Friday, August 28, 2009
Xiaodong Cao (Cornell University)
Differential Harnack Inequalities in the Ricci Flow
2:00 pm / CH 301

Abstract. In this talk, we will talk about differential Harnack inequalities (also known as Li-Yau estimates) for geometric flows. In particular, we will discuss some new Harnack inequalities for the conjugate heat equation and forward conjugate heat equation under the Ricci flow.

Friday, September 11, 2009
Hassan Fathallah (UAB)
Model of the Circadian Clock: Loop Regulation and Transcriptional Integration
2:00 pm / CH 301

Abstract. I will discuss a system for modeling molecular networks and an equation for transcriptional integration and how they advance our undertanding of the dynamics of the Drosophila circadian clock. I will propose an explanation of the paradoxical effects of the clockwork orange molecule and predict the dynamical effects of gain-of-function and null-mutations of the period gene, in particular the per01 mutation.

Friday, September 18, 2009
Rudi Weikard (UAB)
On the Stability of Inverse Resonance Problems
2:00 pm / CH 301

Abstract. Inverse spectral and scattering problems are a classical subject in mathematical physics. In this talk are particular variant, the inverse resonance problem is presented. Since, in practical settings, one can typically not expect to obtain all the necessary data and since, in any case, recovery algorithms cannot make use of all data even if they were available, we investigate which information may be contained from finite noisy data. Results were obtained jointly with Malcolm Brown, Ian Knowles, Marco Marletta, Serguei Naboko and Roman Shterenberg.

Friday, September 25, 2009
Thomas Brazil (Systar corporation)
The Increasing Importance of Predictive Analytics in Monitoring Strategic Business Processes
2:00 pm / CH 301

Abstract. Business Activity Monitoring has been around for several years, but we have recently added various algorithms around forecasting and Correlation. This provides businesses the ability to be proactive in their monitoring in order to ensure sufficient actionable lead-time that can prevent business-impacting conditions. The importance of mathematical functions and algorithms in our software and industry is becoming more important than ever, as we search for new and better ways to enhance operational efficiency for our customers.

Friday, October 02, 2009
Christer Bennewitz (Lund University)
Some Schauder bases in L^p(0,1)
2:00 pm / CH 301

Friday, October 09, 2009
David Damanik (Rice University)
Cantor Spectra and Gap Labeling
2:00 pm / CH 301

Abstract. We describe scenarios in which Cantor spectra appear in the study of Schr"odinger operators and describe old and new proofs of this fact. We will also discuss the labeling of the gaps of the spectrum of a Schr"odinger operator by means of canonical labels and how this is related to stronger versions of Cantor spectrum results.

Friday, October 30, 2009
Konstantin Makarov (University of Missouri-Columbia)
On non-unitary representations of the Weyl commutation relations
2:00 pm / CH 301

Abstract. The Stone-von Neumann Uniqueness Theorem classifies the strongly continuous unitary representations of the Weyl canonical commutation relations $U_tV_s=e^{ist}V_sU_t$, in a separable Hilbert space. In this talk we discuss representations of the commutation relations assuming that $V_s$ is a strongly continuous semi-group of contractions. In this setting we also provide a complete classification of irreducible representations (up to unitary equivalence) in the case where the generator $A$ of the contractive semi-group $V_s=e^{isA}$ is a (quasi-selfadjoint) extension of a symmetric operator with deficiency indices $(0,1)$, $(1,0)$, and $(1,1)$, respectively. This is a joint work with Eduard Tsekanovski.

Friday, November 06, 2009
Oleg Safronov (University of North Carolina-Charlotte)
Absolutely continuous spectrum of Schrodinger operators whose negative eigenvalues tend to zero sufficiently fast
2:00 pm / CH 301

Abstract. We are going to discuss a relation between different parts of the spectrum of a Schrodinger operator. We will establish a certain conservation law claiming that one can judge about the quality of the positive spectrum by looking at the negative eigenvalues. The unusual side of the situation is that instead of studying one operator one needs to consider two of them: one with a potential V and the other with -V.

Friday, November 13, 2009
Alessandro Veneziani (Emory University)
Some recent numerical methods in electro-cardiology
2:00 pm / CH 301

Abstract. Numerical simulation of the electrical potential in the heart is still a challenging problem. On one hand the equations commonly used, the so-called Bidomain model, have mathematical features that make their numerical solution pretty expensive. On the other hand, reliability of the results demands for fine meshes for the space and time variables. Numerical effectiveness is required also in view of the coupled solution of electrical, structural and fluid dynamics. In the last 15 years many advances have been done both in the mathematical analysis and the set up of numerical methods (see e.g. [1, 7, 6]). In this talk, we will discuss some methods recently proposed in collaboration with medical doctors at the School of Medicine of Emory University. In particular we will consider:

an optimal and robust preconditioner for the Bidomain model, based on the simplified Monodomain system [5];

time advancing accurate and adaptive methods for the ionic models, that generalize the classical Rush- Larsen first order scheme [2];

domain decomposition and model adaptive methods, solving the accurate Bidomain system in the parts of the computational domain where more accuracy is required [3, 4]. We will briefly consider also the problem of moving domains by tracking the motion from images.

  • [1] P. Colli Franzone and G. Savare, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, in Evolution equations semigroups and functional analysis, A. Lorenzi and B. Ruffa, eds., 2002, pp. 218-240.
  • [2] M. Perego and A. Veneziani, An efficient generalization of the Rush-Larsen method for solving electro- physiology membrane equations, submitted (2009)
  • [3] L. Mirabella, F. Nobile and A. Veneziani, An a posteriori error estimator for model adaptivity in electrocardiology, in preparation (2009)
  • [4] L. Gerardo Giorda, M. Perego and A. Veneziani, Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology, in preparation (2009)
  • [5] L. Gerardo-Giorda, L. Mirabella, F. Nobile, M. Perego and A. Veneziani, A model precondi- tioner for the Bidomain problem in electrocardiology, J. Comp. Phys., 228 (2009) 3625-3639.
  • [6] A. Pullan, M. Buist, and L. Cheng, Mathematical Modelling the Electrical Activity of the Heart, World Scientific, Singapore, 2005.
  • [7] F. B. Sachse, Computational Cardiology, Springer, Berlin, 2004.

Friday, November 20, 2009
Ira Herbst (University of Virginia)
The analyticity radius for solutions of the Navier-Stokes equations in R^3.
2:00 pm / CH 301

Abstract. I will begin by reviewing some known results for existence, uniqueness, analyticity, and stability for "strong" solutions of the Navier-Stokes equations. For a slightly more general system I will indicate optimal results for the rate of growth of the region of analyticity of global solutions of these equations. This is joint work with Erik Skibsted of Aarhus, Denmark.

Wednesday, January 27, 2010
Maxim Zinchenko (Western Michigan University)
Nonlinear Fourier Analysis
10:00 am / CH 458

Abstract. In this talk, I will give an overview of spectral theory as a nonlinear analog of Fourier analysis. There are several classes of nonlinear differential equations (among which are KdV and nonlinear Schr\"odinger equations) that could be solved with the help of spectral theory. As an illustration, I will discuss a system of exponentially interacting particles known as the Toda lattice whose solution is based on the spectral theory of Jacobi matrices. In addition, I will discuss recent developments in nonlinear analysis and, in particular, present a nonlinear analog of Parseval's identity and a nonlinear version of the Riemann--Lebesgue lemma.

Friday, January 29, 2010
Anna Skripka (Texas A&M University)
Non-commutative Taylor formulas
9:30 am / CH 458

Abstract. The talk will discuss Taylor-type approximations for functions of an operator argument that originate from problems of theoretical physics. (In particular, Schrodinger operators are of interest.) Traces of the remainders of these approximations can be expressed via spectral shift functions (SSF), which provide information about a quantitative change of the spectrum of the operator argument under the influence of a perturbation. The first order SSF was introduced by Lifshits and Krein in 1953; it controls only the case of the trace class perturbations. In order to treat more general perturbations, one needs to consider modified, higher order, spectral shift functions. The second order SSF was introduced by Koplienko in 1984; it applies in the case of Hilbert-Schmidt perturbations. Existence of the SSFs of order greater than 2 (or, equivalently, some powerful estimates for the higher order Taylor remainders) was recently established in joint work with D. Potapov and F. Sukochev. Our result is operator theoretic; its proof is based on multiple operator integration, interpolation, and other non-commutative analysis techniques.

Monday, February 1, 2010
Richard Oberlin (University of California, Los Angeles)
A variation-norm Carleson Theorem
10:30 am / CH 458

Abstract. The Carleson-Hunt theorem shows that for every p-integrable function f on the circle, 1 < p < infinity, the Fourier series of f converges to f almost everywhere. We give an extension of this theorem which provides quantitative information about the rate of convergence, and we discuss some applications and parallel results. Joint work with F. Nazarov, A. Seeger, T. Tao, C. Thiele, and J. Wright.

Friday, February 05, 2010
Nidhal Bouaynaya (University of Arkansas)
Analysis of Proteomics and Genomics Based on Signal Processing, Communication and Control Theory
2:00 pm / CH 445

Abstract. Over the past half a century we have undergone a revolution in our ability to archive, exchange, and control information. Communication of biological systems took a head start 3.5 billion years ago. However, for all the strengthened efforts that are directed towards the study of complex communication engineered systems, remarkably little is known about the broad role of information in biological systems. In this talk, we develop a communication model of the genetic information storage and transmission system, and foster this model into educated intervention in the cell dynamics. In particular, we will show that (i) the highly redundant structure of the genetic codeword (i.e., DNA) maintains a fine balance between two competing yet complementary forces: stability and adaptability; and (ii) it is possible to intervene in the cell in order to change its dynamics in a desirable fashion. The stability role of the genetic codeword is evaluated by deriving the optimal exon length distribution, which minimizes the probability of error in Eukaryotic genomes. Experimental results on various Eukaryotic organisms spanning the phylogenetic tree show that the optimal distribution accurately fits the biological data. The optimal intervention problem is formulated as an inverse perturbation problem. The analytical solution to this inverse problem provides a minimally-perturbed system characterized by a unique attractor corresponding to the desired distribution of cellular states. The criteria adopted for optimality, or minimal change to the gene regulatory interactions are: (a) potential adverse effects on the patient, and (b) length of treatment for the patient.

Wednesday, February 10, 2010
Juhi Jang (Courant Institute)
Vacuum in Gas and Fluid dynamics
3:00 pm / CH 458

Abstract. The study of vacuum is important in understanding the motion of a gaseous star or shallow water. The mathematical difficulty is the degeneracy caused by vacuum. In this talk, I will review some interesting problems of vacuum states arising in gas and fluid dynamics, and present the current status in the understanding of compressible Euler flows near vacuum. My contribution to the subject is joint with Nader Masmoudi.

Friday, February 19, 2010
Spyridon Alexakis (University of Toronto)
A black hole uniqueness theorem
2:00 pm / CH 445

Abstract. I will discuss recent joint work with A. Ionescu and S. Klainerman on the black hole uniqueness problem. A classical result of Hawking (building on earlier work of Carter and Robinson) asserts that any vacuum, stationary black hole exterior region must be isometric to the Kerr exterior, under the restrictive assumption that the space-time metric should be analytic in the entire exterior region. We prove that Hawking's theorem remains valid without the assumption of analyticity, for black hole exteriors which are apriori assumed to be "close" to the Kerr exterior solution in a very precise sense. Our method of proof relies on certain geometric Carleman-type estimates for the wave operator.

Friday, March 05, 2010
Alexander Kiselev (University of Wisconsin-Madison)
Surface quasi-geostrophic equation
2:00 pm / CH 445

Abstract. The surface quasi-geostrophic (SQG) equation is motivated by atmospheric science. It models propagation of temperature fronts in rotating fluid subject to gravity force. The equation has simple structure, but its solutions display a wide variety of complex behaviors. In particular, the SQG equation is probably the simplest equation of fluid dynamics for which the question of global existence of smooth solutions for smooth initial data remains open. I will discuss some recent advances in understanding this equation and methods that have been developed for this purpose. These methods involve a mix of PDE, Fourier analysis and dynamical systems techniques.

Wednesday, March 24, 2010
Sumio Yamada (Tohuku University)
Geometry of Teichmuller-Coxeter complex
10:00 am / CH 458

Abstract. In this talk, I will present a construction of a simplicial complex where the simplex is the Teichmuller space of hyperbolic surfaces. The construction is made possible due to a synthetic geometry provided by a distant function called Weil-Petersson distance. The Coxeter theory which was instrumental in the study of classical Lie groups appearingnaturally in a context which is very far removed from linear groups is of some interest.

Friday, April 02, 2010
Philip Maini (Oxford University)
Modelling aspects of tumour growth
2:00 pm / CH 445

Abstract. Tumour growth emerges from a complex interaction of processes acting across many different scales. In this talk we will consider some of these processes, including vascular adaptation and angiogenesis, somatic evolution and the acid-mediated invasion hypothesis. We will use hybrid cellular automaton and partial differential equation models to study tumour growth and invasion in the context of these processes.

Monday, April 05, 2010
Julie Rowlett (Hausdorff Center for Mathematics, Bonn, Germany)
Interactions between quantum and classical mechanics on asymptotically hyperbolic manifolds
1:00 pm / CH 445

Abstract. Asymptotically hyperbolic manifolds are of particular interest in physics because they generalize the Poincar\'e-Einstein manifolds which arise in anti-de Sitter--conformal field theory correspondence. In this talk, we'll recall the definition of asymptotically hyperbolic manifolds and see some examples. After a brief discussion of their spectral theory and dynamics, I will present a prime orbit theorem and a ``dynamical wave trace formula.'' Based on the prime orbit theorem and the trace formula, we will determine a relationship between the existence of pure point spectrum and the topological entropy of the geodesic flow. We can interpret this physically as an interaction between the quantum and classical mechanics on asymptotically hyperbolic manifolds.

Tuesday, April 06, 2010
Stephen Shea (St. Anselm College, Manchester, NH)
Finitary Isomorphisms and the Finitary Factors Conjecture
10:00 am / CH 458

Abstract. In 1969, Ornstein proved that entropy is a complete isomorphism invariant for Bernoulli schemes. He then proved that entropy is a complete isomorphism invariant for factors of Bernoulli schemes. In 1979, Keane and Smorodinsky proved that entropy is a complete finitary isomorphism invariant for Bernoulli schemes. We know that not all factors of Bernoulli schemes are finitarily isomorphic to Bernoulli schemes. A natural question to ask is whether there exists a finitary equivalent to Ornstein's factor theorem. It was conjectured that entropy is a complete finitary isomorphism invariant for finitary factors of Bernoulli schemes. I will give an introduction to the finitary theory and discuss methods for proving discrete stationary stochastic processes are finitarily isomorphic. I will finish by presenting recent progress towards a disproof of the above conjecture.

Friday, April 09, 2010
Dilhani Uswatte (Hoover City Schools)
Promoting Positive Attitudes Towards Mathematics and Improving Quantitative Literacy In Education
2:00 pm / CH 204

Dilhani J. Uswatte, a 2009 Milken National Educator Award recipient and inductee into the Alabama Teacher Hall of Fame, will discuss the strategies she uses to inspire a positive attitude towards mathematics and improve quantitative literacy in her 8th grade classroom. She will share examples of student work, lesson plans, general teaching practices to prepare students for the 21st century, and her experiences with the Greater Birmingham Mathematics Partnership.

Friday, April 16, 2010
Marcus Khuri (Stony Brook University)
Quasi-Local Mass and The Static Extension Problem in General Relativity
2:00 pm / CH 445

Abstract. There are several competing definitions of quasi-local mass in General Relativity. A very promising and natural candidate, proposed by R. Bartnik, seeks to localize the total or ADM mass. Fundamental to understanding Bartnik's construction is the question of existence and uniqueness for a canonical geometric boundary value problem associated with the static vacuum Einstein equations. In this talk we will report on joint work with M. Anderson, which confirms that existence holds (under a nondegeneracy condition) but also shows that uniqueness fails. The possible implications of this result will be discussed.

Friday, April 23, 2010
Christian Hainzl (UAB)
From B-C-S theory to Ginzburg-Landau via a semiclassical limit
2:00 pm / CH 445

Abstract. A priori the Bardeen-Cooper-Schrieffer theory of superconductivity and the Ginzburg-Landau theory are totally different, the BCS theory is of microscopic nature whereas the G-L theory deals with macroscopic quantities. However, we are able to prove that G-L can be recovered from BCS in a certain limit.