**Friday, August 29, 2008 **

Hassan Fatallah University of Alabama at Birmingham

"New Mathematical Model Reveals Positive and Negative Loops Causing the Paradoxical Effects of Clockwork Orange Mutations "

2:00 pm / CH-301

Abstract. Clockwork Orange (CWO) is a key negative regulator of central elements of the molecular network of the Drosophila circadian clock; it represses the transcription of per, tim, vri, pdpd1, and cwo. Paradoxically, cwo-mutant flies exhibit lower levels of per, tim, pdp1, and vri and higher levels of cwo mRNAs as compared to wild type. We introduce a new system of ordinary differential equations to model the dynamics of this molecular network. Simulations generate 24-hour rhythmic oscillations and entrainment in response to time shifts. In silico targeted deletions reveal that the paradoxical effects of cwo mutations are caused by a positive loop containing vri and the negative cwo autoregulatory loop.

**Friday, September 5, 2008 **

Vladimir Oliker. Emory University

"The Aleksandrov problem and Monge-Kantorovich optimal transport of curvature on S^n"

2:00 pm / CH-301

Postponed to October 31, 2008, 2pm CH-301

Abstract. In this talk I will describe a variational solution to a problem of A.D. Aleksandrov concerning existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Remarkably, in variational formulation this problem turned out to be closely connected with the Monge-Kantorovich theory of optimal mass transport on S^n.

**Friday, September 12, 2008 **

Shihshu Walter Wei. University of Oklahoma.

"p-Harmonic maps and 1-forms, generalized 1-harmonic equations, topology and algebra""

2:00 pm / CH-301

Abstract. P-Harmonic maps are natural generalizations of linear transformations, analytic functions, solutions of (systems of partial differential) Cauchy-Riemann equations, conformal mappings, geodesics, minimal surfaces, harmonic maps (in which p=2) and many more. In fact, p-harmonic geometry is a birthplace of many other branches of mathematics. There are numerous relations among p-harmonic maps, geometric flows, Lie groups and other areas of mathematics and sciences. We will explore some of them. In particular, we wish to discuss some recent progress in generalized 1-harmonic equations, p-harmonic 1-forms, the inverse mean curvature flows, Finsler geometry, topology and algebra.

**Friday, September 19, 2008 **

Maxim Zinchenko. California Institute of Technology.

"Szego-type Theorems in Spectral Theory of Finite Gap Jacobi Matrices"

2:00 pm / CH-301

Abstract. In spectral theory for Jacobi matrices or approximation theory for orthogonal polynomials, the goal is to relate information about the recurrence coefficients (aka Jacobi parameters) to information about the measure of orthogonality (aka spectral measure). The most studied case is the case of Jacobi matrices with the essential spectra consisting of a single interval on the real line. An interesting question that we would like to address is what happens when gaps occur in the essential spectrum? In particular, in this talk we will discuss variants of Szeg\H{o}-type theorem and Szeg\H{o}-type asymptotics for Jacobi matrices with essential spectra given by finite unions of closed intervals. These results can be viewed as perturbation results for Jacobi matrices with periodic and almost periodic Jacobi parameters. The talk is based on joint work with Jacob Christiansen and Barry Simon. Our results rely on potential theory and analytic function theory, with an important link to Riemann surfaces and Fuchsian groups..

**Friday, September 26, 2008 **

Christian Remling. University of Oklahoma.

"The absolutely continuous spectrum of Jacobi matrices "

2:00 pm / CH-301

Abstract. We are interested in general properties of Jacobi matrices (of Schr"odinger operators, if you prefer) with some absolutely continuous spectrum. It turns out that there are special building blocks with rather peculiar properties that must be used to produce any kind of absolutely continuous spectrum. I'll discuss this result and some of its ramifications.

**Friday, October 3, 2008 **

Alexander Blokh. University of Alabama at Birmingham.

"Fatou-Douady-Hubbard-Shishikura inequality and wandering continua"

2:00 pm / CH-301

Abstract. Let P be a polynomial of degree d with a connected Julia set J. Let \nu be the number of cycles of Fatou domains plus the number of Cremer periodic orbits of P. An important Fatou-Douady-Hubbard-Shishikura inequality states that \nu\le d-1. A continuum Q\subset J is said to be of \emph{finite valence v} if the number of components of J\Q equals v. A \emph{wandering collection} (\emph{of continua}) is a collection of continua whose all forward images are pairwise disjoint. Given a \emph{non-empty} wandering collection \Ga of non pre-critical continua with finite valences M_1>2, \dots, M_k>2, we prove that \sum_\Ga (M_i-2)+\nu\le d-2.

**Friday, October 10, 2008**

W. D. Evans. Cardiff University

"Representation of compact linear operators in Banach spaces and nonlinear eigenvalue problems"

2:00 pm / CH-301

Abstract. Let X and Y be Banach spaces with strictly convex duals, and let T be a compact linear map from X to Y. I shall talk about recent work with David Edmunds and Desmond Harris in which we show that a certain nonlinear equation, involving T and its adjoint,, has a normalised solution (an "eigenvector") corresponding to an "eigenvalue", and that the same is true for each member of a countable family of similar equations involving the restrictions of T to certain subspaces of X. The action of T can be described in terms of these "eigenvectors", the resulting expansion reducing to the familiar spectral representation of the modulus of T when X and Y are Hilbert spaces. There are applications to the p-Laplacian, the p-biharmonic operator and integral operators of Hardy type.

**Friday, October 17, 2008**

Alexander Elgart. Virginia Tech.

"On the localization of a matrix valued Anderson Hamiltonian"

2:00 pm / CH-301

Abstract. Transport properties of conducting materials at low temperature T are determined by an interplay between the interaction of the itinerant electrons with each other and the quenched disorder which creates a random potential acting on these electrons. In the absence of the electron-electron interaction the most dramatic phenomenon is the Anderson localization: There are regimes (high disorder, extreme energies) where the spectrum is almost surely dense point, with exponentially localized eigenfunctions. The methods used to establish this type of results (multi-scale technique and fractional moments approach) assume certain properties of the random potential: monotonicity for the first one and disorder at every site of the lattice for the second one, the so called covering condition. We will discuss a realistic model where both of these conditions are not met. In particular, we will discuss the way to prove the regularity of the Green functions in this context, which is a central component of the fractional moment method.

**Friday, October 24, 2008 **

Junping Shi. College of William and Mary.

"Uniqueness of radial solution of variational elliptic systems"

9:00 am / CH-458

Abstract. I will give a preliminary report on the uniqueness of radial symmetric solutions of cooperative elliptic systems with either Hamiltonian or gradient structure. We consider both the solutions on a finite ball or on the entire space. Our approach is based on shooting method of related ordinary differential equations, and we classify the behavior of the all solutions according to the maximum values. Our method is a quite general one, and its applications include many important systems arising from physics and biology. This is a joint work with Jann-Long Chen of Central University and Changshou Lin of Taiwan University.

**Friday, October 31, 200**8

Vladimir Oliker. Emory University

"The Aleksandrov problem and Monge-Kantorovich optimal transport of curvature on S^n"

2:00 pm / CH-301

Abstract. In this talk I will describe a variational solution to a problem of A.D. Aleksandrov concerning existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Remarkably, in variational formulation this problem turned out to be closely connected with the Monge-Kantorovich theory of optimal mass transport on S^n.

**Friday, November 7, 2008**

Dhruba Adhikari. Mississippi University for Women

"Some Topological Degree Theories and Applications"

2:00 pm / CH-301

Abstract. First, a brief introduction to the development of various topo- logical degree theories and their generalizations will be presented. Then as an application of the Browder and Skrypnik degrees, a result concerning the existence of nonzero solutions of operator equations of the form Tx + Cx = 0 in reexive Banach spaces will be given. Here, T and C are certain operators of monotone type. Lastly, applications to invariance of domain and eigenval- ues of recently developed degree theory, which generalizes the Berkovits-Mustonen degree theory, will also be given.

**Friday, November 14, 2008 **

Christian Hainzl. University of Alabama at Birmingham.

"Preventing the Big Bang in Dirac-Einstein cosmology"

2:00 pm / CH-301

Abstract. I will present in a very elementary way how to prevent the big bang (crunch) singularity by coupling Einstein's equations to Dirac spinors. This is joint work with Felix Feinster.

**Friday, November 21, 2008 **

Peter Bates. Michigan State University

"Some Mathematical Problems Arising in Materials Science"

2:00 pm / CH-301

Abstract. Equations for a material that can exist stably in one of two homogeneous states are derived from a microscopic or lattice viewpoint with the assumption that the evolution follows a gradient flow of the free energy with respect to some metric. Alternatively, Newtonian dynamics can be considered. The resulting lattice dynamical systems are analyzed, as are equations on the continuum where the lattice interaction energy is viewed as an approximation to a Riemann integral. These equations are lattice or nonlocal versions of the Allen-Cahn, Cahn-Hilliard, Phase-Field, or Klein-Gordon equations. Some results presented here provide for the well-posedness of the equations, while others give asymptotics or quantitative behavior of special solutions, such as traveling waves or pulses.

**Friday, January 16, 2009 **

Ruth Parker. Mathematics Education Collaborative (MEC),

"Algebra for All Eight Graders: Is It Progress or Folly? "

1:30-3:00 pm / CH-458

Abstract. Is the national movement to ensure that every 8th grader takes algebra a positive development or a potential disaster in waiting? It depends. In this session we will examine the detrimental effects of algebra taught poorly and prematurely, and consider an alternative approach to teaching algebraic reasoning across the grades that positively impacts student learning, develops problem solving skills, and results in students with productive mathematical dispositions.

**Friday, January 23, 2009 **

Yoshimi Saito. University of Alabama at Birmingham,

"Dirac-Sobolev and Sobolev Spaces"

2:00 pm / CH-301

Abstract. PDF

**Friday, January 30, 2009 **

Mónica Clapp. Universidad Nacional Autónoma de México

"Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size"

2:00 pm / CH-301

Abstract. PDF

**Friday, Feburary 6, 2009 **

Haomin Zhou. Georgia Institute of Technology

"Images, PDEs and Wavelets"

2:00 pm / CH-301

Abstract. (Based on Joint work with Tony Chan at UCLA/NSF and Jackie Shen at Barclays) It is well evident that both wavelets and PDEs have profound impacts on imaging sciences. The success relies on their remarkable mathematical properties, many of which are complimentary to each other. In this talk, I will present an overview of our work along the direction of merging them to further improve the performance and model new applications in image processing. More precisely, the first part of the talk will be on an adaptive ENO wavelet transform designed by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet transforms retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining arbitrary high order accuracy uniformly and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have also shown the stability of the ENO-wavelet transform and obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. The second part of the talk is on using a variational framework, in particular the minimization of total variation (TV), to select and modify the retained standard wavelet coefficients in the wavelet thresholding procedure so that the reconstructed images have fewer oscillations near edges. Applications in image compression, denoising, wavelet inpainting will be mentioned. Connections with other related work such as compressed sensing (CS) will be commented.

**Friday, February 13, 2009 **

Nandor Simanyi. University of Alabama at Birmingham

"Sums of squares and orthogonal integral vectors"

2:00 pm / CH-301

Abstract. Two vectors in $Z3$ are called twins if they are orthogonal and have the same length. The talk describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions, but elementary, geometric proofs exist for most of the results as well. The presented results are fruits of my collaboration with Lee M. Goswick, Emil W. Kiss, and Gabor Moussong.

**Friday, February 20, 2009 **

Heinz Siedentop. Ludwig-Maximilians-Universitat Munchen

"The Ground State Energy of Heavy Atoms According to Muller"

2:00 pm / CH-301

Abstract. Hohenberg and Kohn asserted the existence of a universal functional $F$ of the density $\rho$ such that the minimum $F[\rho] +\int V\rho$, $V$ being the external potential, is equal to the ground state energy of the corresponding Schr\"odinger operator. This universal density functional, although not known explicitly, has generated a wave of investigations leading to ever better approximations, and, eventually, earned Walter Kohn the Nobel price in chemistry. The prototype of a density functional is the Thomas-Fermi functional. -- Later Gilbert observed that there is a corresponding functional of the one-particle density matrix whose kinetic energy part is explicitly known. The most well known example is the Hartree-Fock functional. However, the mother of a recent deveopment of the subject became a functional introduced by M\"uller. In this talk we will review the above desribed development, and introduce and discuss the M"uller functional including its relation to quantum mechanics.

**Friday, February 27, 2009 **

Ratnasingham Shivaji Mississippi State University

"Positive Solutions for n by n Elliptic Systems with Combined Nonlinear Effects"

2:00 pm / CH-301

Abstract. PDF

**Friday, March 20, 2009**

Thierry Colin Universite Bordeaux 1,

"Mathematical models for tumor growth and data assimilation"

2:00 pm / CH-301

Abstract. In this talk I will present a mathematical model for solid tumor growth including four population of cells (proliferating, quiscent, necrosis, healthy) and the describtion of the angiogenesis process. I will show how mechanical effects of the tissues can be taken into account in this model. At the end I will present to applications concerning the effects of inhibitors of MMP and of an anti-angiogenic drug.

**Friday, March 27, 2009 **

Tomio Umeda University of Hyogo

"Eigenfunctions at thresholds energies of Dirac operators with vector potentials"

2:00 pm / CH-301

Abstract. The talk will be devoted to investigation of the asymptotic behaviors of the Dirac operator $H=(J\(Balpha (J\(Bcdot (D -A(x)) + m (J\(Bbeta$ at the threshold enegies $(J\(Bpm m$. It turns out in the discussion that zero modes (i.e., eigenfunctions corresponding to the zero energy of the Weyl-Dirac operator $(J\(Bsigma(J\(Bcdot (D -A(x))$ surprisingly plays a crucial role. The core of our discussion is a new result on supersymmetric Dirac operators which will be proposed in the talk. Some related problems will also be discussed. (Joint work with Yoshimi Saito.)

**Friday, April 3, 2009 **

Ivan Blank Kansas State University

"Sharp Results for the Regularity and Stability of the Free Boundary in the Obstacle Problem"

2:00 pm / CH-301

Abstract. The problem of finding the smallest superharmonic function which lies above a given obstacle and which has prescribed boundary data is called the obstacle problem. After finding this minimizer, it is natural to study the regularity of the boundary of the set where the solution makes contact with the obstacle. In 1977 Caffarelli showed that for a very natural class of obstacles, when this (free) boundary is not smooth, it has to have a very specific geometry. This talk explores what happens to this result when we weaken Caffarelli's hypotheses.

**Friday, April 10, 2009 Cancelled. **

**Friday, April 17, 2009**

Jerry L. Bona University of Illinois at Chicago

"Large Waves in the Ocean"

2:00 pm / CH-301

Abstract. After a brief historical discussion, the lecture will turn to mathematical models of oceanic wave motions. In view are tsunamis, rogue waves and, in the near shore zone, wave bottom interaction.

**Friday, April 24, 2009 **

Luca Capogna. University of Arkansas

"Regularity of a class of minimal graphs in Heisenberg groups"

2:00 pm / CH-301

Abstract. Minimal surfaces in the sub-Riemannian geometry of the Heisenberg group are critical points of the sub-Riemannian area functional. In a recent paper, Cheng, Hwang and Yang construct Lipschitz minimizers via a Riemannian approximation technique. In this talk I will describe a theorem due to Citti, Manfredini and myself, proving that away from the characteristic set such minimizers are smooth if the dimension of the Heisenberg group is not three. In particular we prove that any Lipschitz intrinsic graphs in the Heisenberg groups Hn, with n > 1, which are vanishing viscosity solutions of the minimal surface equation are smooth. I will also describe some regularity results for vanishing viscosity solutions in the first Heisenberg group $n=1$ (also in collaboration with Citti and Manfredini).