for such surfaces which describes the typical shape of a random surface when the mesh size tends to zero. Surprisingly, one can parameterize these limit shapes in a very simple way using analytic functions, somewhat reminiscent of the Weierstrass parameterization of minimal surfaces. This is even more surprising since the limit shapes tend to be facetted, that is, only piecewise analytic. There is a large family of boundary conditions for which we can obtain exact solutions to the limit shape problem using algebraic geometry techniques. This family includes the (well-known) solution to the limit shape of a boxed plane partition'' and has many generalizations.
Thursday, February 26, 2009 - 11:00 , Location: Skiles 269 , Igor Pak , University of Minnesota , Organizer: Guillermo Goldsztein
The study of partition identities has a long history going back to Euler, with applications ranging from Analysis to Number Theory, from Enumerative Combina- torics to Probability. Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find "hidden symmetries", etc. In the talk I will present a modern approach to partition bijections based on the geometry of random partitions and complexity ideas.