Seminars and Colloquia by Series

Thursday, March 18, 2010 - 11:00 , Location: Skiles 269 , Frank Morgan , Department of Mathematics and Statistics, Williams College , Organizer: John McCuan

Light refreshments will be available in Room 236 at 10:30 am.

A single round soap bubble provides the least-area way to enclose a given volume. How does the solution change if space is given some density like r^2 or e^{-r^2} that weights both area and volume? There has been much recent progress by undergraduates. Such densities appear prominently in Perelman's paper proving the Poincare Conjecture. No prerequisites, undergraduates welcome.
Thursday, March 4, 2010 - 11:00 , Location: Skiles 269 , Maarten V. de Hoop , Department of Mathematics, Purdue University , Organizer: Guillermo Goldsztein
Much research in modern, quantitative seismology is motivated -- on the one hand -- by the need to understand subsurface structures and processes on a wide range of length scales, and -- on the other hand -- by the availability of ever growing volumes of high fidelity digital data from modern seismograph networks or multicomponent acquisition systems developed for hydro-carbon exploration, and access to increasingly powerful computational facilities. We discuss (elastic-wave) inverse scattering of reflection seismic data, wave-equation tomography, and their interconnection using techniques from microlocal analysis and applied harmonic analysis. We introduce a multi-scale approach and present a framework of partial reconstruction in connection with limited boundary acquisition geometry. The formation of caustics leads to one of the complications which will be discussed. We illustrate various aspects of this research program with examples from global seismology and mineral physics coupled to thermo-chemical convection.
Thursday, February 18, 2010 - 16:00 , Location: Skiles 269 , Raman Parimala , Department of Mathematics and Computer Science, Emory University , Organizer: Guillermo Goldsztein
Let k be a p-adic field and K/k function field in one variable over k. We discuss Hasse principle for existence of rational points on homogeneous spaces under connected linear algebraic groups. We illustrate how a positive answer to Hasse principle leads for instance to the result: every quadratic form in nine variables over K has a nontrivial zero.
Thursday, February 4, 2010 - 11:05 , Location: Skiles 269 , Sergei Tabachnikov , Penn State University , Organizer: Guillermo Goldsztein

Refreshments at 4PM in Skiles 236

The Pentagram map is a projectively natural iteration on plane polygons. Computer experiments show that the Pentagram map has quasi-periodic behavior. I shall explain that the Pentagram map is a completely integrable system whose continuous limit is the Boussinesq equation, a well known integrable system of soliton type. As a by-product, I shall demonstrate new configuration theorems of classical projective geometry.
Thursday, November 19, 2009 - 11:00 , Location: Skiles 269 , Christine Heitsch , School of Mathematics, Georgia Tech , Organizer: Guillermo Goldsztein
Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. In this talk, we focus on understanding how an RNA viral genome can fold into the dodecahedral cage known from experimental data. Using strings and trees as a combinatorial model of RNA folding, we give mathematical results which yield insight into RNA structure formation and suggest new directions in viral capsid assembly. We also illustrate how the interaction between discrete mathematics and molecular biology motivates new combinatorial theorems as well as advancing biomedical applications.
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 , Abhay Ashtekar , Department of Physics and Institute for Gravitational Physics and Geometry, Pennsylvania State University , ashtekar@gravity.psu.edu , 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.

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