## Mathematics Colloquium 2005 - 2006

**Wednesday, April 26, 2006**

Maeve McCarthy (Murray State University)

Constrained Optimization of Eigenvalues for a Particular Class of Singular Sturm-Liouville Operators

10:30 am - 11:30 am / CH 458

Abstract. We investigate the spectral properties of a certain Sturm-Liouville operator that encompasses many mechanical problems, including the buckling of elastic columns. We consider design problems related to the least eigenvalue of these Sturm-Liouville operators. Specifically, we will maximize height for a class of elastic columns, including annular columns. We will discuss the impact of tapering and singular coefficients from both a spectral and design perspective. Singularities leading to limit-circle and limit-point classifications are possible. We will present criteria for the existence of a purely discrete spectrum and hence a least eigenvalue. Integral constraints will be used to specify our objective class. Classical rearrangement techniques will be used to establish the existence of an optimal design in the presence of two design coefficients.

BACKGROUND

**Friday, April 21, 2006**

Roger Nichols Jr. and Taylor N. Snider(UAB)

The Evolution of the Riesz Representation Theorem (Roger Nichols, Jr.) &

Analysis of Some Typical Density Dependent Population Model (Taylor N. Snider)

2:30 pm -3:30pm / Campbell Hall, Fourth Floor

ABSTRACTS

(Roger Nichols, Jr.) In 1907 Frigyes Riesz, a founder of modern functional analysis, published what is now known as “Riesz’s Representation Theorem for Linear Functionals (in a Hilbert Space).” As functional analysis progressed in the early Twentieth Century, Riesz’s Representation Theorem began to evolve, the culmination being the results Kato calls the “First and Second Representation Theorems” in his book Perturbation Theory for Linear Operators. Here an attempt is made to track this evolution, with a look at the important implications of the theorems along the way.

(Taylor N. Snider) Our goal is to analyze the asymptotic dynamical behavior exhibited by several two-stage density dependent population models. To study this, two-dimensional operating diagrams were generated. In our four parameter models, the operating diagrams indicate the stable and chaotic behaviors when two parameters are fixed and two are allowed to vary. For example, an operating diagram could allow one to conclude that for a wide range of biologically significant parameter values a stable positive equilibrium exists. The equations for the juvenile and adult populations were programmed for each model in C++. Then operating diagrams were generated from the data. Excel was used to produce corresponding orbit diagrams.

BACKGROUND

Roger Nichols and Nic Snider are Fast-Track students in the Mathematics Department at UAB.

Maeve McCarthy received her Bachelors and Masters degrees from the National University of Ireland, Galway. She obtained a PhD in Computational and Applied Mathematics from Rice University. She is an associate professor at Murray State University where she has worked since 1998. She is a co-PI in the Biomathematics initiative there. Her mathematical interests include inverse problems and design problems associated with singular Sturm-Liouville problems, and the identification of parameters in parabolic systems.

**Friday, April 14, 2006**

Brian Smith(UAB)

A Blow-Up Result for the Parabolic Scalar Curvature Equation

2:30 pm / CH 458

Abstract. In order to construct maximal initial data in general relativity one must construct a 3-manifold of nonnegative scalar curvature. The most practical method for doing this involves solving a second order nonlinear parabolic equation in which a radial variable r plays the same role, analytically, as the time variable does in the heat equation or porous medium equation. Unfortunately, nonnegative scalar curvature can easily lead to blow-up of solutions at some finite r = r1. This talk focuses on a class of metrics for which the blow-up occurs “evenly” enough that a change of variables shows that there is actually no blow-up in the constructed metric. The talk will not presuppose too much background in differential geometry or parabolic equations; rather, the necessary concepts will be introduced during the talk.

BACKGROUND

Originally a member of the Fast-track Program in the UAB Mathematics Department, Brian Smith went on to get his PhD in Mathematics from UAB in 2001. Afterwards, Dr. Smith held a visiting position at Cornell University where he subsequently held a VIGRE postdoctoral position. Currently, he is visiting UAB, and is scheduled to begin a 2-year postdoctoral position in the fall under the project “Space-Time-Matter” set up between the Free University of Berlin, Humboldt University of Berlin, and Potsdam University (Germany).

**Friday, April 07, 2006**

Jamie H. Glass (University of Alabama)Cancelled due to weather

2:30 pm / CH 458

Abstract. Ms. Glass will address some of the following issues.

1) What is the role of faculty in teaching in the Math Lab environment? How is this different from the traditional teaching environment?

2) What goes on in the weekly class meetings that replace the traditional lectures? What needs to be prepared by the instructor, and how does an instructor conduct a weekly meeting?

3) What is the role of the tutor in the math lab, and how should a faculty tutor, or student tutor, approach a student question in the lab?

BACKGROUND

Ms. Glass received her BS degree in Mathematics from Jacksonville State University and her MA degree in Math Education from UAB. She has taught at the University of Alabama since 1988, full time since 1990 teaching everything from remedial algebra to Calculus I. She was a course leader for the Intermediate Algebra course in 1999 and therefore was chosen to visit the Math Emporium at Virginia Tech to view the model they were using to teach freshmen level mathematics. She volunteered to pilot three sections of intermediate algebra in the spring of 2000 using purchased software. In the fall of 2000, they forged ahead and put all students taking intermediate algebra in a lab setting to take their course. Since that time, the lab has grown to include five courses ranging from remedial algebra to business calculus. They are in their fifth year using this process and are eagerly looking forward to growing as their student body grows. At the beginning of the second year of operation, she was asked to become the lab coordinator. Her main duties as lab coordinator are: hiring assistants and monitors and scheduling assistants, GTAs, instructors, and monitors for lab hours they are to work, making sure the day to day operation is running smoothly, handling student questions and concerns, making decisions about the enforcement of policies, etc.

**Friday, March 17, 2006**

Ruth Parker (Mathematics Education Collaborative)

Numerical Literacy: Continuing the Conversation

2:30 pm / CH 445

Abstract. Back by popular request, Ruth Parker will address additional common practices that interfere with the development of numerical reasoning. Specifically, this session will illuminate the damage done when knowing basic facts is considered a pre-cursor to problem solving. Dr. Parker will present an alternative approach for teaching the so-called ‘basics,’ and will encourage a discussion about the efficacy of the ideas from the perspective of university-level mathematicians, scientists, engineers, educators, philosophers, poets, and others.

BACKGROUND

Dr. Parker currently directs an NSF-funded project that has, for the past six years, developed and implemented a community-engagement model for improving mathematics instruction in schools. The model has been implemented in educational communities in Oregon, Colorado, and Alabama. Dr. Parker is the CEO of Mathematics Education Collaborative which is a supporting partner in the NSF-funded Greater Birmingham Mathematics Partnership.

**Friday, March 10, 2006**

Chad Wilson (Vesta Insurance Group)

How Actuaries Use Mathematics to Price Property Insurance Products2:30 pm / CH 445

Abstract. A firm in the insurance industry provides a promise to make a payment after an accident, and thus does not know, a priori, the cost of the product it provides. Nevertheless, basic probability theory shows that the firm can provide the product at a reasonable price. In addition, basic statistical analysis reveals that the price varies by characteristics of policyholders, and advanced statistical analysis prescribes exactly how the price should vary in order to maintain profitability. Such analyses are limited in their predictive accuracies, though, by the availability of data. For example, the relatively small number of hurricanes over the past 100 years of record-keeping reduces the accuracy of loss-cost predictions based on historical loss costs. Therefore, to determine the appropriate price of insuring a loss caused by a hurricane, the actuary uses techniques of mathematical modeling.

BACKGROUND

Chad Wilson completed his Master of Science degree in 2001 from UAB by participating in the Mathematics Fast Track Program. He currently researches new methods and refines existing methods of pricing property insurance products for Vesta Insurance in Birmingham, Alabama. Outside of studying for actuarial exams, activities include walking, supporting UAB basketball, and playing Halo 2 online.

**Friday, February 24, 2006**

Lee Meadows (School of Education, UAB)

The Role of Collaborative Learning in Reform of College Teaching

2:30 pm / CH 445

Abstract. Collaborative learning (also called cooperative learning) is showing strong promise as a central feature in the reform of college teaching. Collaborative learning engages today's students, who have grown up connected to one another via technology. Collaborative learning also matches the growing body of cognitive science about how people learn. In this talk, I'll overview collaborative learning, and we'll discuss potential applications to the teaching of mathematics.

BACKGROUND

Dr. Meadows is an associate professor in UAB's School of Education. He is a former high school chemistry and physics teacher, and he holds a PhD in science education from the University of Georgia. His expertise is in the reform of high school science, especially the move to inquiry-based science teaching.

**Friday, February 17, 2006**

Alexander Pushnitski (King's College, London)

The Spectrum of the Weakly Perturbed Landau Hamiltonian and Orthogonal Polynomials in the Complex Domain

2:30 pm / CH 445

Abstract. The subject of the talk is the spectrum of a two-dimensional Schrödinger operator with constant magnetic field and a compactly supported electric field. The eigenvalues of such an operator form clusters around the Landau levels. The eigenvalues in these clusters accumulate towards the Landau levels super-exponentially fast. It appears that these eigenvalues can be related to a certain sequence of orthogonal polynomials in the complex domain. This allows one to accurately describe the rate of accumulation of eigenvalues towards the Landau levels. This description involves the logarithmic capacity of the support of the electric potential. The talk is based on a joint work with Nikolai Filonov from St. Petersburg.

BACKGROUND

Dr. Pushnitski is currently spending the 2005/06 academic year as a Leverhulme Research Fellow at Caltech. His research interests include spectral perturbation theory of selfadjoint operators in Hilbert space, spectral shift function theory, and Schrödinger operators.

**Friday, December 16, 2005**

Andrej Zlatoš (University of Wisconsin-Madison)

Reaction and Diffusion in the Presence of Fluid Flow

10:30 am / CH 458

Abstract. In this talk I will review some recent developments in the area of reaction-diffusion-advection equations. I will concentrate on the phenomenon of quenching (extinction) of flames by a strong flow, as well as on quenching in the presence of various types of non-linear reaction terms. These questions naturally lead to the related problem of estimating the relaxation speed for the solution of the corresponding passive scalar equation, which will also be discussed.

BACKGROUND

Dr. Zlatoš received his doctoral degree in 2003 from California Institute of Technology under the direction of Barry Simon. His research interests include Spectral Theory of Schrödinger Operators and Jacobi Matrices, Orthogonal Polynomials, Partial Differential Equations, Reaction-diffusion Equations, and Discrete Models for Fluid Dynamics. He is currently a Van Vleck Visiting Assistant Professor of Mathematics at the University of Wisconsin-Madison.

Tuesday, December 13, 2005

Tuesday, December 13, 2005

Benjamin Schlein (Harvard University)

Derivation of the Cubic Non-linear Schrödinger Equation from Many-body Quantum Dynamics

10:30 am / CH 458

Abstract. We consider a system of N bosons interacting through a repulsive short range mean field potential. In the limit of large N, we prove that the macroscopic dynamics of the system can be described by the one-particle nonlinear Schrödinger equation.

**Monday, December 12, 2005**

Simone Warzel (Universität Erlangen-Nürnberg and Princeton University)

Spectral Analysis of Anderson-type Operators on Trees

10:30 am / CH 458

BACKGROUND

Dr. Warzel received her doctoral degree in 2003 from the University of Erlangen-Nürnberg under the direction of Hajo Leschke. From 1997 to 2003, she served as a Research and Teaching Assistant at the University of Erlangen-Nürnberg. She had also spent a year at Cambridge University (England) to study Theoretical Physics. Currently she is a Research Fellow at Princeton University.

Friday, November 18, 2005

Ruth Parker (Mathematics Education Collaborative)

How Mathematical Instruction Results in Numerical Illiterates, and What We Can Do About It

2:30 pm / CH 445

Abstract. The talk will address the issue of development of numerical reasoning in K-16 classrooms and will propose specific and dramatic changes in the teaching of number. The teaching of number is an issue of enormous educational importance given its central position in the mathematics education of our nation's youth.

BACKGROUND

Dr. Parker currently directs an NSF-funded project that has, for the past six years, developed and implemented a community-engagement model for improving mathematics instruction in schools. The model has been implemented in educational communities in Oregon, Colorado, and Alabama. Dr. Parker is the CEO of Mathematics Education Collaborative which is a supporting partner in the NSF-funded Greater Birmingham Mathematics Partnership.

Wednesday, November 16, 2005

Klaus Schmitt (University of Utah)

Sub and Super-solutions for Nonlinear Elliptic Problems

2:30 pm / CH 458

Abstract. Perron's method of sub and super-harmonic functions allowed for an elegant solution of the Dirichlet problem with general boundary data for Laplace's equation on very general domains. His ideas were later extended to nonlinear elliptic Dirichlet boundary value problems (and other types of boundary value problems) via the method of sub- and super-solutions. The lecture will survey the evolution and use of these concepts and conclude with some results which allow for a unified treatment of many different types of boundary value problems and also obstacle and unilateral problems.

BACKGROUND

Dr. Schmitt received his doctoral degree from the University of Nebraska in 1967. Much of his work is on the theory of nonlinear differential equations and inequalities, especially the study of existence, multiplicity, and bifurcation of solutions to boundary value problems for nonlinear elliptic partial differential equations and inequalities. Professor Schmitt has visited UAB several times, and was a plenary speaker at the 1990 UAB International Conference on Differential Equations and Mathematical Physics. He has been a faculty member in the Department of Mathematics at the University of Utah since 1967. Outside interests include hiking and playing tennis.

**Friday, November 04, 2005**

This email address is being protected from spambots. You need JavaScript enabled to view it. (UAB)

Wigner-von Neumann Perturbations of a Periodic Potential: Spectral Singularities in Bands

2:30 pm / CH 396

Abstract. Wigner-von Neumann type perturbations of a periodic one-dimensional Schrödinger operator are considered. The analysis is based on the investigation of the asymptotics of generalized eigenfunctions and subordinacy theory. It is proven that the subordinated solutions and therefore the embedded eigenvalues may occur at the points of the absolutely continuous spectrum satisfying a certain resonance (quantization) condition between the frequencies of the perturbation, the frequency of the background potential and the corresponding quasimomentum. The presentation is based on a joint work with P. Kurasov (University of Lund, Sweden).

BACKGROUND

Dr. Naboko received his Doctor of Sciences’ degree from the Steklov Mathematical Institute of the Russian Academy of Sciences, St. Petersburg (Russia), in 1987. He has been Professor in the Department of Mathematical Physics at St. Petersburg State University since 1991. He works on the Spectral Theory of Selfadjoint and Nonselfadjoint Operators, and Applications of Complex Analysis in Operator Theory and Mathematical Physics. He is currently a Visiting Professor of Mathematics at UAB. Outside interests include mountain hiking, classical music and ancient history.**Friday, October 28, 2005**

Ryoichi Kawai (UAB)

Fluctuations, Dissipation and the Jarzynski Equality

2:30 pm / CH 396

Abstract. Thermodynamics laws govern the behavior of macroscopic quantities in equilibrium. However, they do not state anything about the fluctuation of the quantities. In the linear regime outside equilibrium, Onsager has shown a relation between fluctuation and dissipation (the Fluctuation-Dissipation Theorem). However, it is limited to systems in near equilibrium. Recently, it has been shown that time-reversibility of deterministic or stochastic dynamics implies relations between fluctuation and dissipation in systems far from equilibrium, taking the form of intriguing equalities: the fluctuation theorem, the Jarzynski equality, and the Crooks relation. In this talk, I will attempt to explain these relations using simple kinetic models which can be solved analytically. The results will be compared with molecular dynamics simulations.

BACKGROUND

Dr. Kawai received his PhD in Theoretical Condensed Matter Physics from Waseda University (Japan) in 1985. He worked as a Research Associate at various institutions including Imperial College of London (Mathematics), IBM T. J. Watson Research Center (Physical Science), University of California, San Diego (Chemistry & Biochemistry), and San Diego Supercomputer Center. Currently, he is an Associate Professor of Physics at University of Alabama at Birmingham. His research interests include a variety of subjects from Solid State Physics to Cellular Biology. Essentially, he is interested in any system that shows remarkable properties generated by interaction among many particles, in particular under non-equilibrium conditions.

**Friday, October 21, 2005**

Mohammad Ghomi (Georgia Institute of Technology)

Shades on Illuminated Surfaces

2:30 pm / CH 396

Abstract. We discuss how one can examine the geometry and topology of a surface immersed in Euclidean space by studying the shades formed on that surface when it is illuminated by parallel rays of light. The study of shades on illuminated surfaces is of interest in variational problems in geometric analysis and the "shape from shading" problems in computer vision. We will also discuss some relations to the study of submanifolds without any pairs of parallel or intersecting tangent lines, and give a quick survey of recent results in this area.

BACKGROUND

Dr. Ghomi got his PhD in 1998 under the direction of Joel Spruck at Johns Hopkins University. He has subsequently held positions at University of California at Santa Cruz, University of South Carolina, and Pennsylvania State University. Presently Dr. Ghomi is an Associate Professor at Georgia Tech. His interests range around classical differential geometry, i.e., the study of curves and surface in Euclidean space, and in 2003 he obtained an NSF CAREER award to pursue his studies in this area. His interests outside Mathematics range from classical movies to Architecture and design, especially from the Arts and Crafts period. Also, Dr. Ghomi and his wife greatly enjoy raising their 20 month old son.

**Friday, September 23, 2005**

Nandor Simanyi (UAB)

Rotation Sets of Billiards

2:30 pm / CH 396

Abstract. We have studied the rotation sets of two families of billiards:

(A) The inertia motion of a point in the flat torus minus a strictly convex, compact obstacle with a smooth boundary. Here, by definition, the rotation set consists of all limiting points of the average displacements along orbit segments, when the lengths of these segments tend to infinity.

(B) The inertia motion of a point in a rectangle minus a strictly convex, compact obstacle with a smooth boundary inside the interior of the rectangle. Here the rotation set is, by definition, the set of all limiting values of the average wrapping around the obstacle by orbit segments whose lengths tend to infinity.

I will present an overview of the obtained results (characterizations of several rotation sets of different type) by also including some simple proofs. Though the point of view of studying these objects is of topological nature, the proofs use ideas from geometry, topology, and a bit of combinatorics. The talk should be accessible to graduate and advanced fast-track students.

These are joint results with A. Blokh and M. Misiurewicz.

BACKGROUND

Dr. Nandor Simanyi received his doctoral degree from Roland Eotvos University, Budapest, in 1987, and his subsequent scientific degrees (C. Sc. and D. Sc.; i. e., Candidate of Sciences and Doctor of Sciences) from the Hungarian Academy of Sciences in 1989 and 1995. He works primarily in the theory of non-uniformly hyperbolic dynamical systems. During the 1980s and 1990s he worked as a research professor for the Alfred Renyi Mathematical Institute (Budapest), and as a professor at the University of Szeged, while visiting several universities in the United States. He has been a faculty member in the Department of Mathematics at UAB since 1999. Outside interests include hiking and enjoying classical music.

**Friday, September 16, 2005**

Nikolai Chernov (UAB)

Hilbert's 13th Problem, Kolmogorov's Superposition Theorem, Neural Networks, and Protein Crystallization

2:30 pm / CH 396

Abstract. We present a deep mathematical result - Kolmogorov's solution to Hilbert's 13th problem (without proof) - in relation with a practical scheme for approximating continuous functions of several variables. This scheme, called neural networks, worked surprisingly well on experimental data obtained at the Center for Biophysical Sciences and Engineering (UAB) and Diversified Scientific Inc., Birmingham.

BACKGROUND

Dr. Chernov, who received his PhD in 1984 from Moscow University, joined UAB in 1994. His research interests are in the areas of Dynamical Systems, Probability and Statistics. Outside interests include walking and hiking.

**Friday, September 09, 2005**

Jere P. Segrest and Gilbert Weinstein (UAB)

Novel Protein-Lipid Conformations of High Density Lipoproteins through Molecular Dynamics (J.P.Segrest)

&

Minimal Surfaces with a Free Elastic Boundary (G. Weinstein)

2:30 pm / CH 396

Abstract.

(Jere P. Segrest) We recently completed a series of molecular dynamics (MD) simulations in which phospholipids were incrementally removed from a previously described discoidal high density lipoprotein (HDL) model (1) until phospholipid-free apoA-I (the major protein component of HDL) resulted. The resulting molecular structures are particularly compelling for three reasons: i) The apoA-I amphipathic α-helical double belt twists to approximate the X-ray structure of lipid-free apoA-I independent of four different conditions of particle shrinkage, and, in so doing, simultaneously conforms to the saddle-shaped (Enneper’s minimal surface?) edge of the lipid bilayer. ii) The dramatic changes in the structure of both protein and lipid occur within a few nanoseconds of MD simulation time, an unprecedented timescale for such large, complex supramolecular assemblies. iii) To the best of our knowledge, this is the first example of MD simulation producing fundamentally new unprecedented results. We therefore have some confidence that these studies provide a high resolution molecular view of apoA-I.

(1) Segrest, J. P., Jones, M. K., Klon, A. E., Sheldahl, C. J., Hellinger, M., De Loof, H., & Harvey, S.C. (1999) J. Biol. Chem. 274, 31755-31758.

(Gilbert Weinstein) We propose a model to predict the shape of the HDL particles observed in the MD simulations described above. The model combines a surface with infinite surface tension with a free boundary composed of a fixed-length thin elastic ribbon. The model leads to the minimal surface equation with free boundary coupled to a 1-d elastic ODE on that boundary.

BACKGROUND

Dr. Segrest, who developed the theory of the amphipathic helix, joined UAB in 1974. He is a Professor of Medicine and Director of the Center for Computational and Structural Biology and of the Atherosclerosis Research Unit. He is currently supported by NIH grants. Outside interests include reading classical Greek and Roman history and science fiction and bicycling.

Dr. Weinstein, whose research interests are in the areas of General Relativity, Differential Geometry, and Nonlinear Partial Differential Equations, joined UAB in 1991. His research has been supported by NSF. Outside interests include playing piano.

**Friday, August 26, 2005**

Michael Teubner (University of Adelaide)

Modelling Groundwater

2:30 pm / CH 396

Abstract. Groundwater is used extensively throughout the world as a supply of drinking water, for agricultural purposes, and for industry. During the last century, the pristine nature of many groundwater aquifers has deteriorated due primarily to mankind’s interference with his environment. This has resulted in many aquifers becoming contaminated, thereby reducing their ability for use.

Understanding and being able to predict the movement of water beneath the ground is difficult because it cannot be seen or its motion measured. Groundwater levels can be measured, and these provide information on location and water chemistry. However, an acceptable understanding of the movement of groundwater can only be obtained from modelling a groundwater system by numerically solving the mathematical equations that govern groundwater flow. This is even more so in the case with contaminants present in the groundwater. Modelling can provide information on contaminant transport, potential sources of the contamination, and possible outcomes of appropriate remedial action.

This presentation will consider how a groundwater model is developed and how it can be used to assess groundwater movement.

An additional benefit of groundwater modelling that will be considered is the ability to examine aquifer storage, treatment, and recovery, whereby surface water or treated sewage can be injected into a groundwater aquifer for later use. A number of applications will be presented and discussed, with very little mathematics.

BACKGROUND

Dr. Teubner is currently a Senior Lecturer in Applied Mathematics at The University of Adelaide in South Australia. His teaching responsibilities include computational fluid dynamics, engineering mathematics, numerical methods, and modelling with differential equations. He has a number of graduate students with whom he conducts research into modelling circulation in shallow water free surface flows, groundwater modelling, and inverse modelling. Prior to joining The University of Adelaide, Dr. Teubner spent 16 years as an environmental consultant in the US. External to the University, Dr. Teubner has an olive grove and a small vineyard, where he produces excellent (unfortunately non-export) red wine!