## Mathematics Colloquium 2006 - 2007

Friday, April 20, 2007

Fernando Schwartz(Duke)

On the topology of black holes
3:00 pm - 4:00 pm / CH 405

Abstract. I will give a brief description of Einstein's equations of general
relativity with a special focus on the theory of black holes. I
will talk about Hawking's black hole topology theorem as well as its
generalization by Galloway and Schoen. I will discuss the recent
discovery of "black rings" by Emparan and Reall, which are black
holes with horizon topology S1 x S2. I will then show my recent
work on how to construct a Riemannian version of black rings within
a rigorous mathematical setting.

Friday, April 6, 2007

Tom Barr(Rhodes College-UAB)

Cryptography: 'Secrets' of Secret Writing
3:00 pm - 4:00 pm / CH 405

Abstract. Internet purchases, ATM transactions, and web-based e-mail
interfaces are examples of everyday activities that are possible
because of cryptography, what might be called mathematically-based
methods for securing information from unintended recipients. The
creation and application of cryptographic tools has been ongoing at
least as long as writing has existed, but in the early twentieth
century it became a mathematical discipline and has subsequently
continued to develop as an active area of research and technological
application. In this talk, we will survey historical examples of
cryptography and cryptanalysis, describe an unbreakable cipher,
illustrate approximations to it, and discuss the role of public-key
cryptography in key exchange and other cryptographic protocols.

Friday, March 30, 2007

David Damanik(Rice)

The subcritical almost Mathieu operator
3:00 pm - 4:00 pm / CH 405

Abstract. This talk will discuss Problem 6 on Barry Simon's list of
open Schr\"odinger operator problems for the twenty-first century,
which reads: "Prove for all irrational alpha and lambda<2 that the
spectrum (of the almost Mathieu operator) is purely absolutely
continuous." We explain the history of the problem and how it fits
into the more general quest for a complete identification of the
spectral type, describe some heuristics explaining the complete
picture, and give a brief tour guide to the recent solution of the
problem.

Friday, March 23, 2007

Yanyan Li(Rutgers)

Some Liouville theorems and gradient estimates
3:00 pm - 4:00 pm / CH 405

Abstract. The classical Liouville theorem says that a positive
entire harmonic function must be a constant. We give a fully
nonlinear version of it. This extension enables us to establish
local gradient estimates of solutions to general conformally
invariant fully nonlinear elliptic equations of second order. This
talk will start from a proof of the classical Liouville theorem
using only the comparison principle and the invariance of
harmonicity under Mobius transformations and scalar multiplications.
We will then outline the proof of the comparison principle used in
establishing the new Liouville theorem. Finally we outline the proof
of the gradient estimates via the Liouville theorem.

Friday, March 9, 2007

Bruno Nachtergaele(UC Davis)

The Marvelous Mathematics of Quantum Spin Systems
3:00 pm - 4:00 pm / CH 405

Abstract. Quantum spin systems have been studied in physics as models of
magnetism from the beginning of quantum mechanics. As mathematical
models they seem to possess that magical mixture of simplicity and
complexity just right to be able to inspire an amazing variety of
new mathematical structures and ideas. We will present several of
those marvelous pieces of mathematics drawn from analysis,
combinatorics, probability, and representation theory.

Wednesday, March 7, 2007

A finite rank property of Teichmuller space
1:00 pm - 2:00 pm / CH 458

Abstract. Given a closed surface, all the conformal structures that can be
equipped on the surface constitutes the Teichmuller space of the
surface. For a surface of higher genus, it is the space of all the
hyperbolic metrics defined on the same surface. We will introduce a
distance function on a Teichmuller space, called Weil-Petersson
distance function, and will discuss the convex geometry induced by
the distance function.

Friday, March 2, 2007

Rupert Frank(KTM)

Hardy-Sobolev and Lieb-Thirring inequalities
3:00 pm - 4:00 pm / CH 405

Abstract. We consider the relation between inequalities on eigenvalue moments
of Schr\"odinger-like operators and Sobolev-like inequalities. We
present some new Hardy-Sobolev inequalities for the Laplacian in a
convex domain and for fractional powers of the Laplacian. As a
consequence we deduce that the Lieb-Thirring inequalities on moments
of negative eigenvalues of Schr\"odinger-like operators remain true,
with possibly different constants, when the critical Hardy-weight is
subtracted from the Laplace operator.

The talk is based on joint works with T. Ekholm, with E.H. Lieb and
R. Seiringer, and with R. Benguria and M. Loss.

Friday, February 23, 2007

Chris Mouron(Rhodes College, UAB)

Expansive homeomorphisms and continuum theory
3:00 pm - 4:00 pm / CH 405

Abstract. A homeomorphism $h:X\longrightarrow X$ is expansive if there exists a $c>0$ such that for every $x, y \in X$ there exists an integer $n$ such that $\mbox{d}(h^n(x),h^n(y))\geq c$. Expansive homeomorphisms exhibit
sensitive dependence on initial conditions in the strongest sense in
that no matter how close any two points are, their images will
eventually be a certain distance apart.

A continuum is a compact connected metric space.
A continuum $X$ admits an expansive homeomorphismif there exists an expansive homeomorphism on $X$. A few examples of expansive homeomorphisms will be presented and the necessary topological structure of continua that admit expansive homeomorphisms will be discussed. This talk is intended for a general mathematics audience.

Friday, February 16, 2007

Alfonso Castro(Harvey Mudd College)

The Nehari manifold and superlinear boundary value problems
3:00 pm - 4:00 pm / CH 405
Abstract. The role of the Nehari manifold in the understanding of the solvability of superlinear elliptic boundary value problems will be discussed. Geometric properties of this manifold help explain low Morse index solutions.

Friday, February 2, 2007

Christopher Coffey(UAB)

Biostatistics at UAB
3:00 pm - 4:00 pm / CH 405

Abstract. Biostatistics involves the development and application of statistical techniques to scientific research in health-related fields, including medicine, epidemiology, public health, dentistry, and nursing.  There is currently a serious and growing shortage of biostatisticians with advanced degrees, particularly those with the necessary skills to both develop new statistical techniques and apply existing techniques in a collaborative manner.  To address this shortage, the Department of Biostatistics in the School of Public Health at the University of Alabama at Birmingham (UAB) has undergone a rapid expansion with respect to the number of faculty, postdoctoral fellows, and graduate students during this decade.  As part of this growth, the department has formed two sections, Statistical Genetics and Research Methods & Clinical Trials.  Consequently, the graduate program in the department has been modified to meet the objective of producing marketable doctoral students who earn a PhD in biostatistics with an emphasis in either statistical genetics or clinical trials.  This presentation will summarize the field of biostatistics, review job opportunities available for students who wish to pursue a career in the field, and summarize the educational opportunities that exist within the department of biostatistics at UAB.

Thursday, January 25, 2007

Spectral Asymptotics on Heisenberg Manifolds
10:00 - 11:00 am / CH 458

Abstract. Let R(t) be the error term in Weyl's law for (2n+1)-dimensional Heisenberg manifolds. Based on the Petridis-Toth conjecture R(t)=Oδ(tn-1/4+δ). We discuss new pointwise and moment results that provide evidence for this conjecture in three dimensions and a proof for it in higher dimensions. The methods used also allow a proof of a new fifth moment result in the case of the Dirichelet Divisor problem.

Tuesday, January 23, 2007

Roman G. Shterenberg(Univ of Wisconsin)

Blow up and regularity for Burgers equation with fractional dissipation
10:00 - 11:00 am / CH 458

Abstract. We present a comprehensive study of the existence, blow up and regularity
properties of solutions of the Burgers equation with fractional
dissipation. We
prove existence of the finite time blow up for the power of Laplacian
$\alpha < 1/2$, and global existence as well as analyticity of solution for
$\alpha \geq 1/2$. We also discuss solutions with very rough initial data.

Friday, January 19, 2007

Marta Garcia-Huidobro(Pontificia Universidad Catolica de Chile)

On the uniqueness of positive solutions of a quasilinear equation containing a weighted p-Laplacian, the superlinear case
3:00 - 4:00 pm / CH 405
Abstract. We consider the quasilinear equation of the form $$-\Delta_p=K(|x|)f(u),\quad x\in \RR^n,\quad n>p>1, \leqno( P)$$ where
$\Delta_pu:=\mbox{div}(|\nabla u|^{p-2}\nabla u)$ is the degenerate $p$-Laplace operator and the weight $K$ is a positive $C^1$ function defined in $\RR^+$. We deal with the case in which $f\in C[0,\infty)$ has one zero at $u_0>0$, is non positive and not identically 0 in $(0,u_0)$, and is locally lipschitz, positive and satisfies some
superlinear growth assumption in $(u_0,\infty)$. We carefully study the behaviour of the solution of the corresponding initial value
problem for the radial version of the quasilinear equation, as well as the behaviour of its derivative with respect to the initial value.
Combining, as Cort\'azar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong,  with some separation results,
we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value,
which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the
Dirichlet problem on a ball.

Thursday, January 18, 2007

Mikyoung Lim(Ecole Polytechnique, France)

Reconstruction of inhomogeneities via boundary measurements
2:00 - 3:00 pm / CH 458

Abstract. We establish an explicit asymptotic formula for the steady state voltage perturbations caused by closely spaced small conductivity inhomogeneities . Based on this new formula we design a very elective numerical method to identify the location and some geometric features of these inhomogeneities from a finite number of boundary measurements.

We also obtain  upper and lower bounds on the gradient of solutions to the conductivity problem in the case where two circular conductivity inclusions in two dimensions, or spherical inclusions in the three dimensional case, are very close but not  touching. These bounds depend on the conductivities of the inclusions, their radii, and the distance between them. Their novelty is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate.

Tuesday, January 16, 2007

Lei Zhang(University of Florida)

Compactness of solutions to the Yamabe problem
2:00 - 3:00 pm / CH 458

Abstract. In nonlinear PDEs and geometric analysis there is a well known
Yamabe problem which connects different fields of mathematics and physics. The Yamabe problem is equivalent to finding a positive solution to a PDE defined on compact Riemannian manifolds. Through the works of Yamabe, Trudinger, Aubin and Schoen the existence problem was solved in 1984. Corresponding to the existence problem there is a compactness problem that concerns a uniform estimate for all the solutions. In this talk I shall report the progress on the compactness problem and explain my joint work with Yanyan Li. The talk is intended for a mathematical literate audience.

Friday, January 5, 2007

Boundary Control Approach to Inverse Problems
3:00 - 4:00 pm / CH 405

Abstract. In this talk we describe an approach to
inverse problems (the so-called Boundary Control method) which is
based on
deep connections between controllability and
identification problems and is applicable to a wide range of
linear distributed systems. We link also the Boundary Control
approach and the Titchmarsh-Weyl theory. This provides a natural
interpretation of the $A-$amplitude due to Simon and yields a new
efficient method to evaluate the Titchmarsh-Weyl $m-$function
associated with the Schr\"{o}dinger operator $H=-\partial _{x}^{2}+q\left( x\right)$ on $L_{2}\left( 0,\infty \right)$
with Dirichlet boundary condition at $x=0.$ \vskip1mm \noindent As
an example of the approach we consider the inverse problem for a
graph. We suppose that the graph is a tree (i.e., it does not
contain cycles), and on each edge the Schr\"odinger equation (with
a variable potential) is defined. The Weyl matrix function is
introduced through all but one boundary vertices. We prove that
the Weyl matrix function uniquely determines the graph (its
connectivity and the lengths of the edges together with potentials
on them). If the connectivity of the graph is known, the lengths
of the edges and potentials on them are uniquely determined by the
diagonal terms of either the Weyl matrix function, the response
operator or by the back scattering coefficients. \vskip1mm
\noindent The talk is partly based on joint work with P.~Kurasov,
V.~Mikhailov, and A.~Rybkin.

Friday, December 8, 2006

Thomas Chen(Princeton)

Quantum dynamics in random environments and renormalization
9:30 am - 10:30 pm / CH 458

Abstract. In this talk, recent developments in the analysis of the Schroedinger
dynamics of particles in random environments are surveyed. In
particular, the link between hydrodynamic scaling limits, lower
bounds on the localization lengths of eigenvectors, and non-perturbative
renormalization is explained.

Wednesday, December 6, 2006

Artem Zvavitch(Kent State)

The Busemann-Petty problem for arbitrary measures
1:00 pm - 2:00 pm / CH 458

Abstract. The Busemann-Petty problem asks whether symmetric convex bodies in $R^n$
with smaller (n-1)-dimensional volume of central hyperplane sections
necessarily have smaller n-dimensional volume.

Clearly, the Busemann-Petty problem is a triviality for $n=2$ and the
answer is yes''. Minkowski's theorem shows that an origin-symmetric
star-shaped body is uniquely determined by the volume of its hyperplane
sections. In view of this fact it is quite surprising that the answer to
the original Busemann Petty problem can be negative. Indeed, it is
affirmative if $n\le 4$ and negative if $n\ge 5$.

In this talk we will present a generalization of the Busemann-Petty
problem to essentially arbitrary measure in place of the volume. We also
present applications of the latter result by proving several
inequalities concerning the measure of sections of convex symmetric
bodies in $R^n$.

Friday, December 1, 2006

Simultaneous continuous extension of uniformly continuous functions
3:00 pm - 4:00 pm / CH 405

Abstract. There have been many improvements over the years to the Tietze-Urysohn
Extension Theorem. Dugundji in 1951 proved that if X is a metric space and A a
closed subset then there is a continuous linear extension operator from C(A) to
C(X). Kunzi and Shapiro in 1997 proved Dugundji's theorem for partial functions
of X with variable but compact domains.

Let X be a metric space and Cub the family of uniformly continuous bounded
real-valued functions whose domains are bounded subsets of X. Topologize Cub by
taking as distance between functions the Hausdorff distance between their
graphs.

Theorem 1. Then there is a continuous linear extension operator from Cub to
C(X) where C(X) has the topology of pointwise convergence.

Theorem 2. There is a continuous regular extension operator from Cub to the
space of uniformly continuous bounded  functions on X.

Friday, November 17, 2006

Michael Plum(Univ Karlsruhe)

An existence and enclosure method for non-linear elliptic boundary problems
3:00 pm - 4:00 pm / CH 405

Abstract. The lecture will be concerned with numerical enclosure methods for nonlinear elliptic boundary value problems. Here, analytical and numerical methods are combined to prove rigorously the existence of a solution in some "close"neighborhood of an approximate solution computed by numerical means.

Thus, besides the existence proof, verified bounds for the error (i.e. the difference between exact and approximate solution) are provided.\\ For the first step, consisting of the computation of an approximate solution $\omega$ in some appropriate Sobolev space, no error control is needed, so a wide range of well-established numerical methods (including multigrid schemes) is at hand here. Using $\omega$, the given problem is rewritten as a {\it fixed-point equation} for the error, and the goal is to apply a {\it fixed-point theorem} providing the desired error bound.\\ The conditions required by the chosen fixed-point theorem (e.g., compactness or contractivity, inclusion properties for a suitable subset etc.) are now verified by a combination of analytical arguments (e.g. explicit Sobolev embeddings, variational characterizations etc.) and verified computations of certain auxiliary terms, in particular of eigenvalue bounds for the linearization of the given problem at $\omega$.\\ The method is illustrated by several examples (on bounded as well as on unbounded domains), where in particular it gives existence proofs in cases where no purely analytical proof is known.

Friday, November 10, 2006, 2 Colloquia

Benjamin Schlein(UC Davis)
Dynamics of Bose-Einstein Condensates
3:00 pm - 4:00 pm / CH 405

Abstract. Bose-Einstein condensates have recently been observed for the
first time in experiments: these are states of many body quantum systems,
where all the particles are described by the same one-particle wave
function, the so called condensate wave function. In this talk I am going
to show, starting from microscopic many body quantum dynamics, that the
time evolution of the condensate wave function can be described by a cubic
nonlinear Schroedinger equation commonly known as the Gross-Pitaevskii
equation.

Heinz Siedentop(LMU-Munich)
The Ground State Energy of Heavy Atoms according to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order
11:00 am - 12:00 pm / CH 445

Abstract. It is shown that the ground state energy of heavy atoms is, to
leading order, given by the non-relativistic Thomas-Fermi energy.
The proof is based on the relativistic Hamiltonian of Brown and
Ravenhall which is derived from quantum electrodynamics yielding
energy levels correctly up to order $\alpha^2$Ry.

Friday, October 27, 2006
Tracy Hamilton(UAB)
The Current State of Hartree-Fock Theory, Density Functional Theory and Their Hybrids.
3:00 pm - 4:00 pm / CH 405

Abstract. This talk will first explain the reasons for the growing popularity of hybrid methods which mix the Hartree-Fock (HF) and density functional (DFT) methods.   These schemes typically use numerical integration over a Lebedev grid for the DFT portion, and basis sets (linear combination of atomic orbitals to make molecular orbitals) for Hartree-Fock.  The introduction of a basis set in HF theory transforms an integro-differential equation into a much more computationally suitable matrix form. However, the introduction of a basis set also results in difficult integrals that need to be evaluated.  The pros and cons of the Slater vs. Gaussian functions will be presented in the context of their compactness, accuracy, and state of the art.    Many of the integrals are small in magnitude because the functions are localized to a small region of space, hence relationships such as Schwartz or triangle inequalities are very useful in obtaining upper bounds.  The final topic will address the potential for systematic improvement in post-HF methods, specifically the important first step of integral transformation.

Friday, October 27, 2006
Robert Seiringer(Princeton)
Lieb-Thirring inequalities-recent results
3:00 pm - 4:00 pm / CH 405

Abstract. Lieb-Thirring inequalities are bounds on power sums of the modulus of
negative eigenvalues of Schroedinger-type operators. They play a major
role in many aspects of analysis. After a brief review of the topic, we
present some recent generalizations of such bounds. One generalization is
the inclusion of complex-valued potentials. Another concerns the
possiblility of substracting a Hardy-term; i.e., we show that the
Lieb-Thirring inequalities remain true when the critical Hardy weight is
subtracted from the Laplace operator.

This talk is based on joint work with Rupert Frank, Ari Laptev and Elliott
Lieb.

Friday, October 20, 2006
Stefan Hoops(Virginia Bioinformatics Institute)
COPASI-Software for modeling and simulation of Biochemical networks
3:00 pm - 4:00 pm / CH 405

Abstract. In recent years, simulation and modeling of biochemical networks has become a major methodology in biology. With increasing number of non-specialists becoming involved, a need is growing for software that enables researchers to carry out biochemical modeling without knowledge of sophisticated mathematics. Here we describe COPASI, a new biochemical network simulator that is both user-friendly and powerful. This software continues the directions set by our older and popular Gepasi software. COPASI is the product of an international collaboration between the Mendes and Kummer groups. COPASI is targeted both at experienced biochemical modelers as well as wet lab scientists that are new to modeling. These divergent goals are achieved by providing an intuitive Graphical User Interface (GUI) for interactive modeling tasks and a command line version containing only the simulation engine (SE). The GUI allows non-experts to specify advanced features through intuitive means, while the more technical options are hidden away in advanced dialog boxes. The SE is a powerful processor that includes standard methods, such as solving trajectories with ODE or stochastic algorithms, steady-state solutions, stoichiometric and stability analyses. The SE can be run in batch mode and some of its heavier computations can be parallelized in distributed computing environments. COPASI is able to investigate model properties through parameter scans. It is also able to modify the model parameters to fit a user specified objective functions as well as fitting the model to provided experimental data from steady state and time course experiments simultaneously. For optimization and parameter fitting the user can select from a list of deterministic and stochastic optimization methods. COPASI supports the Systems Biology Markup Language (SBML), enabling it to exchange models with other modeling tools (we are among the founders of the SBML effort). Several other modeling groups have expressed interest in interfacing their software with the COPASI SE, enlarging the reach of our software beyond the users of its GUI.

Contributors:   Sven Sahle, Christine Lee, Anurag Srivastava, Sameer Tupe, Ralf Gauges, Ursula Kummer, and Pedro Mendes

Friday, October 13, 2006
Gunter Stolz(UAB)
Anderson versus displacements: two very different random Schrödinger operators
3:00 pm - 4:00 pm / CH 405

Abstract. The most studied random Schrödinger operators are Anderson models, which describe alloy-type materials by placing a fixed potential at each lattice site and multiplying it with a random coupling constant. Another physically motivated way to model a disordered medium is to randomly displace the single site potential from each point of the lattice. The mathematical investigation of these two models turns out to be quite different. This is mostly due to the fact that the displacement model lacks monotonicity properties of the Anderson model. We will discuss this mainly in the context of determining the almost sure spectral minimum of the displacement model. Here we can handle dimension one, but have mostly open conjectures to offer in higher dimension.

Friday, October 6, 2006
Eric Carlen (Georgia Institute of Technology)
Some problems in Calculus of Variations related to droplet formation
3:30 pm - 4:30 pm / CH 405

Abstract. Nature abhors a vacuum, and loves minimizers, which are often symmetric. The study of natural phenomena often gives rise to variational problems involving a balance between energy and entropy, with the entropy enforcing a "no vacuum condition".  In such problems, the functional being minimized is often non convex, due to this competition between energy and entropy. Thus, determining the properties of the minimizers leads to interesting mathematical questions. In this talk, which will delivered with a general coloquium audience in mind, I shall
discuss some mathematical problems of this type taken from my work with Carvalho, Esposito,
Lebowitz,  and Marra.

Friday, September 29, 2006
James Malaugh (Inductis Corporation)
So what do you do with a PhD in Mathematics
3:00 pm - 4:00 pm / CH 405

Abstract. Entering the non-academic job market with a background in mathematics can at first appear a daunting task. Without the specific training, of say an engineer, you may feel like a very intelligent generalist in a specialty-seeking world. What kind of work can I do? What do I like? Who needs people with degrees in math? Fortunately, there are companies out there willing to take on very smart people, who although they may not be specialized, can show a penchant for learning things quickly and have some degree of technical ability. This talk will focus on my experience as a very inexperienced professional entering the world of consulting, specifically financial services consulting. I will discuss things such as how I got the job, what I do day to day, and how do I use the skills learned during my mathematics education. I will also touch upon some relevant skills and classes that would be useful preparation for someone entering the non-academic job market, not specific to the consulting world. I will also announce an opportunity for students and faculty to participate in a competition where they would get to experience the type of work I do day to day.

BACKGROUND

Jamie graduated from UAB with a PhD in Applied Mathematics in 2003, having started as an undergraduate Mathematics Fast-Track student.  He now works for Inductis in New Providence, New Jersey.  (The name, Inductis, comes from inductive logic – proceeding from particular facts to a general conclusion – and from the Latin induco: to bring forward, to induce, to lead.)

Friday, September 22, 2006
Getting Even: Regulation of ant worker allocation
3:00 pm - 4:00 pm / CH 458

Abstract. Ants need to achieve a variety of tasks in an unpredictable environment, including collection or capture of food and deterring or fighting with competitors. Given their well-known lack of centralized control, we propose that simple algorithms to equalize worker numbers in different patches provides a robust way to achieve these tasks efficiently, although not quite optimally. In particular, we show how this approach may be used by seed harvester ants to harvest slowly-renewing seed resources, by army ants to capture fast-moving insect prey, and by one species in the genus Formica to protect aphid food sources by deterring and defeating a larger species in that same genus. Maintaining an even distribution of workers may provide one basis for the diversity and ecological importance of ants.

Friday, September 15, 2006
Tomio Umeda (University of Hyogo, Japan)
On the zero resonances and the zero modes of the Weyl-Dirac operator
3:00 pm - 4:00 pm / CH 458

Abstract. The Weyl-Dirac operator is obtained if the mass parameter is taken to be zero in the Dirac operator. Zero modes of the Weyl-Dirac operator mean the eigenfunctions of the operator corresponding to the eigenvalue zero. Zero resonances mean the eigenfunctions, in an extended sense, of the operator corresponding to the zero energy. In this talk, I start with a brief review of the zero modes and the zero resonances in terms of its roles in various fields of mathematical physics, including spectral analysis in particular. The zero modes and the zero resonances themselves, however, are not well-understood. Our results are:
(1) a pointwise estimate of the zero modes,
(2) non-existence of the zero resonance.