**Friday, April 25, 2003** **László Erdõs (Georgia Tech)**

Quantum diffusion of the random Schrodinger evolution in a scaling limit

2:30 pm / CH 445**Abstract.** Schrodinger equation with a random potential models electron propagation in an impure medium. For large disorder, the eigenstates are localized and no conductance occurs (Anderson localization). Much less is known about the weak disorder regime where delocalization is expected in three or more space dimensions. In this talk I will consider the quantum evolution on a sufficiently long time scale so that the cumulative effect of impurity scatterings yields a diffusive behavior, i.e. the Schrodinger evolution converges to a heat equation in the scaling limit.

**Friday, April 18, 2003** **Joel Smoller (University of Michigan)**

Cosmology, Black Holes, and Shock Waves Beyond the Hubble Distance

2:30 pm / CH 445**Abstract.** We construct a class of global exact solutions of the Einstein equations that extend the Oppenheimer-Snyder (OS) model to the case of non-zero pressure, "inside a black-hole", by incorporating a shock wave at the leading edge of the expansion of the galaxies, arbitrarily far beyond the Hubble length in the Friedman-Robertson-Walker (FRW) spacetime. Here the expanding FRW universe emerges behind a subluminous blast wave that explodes outward from the FRW center at the instant of the Big Bang. The equation of state p=(1/3)(rho) plays a special role, and only in this case, the shock wave emerges from the Big Bang at the speed of light, decelerating from that time onward. The entropy condition implies that the shock wave must weaken to the point where it settles down to an OS interface, that eventually emerges from the White Hole event horizon of an ambient Schwarzschild spacetime. The entropy condition also breaks the time symmetry of the Einstein equations, selecting the explosion over the implosion. These shock wave solutions indicate a new cosmological model in which the Big Bang arises from a localized explosion occurring inside the Black hole of an asymptotically flat Schwarzschild spacetime. (This is joint work with Blake Temple.) I will strive to make this talk understandable to non-experts.

**Friday, April 11, 2003** **Nikolai Chernov (UAB) and Larry Cosenza (Diversified Scientific, Inc)**

Protein crystallization in 10 dimensions: Regression models for optimization, analysis and interpretation of screening experiments.

2:30 pm / CH 445**Abstract.** Optimization of crystallization experiments presents a challenging problem. Crystal growth is affected by many factors in a very unpredictable manner. In experiments done at Diversified Scientific, Inc., ten factors have been selected to study (screen) the formation of crystals in a laboratory. Screening data are used to model the process by a function of ten variables that would allow us to predict optimal conditions for further experiments. A standard method described in the literature is a least squares fit by linear and quadratic functions. This method has limitations, and we found ways to overcome some of them. Our algorithm makes the most of the regression-based approach to the problem.

**Friday, April 4, 2003** **No talk**

**Friday, March 28, 2003** **Simone Warzel (University Erlangen-Nuremberg)**

Spectral asymptotics for magnetic Schroedinger operators

2:30 pm / CH 445**Abstract.** A quantum particle with negative energy, which moves in Euclidean configuration space subject to a potential well V, may be bound to this well. The number of the corresponding bound states heavily depends on the shape of V. In particular, its behavior near infinity determines the spectral asymptotics, that is, it determines the possible accumulation of the associated energy eigenvalues. In this talk we will survey corresponding results on the spectral theory of Schroedinger operators in case an additional constant magnetic field influences the particle. It turns out that the magnetic field always produces an infinite number of bound states whose spectral asymptotics exhibits a transition from a quasi-classical to a true quantum regime.

**Friday, March 14, 2003** **John Aarsvold (Emory University & and Atlanta VAMC)**

Models and Analysis of Multiple-Pinhole Tomographs: Visualizing Mathematics in Applications

2:30 pm / CH 445**Abstract.** For mathematicians, "tomography" generally connotes "inverse problem". However, design of tomographic systems involves analysis of "forward problems", as well as "inverse problems". This presentation is a discussion of "forward problems", or system models, of a class of novel emission imagers known as multiple-pinhole transaxial tomographs--tomographs of present interest because they have the potential to provide real-time dynamic tomography of the physiology of hearts and brains. The discussion will focus on matrix models generated via computer simulation and on singular-value decomposition (SVD) analysis of the matrix models.

Tractable analytic models of multiple-pinhole tomographs have not been developed. Such models when developed will be related to the attenuated Radon transform, the model of an ideal emission tomograph, but the complexities of multiple-pinhole systems make specification of the related transforms difficult. To date, results obtained on the quest to characterize multiple-pinhole tomographs have been obtained through analysis of matrix models generated via computer simulation. Numerical singular-value decomposition (SVD) (generalized eigenvalue decomposition) has been the primary tool of the analysis.

Conventional transmission computed tomography (CT) is an overdetermined problem. Multiple-pinhole tomography is an underdetermined problem. It is this mathematical context--that of non-square matrices--that suggests the use of SVD analysis. The mathematical context follows from the physical reality that available detector resolution does not match desired object-image reconstructed resolution.

Physically, a multiple-pinhole tomograph only acquires some of the information that defines an object being imaged--some object information is not measured and thus is lost to the imaging process. One aspect of design of multiple-pinhole tomographs is determination of the number and locations of pinholes that lead to acquisition of the object information that is "most desired". Mathematically, the goal is to map from the object to the "measurement" space the information that is the most useful information and to map to the "null" space that which is the least useful--for the diagnostic imaging task being performed.

SVD analysis has proved useful in the characterization of many imaging systems. This has been true for the systems of interest here as well and investigations of various system matrices has led to the asking of the following question. Can the relationship between the singular values and singular vectors of a complex pinhole tomograph, one with a multiple-pinhole coded aperture, and those of a simple pinhole tomograph, one with a single-pinhole aperture, be characterized explicitly, given the simple system is a building block of the complex system? Presented will be a discussion of multiple-pinhole tomographs being constructed, numerical results that lead to the asking of the just stated question, and results obtained to date relevant to the answering of the question.

**Friday, March 7, 2003** **Jason Cantarella (University of Georgia)**

Ropelength Criticality and the "Simple" Clasp

2:30 pm / CH 445**Abstract.** Imagine hanging a loop of rope from hooks in the ceiling of a room. Now toss another rope through the loop, and fix both its ends to hooks in the floor. Tighten both ropes (and move the hooks, if you need to) until the entire configuration is as short as possible.

What is the shape of the length-minimizing configuration?

The problem above is a natural geometric optimization problem; given a space curve with an embedded tubular neighborhood of fixed radius, find the shortest curve which can be obtained by any isotopy of the original curve and tubular neighborhood which keeps the tube embedded.

It is known that solutions to these problems always exist, and that they have a certain amount of regularity. For some simple links, such as the classical Gehring link problem, we know explicit examples of ropelength minimizing curves.

Going further requires the development of a general theory of ropelength criticality, and something analogous to an Euler-Lagrange equation for critical configurations. In this talk, we'll use ideas from the theory of frameworks to construct a criterion for ropelength criticality.

We'll then use our picture to come up with some surprising explicit solutions to the problem above.

**Friday, February 28, 2003** **Ed Tymchatyn (University of Saskatchewan)**

Construction of convex metrics in locally connected spaces

2:30 pm / CH 445**Abstract.** A metric d on a space X is convex if each pair of points of X lie in a shortest arc. More precisely, the metric space (X,d) is convex if each pair of points of X lies in an arc isometric to an interval in the real line with its usual metric. Bing (1949) proved that each Peano continuum (i.e. connected, locally connected, compact metric space = continuous and Hausdorff image of the unit interval [0,1] ) can be metrized with a convex metric. Bing used his result that each Peano continuum can be partitioned into finitely many pieces which are themselves Peano continua of small diameter and which are pairwise disjoint except for common parts of their boundaries. (Partititioning can be regarded as an analogue for continua of triangulation for manifolds). I will give a proof of Bing's convexification theorem and indicate how it may be extended to the non-compact case. If time permits I will sketch a simplified proof of the partitioning theorem.

**Friday, February 21, 2003** **Janusz Prajs (University of Idaho)**

Current research in continuum theory

2:30 pm / CH 445**Abstract.** In this talk we will briefly review several of the areas in continuum theory that have been active during last 7-10 years.

**Friday, February 14, 2003** **Günter Stolz (UAB)**

The Anderson Model of Disordered Media

2:30 pm / CH 445**Abstract.** The Anderson model is used in Statistical Physics to describe the electrical conductivity properties of disordered materials such as alloys or crystals with impurities. The main theoretical challenge is to map out neighboring energy regimes in which such materials act as conductors or insulators, respectively, and thus to rigorously establish a phase transition. Mathematical work on the Anderson model combines various fields such as operator and spectral theory, linear ordinary and partial differential equations, and probability theory. In the talk we will introduce the model, and describe some of the mathematical results as well as open problems. The entire talk will be kept non-technical.

**Friday, February 7, 2003** **Steve Cantrell (University of Miami)**

On predator mediated coexistence

2:30 pm / CH 445**Abstract.** In this talk we consider via a reaction-diffusion model the coexistence of two competing species that are preyed upon by a common predator. In the absence of the predator, one of the prey species competitively excludes the other. The predator is presumed to be able to persist on either of the prey species by itself. If the impact of predation is felt more strongly by the dominant prey species, the inclusion of the predator may result in the coexistence of the three species community.

**Friday, January 31, 2003** **Michael Kiessling (Rutgers University)**

Electromagnetic theory: the Born-Infeld legacy

2:30 pm / CH 445**Abstract.** Quantum-Electrodynamics (QED) requires a mathematical regularizer (a so-called ultraviolet cutoff) which (so it seems) cannot be removed. According to experts in high energy physics (a.k.a. M, n\'ee superstring theory), QED is just an effective field theory in the low-energy regime, which happens to require a cutoff for the omitted high-energy physics. However, as early as 1933, Max Born suggested that QED's ultraviolet problems have been inherited from the classical electromagnetic Maxwell--Lorentz model (which, too, requires an ultraviolet cutoff). Together with Leo Infeld he proposed an intriguing modification of the classical electromagnetic field equations and attempted their quantization. Their program never took off from ground, and work on it has remained a fringe activity of main-stream research. In recent years, I have been able to tie together some of the loose ends in the Born--Infeld program, as a result of which one obtains for the first time a mathematically consistent, relativistic, classical electromagnetic theory without UV regularizer; moreover, this formulation suggests its own quantization.

In my talk, which is accessible to graduate students, I will review the mathematical problems of classical electromagnetic theory and their solution within the classical Born--Infeld program. I will also present the first successful steps from there into the quantum world.

**Friday, January 24, 2003** **Justin Corvino (Brown University)**

The Einstein Constraint Equations

2:30 pm / CH 445**Abstract.** The Einstein equation in general relativity is essentially hyperbolic in character, and admits an initial value formulation. The initial data must itself satisfy a system of geometric equations, the constraint equations, which form an (underdetermined) elliptic system. We discuss some methods to construct solutions of the constraint equations, and along the way remark on connections with geometry, topology and physics.

**Friday, December 6, 2002** **Rudi Weikard (UAB)**

The Radon transform and computed tomography

2:30 pm / CH 405**Abstract.** In 1917 J. Radon wrote a paper which would prove decades later to be very influential as it provides the basis for CT (computed tomography, CAT scans). I will present part of the paper, the history of CT, and discuss some of the more practical problems to be solved before a CT becomes applicable. This may shed some light on the relationship between pure and applied mathematics. - The talk will be accessible to all graduate students.

**Friday, November 22, 2002** **Greg Galloway (University of Miami)**

Null Geometry and the Einstein equations

2:30 pm / CH 405**Abstract.** We present some global results concerning null objects in Lorentzian geometry, such as null geodesics and null hypersurfaces, and describe the usefulness of such results in studying the global behavior of solutions to the Einstein equations. In particular, we discuss a uniqueness result for de Sitter space, and some results concerning the topology of asymptotically de Sitter spacetimes.

**Friday, November 15, 2002** **Bharat Soni (UAB Department of Mechanical Engineering)**

Computational Field Simulations: Perspectives, Visions & Future Directions

2:30 pm / CH 405**Abstract.** The state-of-the-art and state-of-the-practice of Computational Field Simulations (CFS) along with critical barriers and future directions in interdisciplinary education and multidisciplinary research will be presented. Emphasis will be placed on all aspects of CFS involving pre-processing (geometry-mesh generation, boundary conditions and problem set-up), processing (numerical solution of pertinent set of non-linear partial differential equations representative of the science of the field to be simulated), and post-processing (visualization and feature detection). The role of HPC and information technology in the overall problem solving environment and enabling technology tools utilized in CFS will be discussed. The tools and technology developed by the presenter and his group will be presented with computational examples of practical interest.

**Friday, November 8, 2002** **Maciej P. Wojtkowski (University of Arizona)**

Geodesic flows on Weyl manifolds and Gaussian thermostats

2:30 pm / CH 405**Abstract.** We will show that geodesic flows of Weyl connections model systems with forcing and thermostatting, in particular Gaussian thermostats. Weyl manifolds with negative sectional curvatures have geodesic flows with at least some hyperbolicity. It is not known if there are homogenous Weyl manifolds with negative sectional curvature (different from Riemannian symmetric spaces). We will discuss this and other open problems.

**Friday, October 25, 2002** **Alexander Elgart (New York University)**

Adiabatic Theorem of Quantum Mechanics

2:30 pm / CH 405**Abstract.** The adiabatic theorem of quantum mechanics describes the behavior of a non-autonomous system driven by means of a slowly altered external field. We will present the basic results and discuss some of the related topics - the theory of linear response and adiabatic invariants in classical mechanics.

**Friday, October 18, 2002** **Michal Misiurewicz (Purdue University Indianapolis)**

Expanding polymodials

2:30 pm / CH 405**Abstract.** One of the most popular areas of Dynamical Systems is Holomorphic Dynamics. This is partially due to nice pictures that accompany it (see Mandelbrot), but the main reason is that powerful tools of the Analytic Functions Theory can be used to study it. We can ask what remains if we keep the general topological setup but remove the assumption that the system is holomorphic. It turns out that under some additional geometric assumptions much of this beautiful theory survives. We call our maps "expanding polymodials" to stress connections with complex polynomials on one hand and with piecewise expanding (polymodal) interval maps on the other hand.

**Friday, October 11, 2002** **George Howard (UAB Department of Biostatistics) **

Difference and Equivalence Testing: What are they and can they peacefully co-exist in the same randomized clinical trial?

2:30 pm / CH 405**Abstract.** We introduce the idea of mathematics working in the evaluations of major medical decisions (in this case, the evaluation of the introduction of the new invasive therapy of carotid stenting), the idea of a randomized clinical trial, and the two major ways of evaluating the success of such therapies.

**Friday, August 30, 2002** **Alexander Blokh (UAB)**

Milnor attractors

2:30 pm / CH 405