Departmental Colloquium

	Date	Speaker	Talk Information
View details	January 12	Natasha Flyer National Center for Atmospheric Research	It is obvious that if the initial (IC) and boundary condition (BC) do not agree in the corner of the time-space domain for an initial boundary value problem (IBVP), a discontinuity will arise in the solution. What is not so obvious is that in order for the solution to be $C^\infty$, the BC and IC must satisfy the partial differential equation (PDE) and all differentiated forms of it, forming an infinite set of compatibility conditions. The IC, being independent of the BC (otherwise one could pose an initial value problem instead), cannot satisfy both the boundary equation and all the compatibility conditions determined by the PDE on the boundary. As a result, a singularity will arise in some derivative of the solution. Although, the theory of compatibility conditions and the regularity of solutions for IBVPs is well known in the realm of theoretical mathematics, it has essentially gone unnoticed in the numerical community. Yet, its ramifications on the performance of high-order methods are severe. Here, we discuss the nature of IBV solutions for dissipative and dispersive equations and its impact on the convergence and accuracy of high-order methods. Examples will include the heat, linear KDV and Schr\"odinger equations.
View details	January 19	Kartik Prasanna UCLA	The first half of the talk will be an elementary discussion of elliptic curves, their associated L-functions and the structure of the Birch and Swinnerton-Dyer conjecture (BSD) for the behaviour of such L-functions at the central point s=1. In the second half, I will begin by explaining a theorem of Waldspurger on nonvanishing of central values of quadratic twists, and its applications, most notably to BSD. Finally, I will formulate a conjectural "mod p" analog of Waldspurger's theorem and describe some recent results that are related to this problem.
View details	February 2	Sai Kee Yeung Purdue University	Fake projective planes are smooth complex surfaces different from the complex projective plane but has the same Betti numbers as the latter. The first example was constructed by Mumford at late seventies using p-adic uniformization and two more were constructed more recently by Ishida-Kato by related methods. A natural problem is to determine all such surfaces. In this talk I would explain a result of Prasad and myself in which we show that there are exactly twelve classes of fake projective planes and construct explicit examples from each class.
View details	February 7 Special Colloquium	Y. C. (Yen-Ching) Pao University of Nebraska	Mathematical Models for Assessing Elasticity of Heart Muscles Mathematical Models for Assessing the Passive Elasticity and Active Contractility of Heart Muscles* Y. C. (Yen-Ching) Pao, Ph. D. Professor Emeritus of Engineering Mechanics # University of Nebraska Lincoln, Nebraska Finite Element Models are applied for assessing in vivo how the heart wall muscles (myocardium) are stretching during the diastolic, passive phase and contracting during the systolic, active phase of cardiac cycles. The chronologically developed models for (a) the x-ray, by-plane silhouettes, (b) computer-tomographically imaged cross sections, and (c) reconstructed, 3-D true shape of the heart are explained. These passive and active characteristics of the working heart wall muscles, particularly those of the pumping left ventricle during cardiac cycles are iteratively determined using these finite-element models. The heart wall muscles have been considered as a layered (with varying angle of fiber-winding), viscoelastic material. Investigation of regional, left ventricular wall muscles with the bifurcation points of the coronary arterial tree as markers for defining the region and also as nodes of isoparametric element using curvilinear coordinates has led to the study of the burst pressure of obstructed arterial blood vessel. Development of the necessary in-house computer software and algorithms for the finite-element iteration involving a very large system of linear algebraic equation (with a coefficient matrix having non-zero elements sparsely distributed near its main diagonal) will also be discussed. * A collaborative research in the years of 1974~2003 with the scientists at the Mayo Clinic, Rochester, Minnesota # Fellow, the American Society of Mechanical Engineering
View details	February 9	Wei Wu Univ. of Chicago	Effective neural motor prostheses require a method for decoding neural activity representing desired movement. In particular, the accurate reconstruction of a continuous motion signal is necessary for the control of devices such as computer cursors, robots, or a patient's own paralyzed limbs. In this talk, I will present our real-time system for such applications that uses statistical Bayesian inference techniques to estimate hand motion from the firing rates of multiple neurons in a monkey's primary motor cortex. The Bayesian model is formulated in terms of the product of a likelihood and a prior. The likelihood term models the probability of neural firing rates given a particular hand motion. The prior term defines a probabilistic model of hand kinematics. Decoding was performed using a Kalman filter as well as a more sophisticated Switching Kalman filter. Particularly, I will show on-line neural control results in which a monkey exploits the Kalman filter to move a computer cursor with its brain. Aiming at more appropriate and accurate decoding, I will also present further investigations on movement direction encoding under Cartesian, Joint Angle, and Shoulder-Centered coordinate systems.
View details	February 16	Thomas Geisser USC	We study zeta-functions of arithemetic schemes (for example, the Riemann-zeta function), and use them to motivate the Beilinson-Lichtenbaum conjectures. We then explain what is known at this point, and which part of the conjecture remains open.
View details	February 17	Maria Reznikoff Princeton	Sometimes physical systems exhibit ``metastability,'' in the sense that states get drawn toward so--called metastable states and are trapped near them for a very long time. A familiar example is the one--dimensional Allen Cahn equation: initial data is drawn quickly to a ``multi--kink'' state and the subsequent evolution is exponentially slow. The slow coarsening has been analyzed by Carr & Pego, Fusco & Hale, and Bronsard & Kohn. In general, what causes metastability? Our main idea is to convert information about the energy landscape (statics) into information about the coarsening rate (dynamics). We give sufficient conditions for a gradient flow system to exhibit metastability. We then apply this abstract framework to give a new analysis of the 1--d Allen Cahn equation. The central ingredient is to establish a certain nonlinear energy--energy--dissipation relationship. One benefit of the method is that it shows that exponential closeness to the multi--kink state is not only propagated, but also generated. This work is joint with Felix Otto, University of Bonn.
View details	February 22	Michael Westdickenberg Bonn Univ.	Since weak solutions of hyperbolic conservation laws may be nonunique, typically an entropy condition is imposed in order to obtain uniqueness. We discuss how the entropy condition implies regularity and structure of solutions of scalar conservation laws. For the one-dimensional system of isentropic Euler equations we show how the entropy condition gives global existence of solutions with natural bounds.
View details	February 23	Martin Weissman Univ. of California, Berkeley	I will begin by discussing a classical problem in number theory: representing integers as sums of perfect squares. During the first part of the talk, I will try to place this problem in a wider historical context, and emphasize how a hidden symmetry can be exploited to find arithmetic formulae. In the second part of my talk, I will discuss generalizations of these problems made during the 20th and 21st centuries. From a modern standpoint, the classical problems are arithmetic embedding problems of type A_1 (refering to the simple Lie algebra or Dynkin diagram). I will mention problems of type C_n for all n, before discussing my own work on embedding problems of type D_4 and triality.
View details	March 2	Brian Conrad Univ. of Michigan	It is a classical and extremely difficult problem to prove theorems about prime values of irreducible polynomials over the integers. For example, it is still not known if there are infinitely many primes of the form n^2 + 1. There is a long history of analogies between the integers and polynomials (in one variable) over a finite field, and so one can formulate an analogous problem in this other setting. It turns out that there are some surprises! We illustrate the unexpected behavior by means of some simple explicit examples, and we discuss some new theorems that are used to predict such new phenomena. Finally, we discuss asymptotic results as the finite field and polynomial being specialized are allowed to vary.
View details	March 7	—	Archival event entry migrated from the legacy colloquium website.
View details	March 23	John Willis Cambridge	The theory and physical origin of strain-gradient plasticity will be briefly outlined, and a "deformation theory" (as opposed to "flow theory") version will be developed. A distinctive aspect of the theory is that it requires an additional boundary condition, or condition across any interface. This may be turned to advantage by introducing an "interfacial potential" that penalises the development of plastic flow at an interface, by impeding the motion of dislocations. A scale-dependent hardening effect in any material such as a composite or a polycrystal is generated thereby. Treatment of such materials as having random microgeometry renders exact solution intractable but approximations (which in some cases are bounds) can be developed via a variational formulation. This will be illustrated with simple examples.
View details	March 30	Lior Pachter Univ. of California, Berkeley	The Drosophila Genome Project is a large scale research effort whose aim is to sequence, compare and contrast 12 Drosophila genomes with the goal of significantly advancing comparative genomics methods. We will provide an overview of our recent work on annotation and alignment of the genomes, which focuses on the related problems of transposable element identification, gene finding and multiple sequence alignment. In particular, we emphasize the importance of robust alignment methods, and their relevance for identifying functional elements in genomes. A key concept is the alignment polytope, which will be explained and illustrated.
View details	April 6	Yeng-Ching Pao Univ. of Nebraska	view
View details	April 13	Andy Wissink Eloret Corp. and NASA Ames Reseach Center	view
View details	April 20	David Kinderlehrer Carnegie Mellon Univ.	The Monge-Kantorovich mass transport theory is presently in a state of feverish development. Originally conceived as an optimization problem for relocating debris - but really about statistics - it has emerged as method for studying many varieties of evolution behavior and, as a consequence, non-equilibrium systems. For example, and indeed our example, will take a look at protein motors, among the primary mechanisms of intracellular transport in eukarya. In this expository discussion, we shall concentrate on how we came to investigate these issues and and what we wish to accomplish. (joint work with Michel Chipot, Stuart Hastings, Michael Kowalczyk, and Bryce McLeod) FALL 2005

Date

Speaker

Talk Information

January 12

Natasha Flyer
National Center for Atmospheric Research

It is obvious that if the initial (IC) and boundary condition (BC) do not agree in the corner of the time-space domain for an initial boundary value problem (IBVP), a discontinuity will arise in the solution. What is not so obvious is that in order for the solution to be $C^\infty$, the BC and IC must satisfy the partial differential equation (PDE) and all differentiated forms of it, forming an infinite set of compatibility conditions. The IC, being independent of the BC (otherwise one could pose an initial value problem instead), cannot satisfy both the boundary equation and all the compatibility conditions determined by the PDE on the boundary. As a result, a singularity will arise in some derivative of the solution. Although, the theory of compatibility conditions and the regularity of solutions for IBVPs is well known in the realm of theoretical mathematics, it has essentially gone unnoticed in the numerical community. Yet, its ramifications on the performance of high-order methods are severe. Here, we discuss the nature of IBV solutions for dissipative and dispersive equations and its impact on the convergence and accuracy of high-order methods. Examples will include the heat, linear KDV and Schr\"odinger equations.

Spring 2006

Special Colloquium