Seminars and Colloquia by Series

Generalized Borcherds Products and Two number theoretic applications

Series
School of Mathematics Colloquium
Time
Thursday, October 7, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Ken Ono – University of Wisconsin at Madison and Emory University –
n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics: 1) Parity of the partition function 2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.

The Aleksandrov problem and optimal transport on $S^n$

Series
School of Mathematics Colloquium
Time
Thursday, September 2, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
249 Skiles
Speaker
Vladimir Oliker – Emory University –
The purpose of this talk is to describe a variational approach to the problemof A.D. Aleksandrov concerning existence and uniqueness of a closed convexhypersurface in Euclidean space $R^{n+1}, ~n \geq 2$ with prescribed integral Gauss curvature. It is shown that this problem in variational formulation is closely connected with theproblem of optimal transport on $S^n$ with a geometrically motivated cost function.

Noncommutative geometry and the field with one element

Series
School of Mathematics Colloquium
Time
Tuesday, April 27, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matilde Marcolli – Caltech –
There are presently different approaches to definealgebraic geometry over the mysterious "field with one element".I will focus on two versions, one by Soule' and one by Borger,that appear to have a direct connection to NoncommutativeGeometry via the quantum statistical mechanics of Q-latticesand the theory of endomotives. I will also relate to endomotivesand Noncommutative Geometry the analytic geometry over F1,as defined by Manin in terms of the Habiro ring.

Continuous Solutions of Hyperbolic Conservation Laws

Series
School of Mathematics Colloquium
Time
Thursday, April 1, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Contantine Dafermos – Brown University
The lecture will outline how the method of characteristics can be used in the context of solutions to hyperbolic conservation laws that are merely continuous functions. The Hunter-Saxton equation will be used as a vehicle for explaining the approach.

From Soap Bubbles to the Poincare Conjecture

Series
School of Mathematics Colloquium
Time
Thursday, March 18, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Frank Morgan – Department of Mathematics and Statistics, Williams College

Please Note: 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.

Inverse scattering and wave-equation tomography - Imaging Earth's deep interior

Series
School of Mathematics Colloquium
Time
Thursday, March 4, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Maarten V. de Hoop – Department of Mathematics, Purdue University
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.

A Hasse principle for homogeneous spaces over function fields of p-adic curves

Series
School of Mathematics Colloquium
Time
Thursday, February 18, 2010 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Raman Parimala – Department of Mathematics and Computer Science, Emory University
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.

Pentagrama Myrificum, old wine into new wineskins

Series
School of Mathematics Colloquium
Time
Thursday, February 4, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sergei Tabachnikov – Penn State University

Please Note: 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.

Strings, Trees, and RNA Folding

Series
School of Mathematics Colloquium
Time
Thursday, November 19, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Christine Heitsch – School of Mathematics, Georgia Tech
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.

DYNAMICAL NETWORKS, ISOSPECTRAL GRAPH REDUCTION

Series
School of Mathematics Colloquium
Time
Thursday, November 5, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Lyonia Bunimovich – Georgia Tech
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.

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