**Wednesday, April 13, 2005 **

Jeff Morgan (University of Houston)

Analysis of a Reaction-Diffusion Systems Associated With Atmospheric Modelling

2:30 pm / CH 435

Abstract. We give an overview of aspects of chemical reaction-diffusion systems. Then we consider the analysis of a weakly coupled system of semilinear partial differential equations used to describe photochemical generation and atmospheric dispersion of ozone and other pollutants.

**Friday, April 8, 2005 **Douglas H. Comstock (UAB)

A population model in search of a model population

3:00-3:25 pm / CH 435

Abstract. The purpose of this project is to model a population with two stages, that is, adults and juveniles, at least one of which interferes with the other in some way. One type of interference to consider is that of the adults in a given species cannibalizing the juveniles. This type of behavior is evident in some species of fish that give live birth, and some insects. The simplest form of this model is given by using a linear equation for the population of juveniles at time t+1: Jt + 1 = c*At

And an equation for the adult population at time t+1 that incorporates a Ricker type term for the transition of juveniles to adults:

At + 1 = b*Jt*exp(-k*At) + a*At

Here the Ricker type term accounts for the cannibalistic behavior. Incorporating several parameters is necessary in order to obtain a data fit to a particular species. The beauty is in the fact that these parameters can be varied under controlled conditions in order to investigate the types of bifurcations that can be seen in the model. This approach to population modelling was inspired by the approach used by the writers of Chaos in Ecology. The particular types (and sequences) of bifurcations that can occur in such a model is still the focus of some investigation. In biological terms, these bifurcations are the response of the population to disturbances caused by external factors. These external factors range from varying food supply to directly adding or removing individuals from the population. This model is still in search of a population to which it can be applied. Also it is important to note that as of yet the model is purely deterministic, and that after an application is selected and data has been fit, there will be additional stochastic elements incorporated into the model.

**Jeanine E. Sedjro (UAB) **

Is pollution necessary to maintain genetic diversity within the peppered moth population?

2:30-2:55 pm / CH 435

Abstract. This research is inspired by the classical story of the peppered moths used as an example of natural selection. We classify the peppered moth population in three subpopulations according to their genotypes, the AA and aa moths which are homozygous for the color gene and the Aa moths which are heterozygous for the color gene. AA and Aa moths have dark color and aa moths have light color. The goal of this research is to find a suitable pollution level at which all the subpopulations survive. We set the bird predation rate on light moths and dark moths to be an increasing and a decreasing linear function respectively of the pollution level. We predict the AA and Aa moths to be extinct at low pollution levels and to survive at high pollution levels whereas the aa moths survive at low pollution levels and are extent at high pollution levels.

**Friday, March 25, 2005 **

Robert Szoke (Purdue University)

Complexifications: analytic, algebraic, geometric

2:30 pm / CH 435

Abstract. Complexification of a real manifold tries to imitate the case how the real Euclidean space sits in the same dimensional complex Euclidean space, as the set of points, with all coordinates real. To associate a complex manifold in a natural way to a real manifold makes it possible to use complex methods to study real manifolds. The very first application of such ideas was Grauert's theorem to show that every real-analytic manifold admits a real analytic embedding to some Euclidean space. In this lecture we shall give a survey how one can try to construct natural complex manifolds, starting with a real manifold, using analytic, algebraic ideas, or using some extra structures on the real manifold (Riemannian metrics, connection), what is know about these complexifications and finally we shall discuss some conjectures about these structures.

**Friday, March 4, 2005 **

Jyotsna Prajapat (UAB)

On a class of elliptic problems in R2: symmetry and uniqueness results

2:30 pm / CH 435

Abstract. In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a radial weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the givne problem, as established by Chanillo and Kiessling.

The problem we consider stems from different areas of analysis and geometry. We are particularly interested in it since it occurs as a limiting equation in blow-up analysis of Chern-Simmons theory. On the other hand, the existence of non-radial solutions was observed in results related to statistical mechanics

**Friday, February 25, 2005 **

Hans Kaper (Argonne National Laboratory)

Reduction Methods for Systems of Ordinary Differential Equations

2:30 pm / CH 435

Abstract. We are interested in systematic methods to reduce the dimension of large systems of ODEs that involve fast and slow dynamics. The fast dynamics take the orbits close to an invariant low-dimensional manifold, and the slow dynamics take over as the state of the system approaches the manifold. One can achieve considerable savings in the study of the long-term dynamics of such systems if one knows the location and properties of the slow manifold. Many reduction methods have been proposed for this purpose, especially in the chemical kinetics literature. In this talk I will discuss some of these methods and their mathematical properties.

**Wednesday, February 23, 2005 **

Fernando Schwartz (Cornell University)

The Yamabe problem on noncompact manifolds with boundary

2:15 pm / CH 458

Abstract. The celebrated Riemann mapping theorem states that any simply connected region in the plane is conformally equivalent to the unit disc U. If we think of U as a surface of zero Gauss curvature and constant curvature on the boundary, a possible generalization of this theorem to higher-dimensional manifolds comes from studying the following:

For n > 3, let (Mn,∂M,g) denote a Riemannian manifold with boundary and metric g. Find a metric g conformally equivalent to g so that (Mn,∂M,g) has zero scalar curvature and such that ∂M has constant mean curvature.

More generally, one can address the problem of finding conformally related metrics with prescribed behavior on the boundary. Namely:

Let f: ∂M → R be smooth. Find g conformally equivalent to g that is scalar flat (i.e. has zero scalar curvature) and has mean curvature ≡f on ∂M.

In this talk we will show that for (a large class of) noncompact manifolds with boundary the following holds:

Theorem. Any smooth function on ∂M can be realized as the mean curvature of a scalar flat metric conformally related to g.

The proof of this theorem follows from a more general PDE theorem that we will present.

**Friday, February 18, 2005 **

Mayumi Sakata (William Jewell College)

Almost All Palindromes Are Composite

2:30 pm / CH 435

Abstract. We study the distribution of palindromic numbers, with respect to a fixed base g≤2, over certain congruence classes using a basic sieve method and exponential sums. We drive a nontrivial upper bound for the number of prime palindromes n<x as x goes to infinity, and our result shows that almost all palindromes are composite.

**Friday, February 11, 2005 **

Peter Hislop (University of Kentucky)

An overview of random Schrodinger operators

2:30 pm / CH 435

Abstract. This talk is intended to be an introduction to, and an overview of, some of the highlights of the theory of random Schrodinger operators. These can be roughly grouped into spectral and dynamical results and include the almost sure invariance of the spectrum, Anderson spectral and dynamical localization, and properties of the density of states. Some of the major open problems in the field will be discussed.

**Friday, February 4, 2005 **

Peter Balint (Technical University of Budapest)

Hyperbolicity in soft billiards

2:30 pm / CH 435

Abstract. Given a planar/spatial configuration of circles/spheres, one may consider the motion of a point particle that travels along straight lines and experiences elastic collisions with the circular/spherical scatterers. This model, introduced a century ago by H. Lorentz to describe the motion of an electron in some material (the scatterers represent atoms) is one of the main motivations for studying billiard dynamical systems. These systems are characterized by hyperbolic (chaotic) phenomena, which is, roughly speaking, strong sensitivity to perturbations at initial conditions.

With the above historical remark in mind, the following generalization is natural: interaction with the scatterers is no longer hard core, the point particle may enter them and in the interior its motion is governed by some circularly/spherically symmetric potential. This boils down to the study of soft billiard systems. As to the planar case, it was observed more than thirty years ago and since then has been studied in detail that, for certain softening of the potential, the hyperbolic nature of the billiard dynamics persists. Recently similar phenomena have been identified in the three dimensional case as well.

In this talk, I would like to describe the mechanism behind the presence of hyperbolicity, both in two and in three dimensions. I would also like to comment briefly on the question whether hyperbolicity implies strong statistical properties - like fast relaxation to equilibrium - in these systems.

All the new results I am going to mention are joint with Imre Peter Toth.

**Friday, January 28, 2005 **

Leonid Bunimovich (Georgia Tech)

Deterministic walks in random environments

2:30 pm / CH 435

Abstract. Deterministic walks in random environments (DWRE) occupy an intermediate position between purely random(generated by random trials) and purely deterministic (e.g. generated by maps) models of transport in systems of (generalized) particles. Such models appeared in many applications where a role of particles was played by signals, waves, robots, read/write heads of Turing machine, etc, and, of course, by particles. In DWRE these objects move on some graph G by hopping between its vertices according some deterministic rules (scatterers). These scatterers are randomly distributed over the vertices of G. We present several examples of unusual and fascinating dynamics of such models as well as the simplest deterministic models of diffusion, sub- and super-diffusion. This talk will be accessible to undergraduates familiar with the (very) basics of probability theory.

**Tuesday, January 25, 2005 **

Alexandru Tamasan (University of Toronto)

Inverse Scattering for ∂ Equations and Applications

11:00 am / CH 458

Abstract. The works of Nachman and Ablowitz, and Henkin and Novikov introduced inverse scattering methods to the parameter identification problems. In there, the linear Schrodinger equation in the physical space is paired with a pseudo-analytic equation in the complex space of the parameter. Another method, due to Beals and Coifman pairs a first order (∂-system in the physical space with a pseudo-analytic matrix equation in the parameter space.

I will present an inverse scattering method in which the ∂-equation

∂ u — q û = 0

in the physical space is paired with another ∂-equation in the parameter space. Due to the symmetry between the scattered solutions in the physical space and the ones in the parameter space, we are able to present a non-linear analog of the Fourier inversion formula.

As an application I will revisit a problem proposed by Cheng an Yamamoto, and show how to reconstruct the convection coefficients b1 and b2 in Δ u + b1 ux + b2 uy = 0 from measurements on the boundary,

**Tuesday, January 18, 2005 **

Christian Hainzl (University of Copenhagen)

A non-linear model for relativistic electrons interacting with Dirac's vacuum

11:00 am / CH 458

Abstract. We study the Bogolubov-Dirac-Fock model, which is a mean-field theory deduced from QED. Contrary to usual Dirac-type theories the associated BDF-functional is bounded from below. Its elements are infinite rank projectors describing the electrons, the observable ones as well as those filling up the Dirac sea. We prove that for any ultra-violet momentum cut-off Λ the BDF-functional attains its minimum. The minimizer fulfills a self-consistent equation representing the polarized vacuum. Moreover we show that for Λ to infinity the theory gets "nullified" as predicted by Landau. Furthermore the BDF-functional can be minimized under the constraint of a fixed charge q allowing the description of atoms and molecules. Finally we also state the existence of global-in-time solutions to the associated time-dependent equation.

**Friday, January 14, 2005 **

Xiaofeng Sun (Harvard University)

Analytic and Geometric Aspects of Moduli Space of Riemann Surfaces

10:00 am / CH 458

Abstract. I will discuss my recent joint works with Prof. Kefeng Liu and Prof. Shing-Tung Yau.

We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces, and study their curvatures and boundary behaviors by using the singular perturbation techniques from partial differential equations. These new metrics have Poincare growth near the boundary of the moduli space and have bounded geometry. Based on the detailed analysis of these new metrics, we obtain good understanding of all of the known classical complete Kahler metrics, in particular the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford.

By studying the Monge-Ampere equation together with the Kahler-Ricci flow on complete non-compact manifolds, we derive C^k estimates directly without using the C^0 estimate. Based on these analysis, we prove that the Kahler-Einstein metric has strongly bounded geometry.

Another corollary is a proof of the equivalences of all of the known classical complete metrics to these new metrics, in particular Yau's conjectures on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.

**Wednesday, January 12, 2005 **

Hao Fang (Institute for Advanced Study, Princeton)

Analytic torsion on Calabi-Yau manifolds

11:00 am / CH 458

Abstract. We study spectral invariants of torsion type on Calabi-Yau manifolds of dimension three. We define a new holomorphic invariant modifying the so-called BCOV torsion and explore its applications in the study of moduli space of Calabi-Yau manifolds. Using both analytical and algebraic tools, we determine the asymptotic behavior of the new invariant near a singular Calabi-Yau variety, especially for a Calabi-Yau variety with only one ODP singularity, which leads to a current curvature formula. As an application, We verify a conjecture of Beshadsky-Ceccotti-Ooguri-Vafa. In many examples, we link our metric invariant to automorphic forms. This is a recent joint work with Z. Lu and K.-I. Yoshikawa.

**Tuesday, December 14, 2004 **

Alexander Elgart (Stanford University)

Localization for random Schr\"odinger operators and Lifshitz tails

11:00 am / CH 458

Abstract. The addition of disorder can have a profound effect on the spectral and dynamical properties of a self-adjoint differential operator. This phenomenon, known as Anderson localization (named after P. W. Anderson, who first proposed it as a mechanism for localization in 1958 and won the Nobel prize for this work) was initially discussed in the context of the conduction properties of metals, but the mechanism is of relevance in a variety of other situations. The theory of random operators has seen rapid growth in the last decade. The main open questions in the field revolve around the existence of extended states in dimensions greater than or equal to three, and localization at all energies in two dimensions. In this talk, I will present the (Aizenman - Molchanov) method of fractional moments, as well as a new result on localization in dimension three at weak disorder (Lifshitz tails) regime.

**Friday, December 10, 2004 **

Arpad Benyi (University of Massachusetts, Amherst)

Bilinear pseudodifferential operators: towards a comprehensive theory

10:45 am / CH 458

Abstract. One of the principal motivations for the study of pseudo-differential operators is their wide applicability to partial differential equations. We start by illustrating this with some heuristics about the constructions of approximate inverses to elliptic partial differential operators. From here, we make the transition to bilinear pseudo-differential operators, which can be thought of generalizations of the product of two functions and their derivatives. We introduce several classes of these operators and present some recent results about their boundedness on products of Lebesgue, Sobolev and other spaces of smooth functions.

**Friday, November 19, 2004 **

Henry van den Bedem (Stanford University)

Modeling Missing Fragments in Incomplete Protein Models

2:30 pm / CH 445

Abstract. X-ray crystallography is a widely used experimental technique to obtain atomic coordinates of 3D protein structures. Once a sequence of X-ray diffraction images is collected from a sample in crystalline form, a distribution of electronic charge in R^3 of the atoms constituting the macromolecule can be calculated. A protein model is then obtained by placing atoms such that electronic charge is best accounted for. Over the past few years there have been tremendous advances in automated model building techniques, and in many cases it is now possible to proceed from diffraction data to an initial model of a new protein structure in a few hours. However, the degree of completeness of these initial models, i.e. the fraction of atoms or residues correctly placed, varies widely depending on the quality of the experimental data and rarely reaches 100%. The subsequent step of manual completion of a partial protein model, i.e. building missing fragments, is time-consuming and prone to subjective interpretation.

We have developed a method that combines fast, inverse kinematics algorithms with a real space, torsion angle refinement procedure in a two stage approach to automatically fit missing main-chain fragments into the electron density between two anchor points. The first stage samples a large number of closing conformations, guided by the electron density. These candidates are ranked according to density fit. In a subsequent refinement stage, optimization steps are projected onto a carefully chosen subspace of conformation space to preserve rigid geometry and closure. Experimental results show that fitted fragments are in excellent agreement with the final, refined structure for lengths up to 12-15 residues in areas of weak or ambiguous electron density, even at medium to low resolution.

**Friday, November 12, 2004 **

Alain Joye (Université de Grenoble I, France)

A molecular Landau-Zener formula in the time-dependent Born-Oppenheimer approximation.

2:30 pm / CH 445

Abstract. Consider the Schroedinger equation generated by a slowly varying time-dependent two-level Hamiltonian with isolated eigenvalues. The Landau-Zener formula gives the transition probability between the corresponding eigenstates in the adiabatic limit.

A molecular Hamiltonian describing heavy nuclei and light electrons is naturally split into the kinetic energy of the nuclei and the electronic Hamiltonian depending on the position of the nuclei. The time-dependent Born-Oppenheimer approximation allows to treat the nuclei semiclassically and the electrons adiabatically. It is thus tempting to replace the semiclassical nuclear degrees of freedom by classical trajectories, so that the electronic transitions are described by means of an effective adiabatic time-dependent electronic Hamiltonian and the corresponding Landau-Zener formula.

In this talk we shall revisit this scenario by considering time-dependent solutions to the molecular Schroedinger equation for two-level electronic Hamiltonians. We will describe the piece of the solution that makes a transition between the electronic energy levels in the Born-Oppenheimer approximation and show how the Landau-Zener formula must be modified in order to yield the corresponding transition probability.

This is joint work with George Hagedorn.

**Friday, November 5, 2004 **

John Neuberger (University of North Texas)

Continuous Newton's Method and a new Nash-Moser Inverse Function Theorem

2:30 pm / CH 445

Abstract. We indicate how continuous Newton's method leads to a somewhat unusual kind of dynamical system in the case of complex polynomials. A point of view developed for polynomials leads to new, vastly simpler, inverse function theorems of the Nash-Moser type. The importance of such inverse function results in partial differential equations is discussed.

**Friday, October 29, 2004 **

Jean Bellissard (Georgia Institute of Technology)

Coherent and Dissipative Transport in Quantum Systems

2:30 pm / CH 445

Abstract. The theory of transport in Quantum Systems is still an open mathematical problem to a great extent. Transport in aperiodic solids, such as quasicrystals or semiconductors at low temperature, the theory of the Quantum Hall effect and analogies with atomic physics have been a strong motivation to revisit the theory on better mathematical grounds. After an introduction motivating the problems, various rigorous results on coherent transport will be reviewed. Then the dissipative part will be described in more detail: the Relaxation Time Approximation (RTA) will lead to anomalous Drude formula, and the Lindblad operator formalism will be described to represent the Markov Approximation. Proofs of Kubo's formula for transport coefficients follow. Open problems and partial results in justifying this formula will be described.

**Friday, October 22, 2004 **

Marco Marletta (Cardiff University, Cardiff, Wales)

Spectral concentration and resonances of a λ-rational Sturm-Liouville problem

2:30 pm / CH 445

Abstract. We examine the resonances and points of spectral concentration of the λ-rational problem

-y'' + [q - λ - t w / (u - λ)] y = 0

with boundary conditions

y(0) cos α + y'(0) sin α =0, y(1) cos β + y'(1) sin β =0.

In particular we study the dynamics of the resonance/eigenvalue λ(t) showing that an embedded eigenvalue can evolve into a resonance and that eigenvalues which are absorbed by the essential spectrum give rise to resonance points. A connection is also established between resonances and points of spectral concentration. Finally some numerical examples are given which show that each of the above theoretical possibilities can be realized.

**Friday, October 15, 2004**

Sergey Naboko (Saint-Petersburg University, Russia)

Jacobi Matrices: spectral phase transition phenomenae

2:30 pm / CH 445

Abstract. We consider some classes of selfadjoint Jacobi matrices (=infinite tridiagonal matrices) with power-like growing entries. The main problem under investigation is the spectral behaviour of the families of such matrices with respect to the changing of some "physical" parameters included into the formulae for matrix entries. Examples will be considered in which a jump from discrete spectrum to purely absolutely continuous spectrum is observed if the parameters cross so-called crititcal surfaces. The main tool here is the asymptotic analysis of the corresponding orthogonal polynomials related to the Jacobi Matrices in a classical way.

**Friday, October 8, 2004 **

Ronghua Pan (Georgia Institute of Technology)

Darcy's law in isentropic porous media flows

2:30 pm / CH 445

Abstract. As an experimental law, Darcy's law plays an important role in the investigation on compressible flows through porous medium. It was conjectured that Darcy's law can be verified by basic balance laws of mechanics time asymptotically. Previous attempts are able to justify the conjecture for small smooth flows away from vacuum. I will show a proof to the conjecture valid for all physical flows. This is joint works with F. Huang and P. Marcati.

**Friday, September 24, 2004 **

Yoshimi Saito (University of Alabama at Birmingham)

The Euler equation for the Hardy operator

2:30 pm / CH 445

Abstract. Let T be a bounded linear operator from a Banach space X into another Banach space Y. Then, for n>0, the n-th approximation number an=an(T) of T is defined by

an = infrank(P)<N< i> ||T - P||

where ||T - P|| denotes the operator norm of T-P and the infimum is taken over all linear operator P from X to Y with their rank smaller than n. Of course, the approximation numbers should reflect some properties of the operator T. For example, T is compact if and only if an converges to 0.

Many works have been done in the case that T is the Sobolev embedding operator for a domain Ω in Rm (or, more exactly, T is one of the variants of the the Sobolev embedding operators). Even in this case not much has been known for a general domain Ω. In the simplest case that Ω=I, where I is a finite interval, we can now compute the approximation numbers for almost all Sobolev embeddings. So called extended trigonometric functions play crucial role in the analysis. The (generalized) Hardy operator T from Lp(I) to Lq(I) is defined by

Tf(x) = v(x) ∫ax u(t) f(t) dt.

There have been many works to evaluate the approximation numbers for the Hardy operator and obtain the asymptotic behavior of an by approximating the functions u and v by simpler function such us step function. In this talk, after giving the background of problem briefly, we are going to present another method which may be called a differential equation approach. We are introducing some central concepts such as the Euler equation for T, extended trigonometric functions, the Prufer transform. Then we discuss how to show the `eigenvalues' of the Euler equation are the same as the approximation numbers (or not). Little background knowledge is necessary for this talk other than elementary knowledge of ordinary differential equations and functional analysis. Indeed most of the arguments are on the level of calculus.

**Friday, September 10, 2004 **

Peter O'Neil (University of Alabama at Birmingham)

An overview of Graph Theory

2:30 pm / CH 445

Abstract. A selection of important results and open problems from graph theory, including Ramsey theory, the reconstruction problem, the four-color problem, genus and chromatic number of a graph, crossing number, and planarity theorems.

**Friday, August 27, 2004 **

Michael D Teubner (University of Adelaide, Adelaide, Australia)

Modelling Electric Potential within a Human Head

2:30 pm / CH 445

Abstract. The ability to locate the source of electrical activity within the human brain as a result of external stimuli, trauma, or medical conditions such as schizophrenia is important for appropriate medical treatment. This presentation will discuss the development of a mathematical model to simulate the movement of electrical potential within the brain and a novel technique for locating the source of electrical potential using an inverse technique. The distribution of electrical potential within the human head can be represented using the Diffusion Equation. The finite difference technique has been applied to this equation using magnetic resonance imaging (MRI) data from a real head and assuming a relationship between tissue type and electrical conductivity within the brain. The resulting forward model has been compared favorably with a known analytic solution. The forward model forms the basis of an inverse model, wherein measured electroencephalography (EEG) data on the scalp are used together with linear and non-linear response functions and non-linear regression to attempt to locate the source of the electrical potential. The mathematical foundations of this presentation will be limited; the major focus will be in showing how relatively simple numerical techniques can be applied very effectively to cognitive neuroscience.