Colloquium: Friday, May 6, 2011

Spectral Theory and Weyl Asymptotics for Perturbed Krein Laplacians
Fritz Gesztesy (University of Missouri)
2:00 pm / CH 445

Abstract. We study spectral properties for the Krein-von Neumann extension of the perturbed Laplacian \(-\Delta+V\) defined on \(C^\infty_0(\Omega)\), where \(V\) is measurable, bounded and nonnegative, in a bounded open set \(\Omega\subset\mathbb{R}^n\) belonging to a certain class of nonsmooth domains which contains all convex domains, along with all domains of class \(C^{1,r}\), \(r>1/2\). In this context we establish the Weyl asymptotic formula for the non-zero eigenvalues of the Krein-von Neumann extension, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian is spectrally equivalent to the buckling of a clamped plate

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Colloquium: Friday, April 22, 2011

Hardy's and related inequalities
Michael Loss (Georgia Institute of Technology )
2:00 pm / CH 445

Abstract. 
In this talk I review some recent progress concerning Hardy - and Hardy-Sobolev-Maz'ya inequalities, some of them in their sharp form. Besides being interesting inequalities in their own right, they pose hard problems in the calculus of variations. The discussion will be interspersed with some applications. Some of the work I report on is joint with Rafael Benguria, Rupert Frank and Craig Sloane.

Colloquium: Friday, April 15, 2011

The Ray-Singer conjecture for singular manifolds
Xianzhe Dai (University of California, Santa Barbara)
2:00 pm / CH 445

Abstract. The Reidemeister torsion (R-torsion) is a combinatorial invariant introduced by Reidemeister in 1935. It is a secondary invariant associated to the Euler characteristic and is the first topological invariant which distinguishes homotopy equivalent spaces. The analytic torsion is introduced by Ray and Singer in the 70's as an analytic analog of the R-torsion. The Ray-Singer conjecture, which is proven independently by Cheeger and Mueller, says that the analytic torsion equals the R-torsion for closed manifolds. Recent interesting application of the Cheeger-Mueller theorem includes detecting torsion homology classes of hyperbolic manifolds. Thus it will be both interesting and desirable to extend it to singular manifolds. We will discuss the recent understanding along this direction.

Colloquium: Friday, April 1, 2011

Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities
Yanyan Li (Rutgers Unversity)
2:00 pm / CH 445

Abstract: We present some results on the behavior of positive solutions in a punctured ball of general second order fully nonlinear conformally invariant elliptic equations. We prove that such a solution, near the puncture, is asymptotic to some radial solution of the same equation in the punctured Euclidean space. This is a joint work with Z.C. Han and E. Teixeira.

Colloquium: Friday, March 25, 2011

Mathematical Issues in Visual Transduction
Emmanuele DiBenedetto (Vanderbilt University)
2:00 pm / CH 445

Abstract: Visual transduction is the process by which photons of light are converted into electrical signals by diffusion of the second messengers Calcium and cGMP (cyclic guanosine monophosphate) in the cytoplasm of the Rod Outer Segment (ROS). A mathematical model of such a transduction is presented, that accounts for the layered geometry of the Rod Outer Segments and the incisures born by the discs. The model provides an explanation for the role of incisures, believed as evolutionary residues. The model also explains the biological/structural reasons for the high filelity of the photoresponse, despite the fact that reception of photons of light is a process with several random components.

Colloquium: Friday, March 4, 2011

Topological Models for Julia Sets
Clinton P. Curry (Stony Brook University)
2:00 pm / CH 445

Abstract: The dynamics of complex polynomials is concentrated on a fully invariant subset of the plane called the Julia set. It is of intrinsic interest to describe the Julia set topologically. I will describe recent advances in terms of what we call finest models, and describe difficulties in extending our understanding to more general kinds of functions (for example, quotients of polynomials). The topics discussed will include joint work with Alexander Blokh, John C. Mayer, and Lex Oversteegen of UAB, and E. D. Tymchatyn of the Universoty of Saskatchewan.

Colloquium: Friday, February 25, 2011

Lieb-Robinson Bounds and Applications
Robert Sims (University of Arizona)
2:00 pm / CH 445

Abstract: Locality is an essential tool in analyzing various physical systems. For non-relativistic systems generated by a Hamiltonian dynamics, it is well-known that the time evolution does not generally preserve local structures, i.e., there is no strict equivalent to a finite speed of light. In 1972, Lieb and Robinson demonstrated that the dynamics associated with certain non-relativistic systems has an approximate local structure. In particular, such systems have an associated, finite group velocity. We will discuss this result, several recent generalizations, and a variety of interesting applications.

Colloquium: Friday, February 18, 2011

Special Functions and Integrable Systems
Alexander Its (Indiana University-Purdue University Indianapolis)
2:00 pm / CH 445


Abstract: The recent developments in the theory of integrable systems have revealed its intrinsic relation to the theory of special functions. Perhaps the most generally known aspects of this relation are the group-theoretical, especially the quantum-group theoretical, and the algebra-geometrical ones. In the talk we will discuss the analytic side of the Special Functions-Integrable Systems connection. This aspect of the relation between the two theories is less known to the general mathematical community,although it goes back to the classical works of Fuchs, Garnier and Schlesinger on the isomonodrony deformations of the systems of linear differential equations with rational coefficients. Indeed, the monodromy theory of linear systems provides a unified framework for the linear(hypergeometric type) and nonlinear (Painlev\'e type) special functions and, simultaneously, builds a base for the new powerful technique of the asymptotic analysis - the Riemann-Hilbert method. In this survey talk, which is based on the works of many authors spanned over more than two decades, the isomonodromy point of view on special function will be outlined. We will also review the history of the Riemann-Hilbert method as well as its most recent applications in the theory of orthogonal polynomials and random matrices.

Colloquium: Friday, February 4, 2011

The Liouville theorem for subharmonic functions, and generalizations

James Serrin (University of Minnesota)
2:00 pm / CH 445 

 Abstract. 'The standard Liouville theorem states that every bounded entire complex analytic function is a constant. We discuss various classical improvements of this result, including whether it remains true for harmonic and subharmonic functions. This leads to a quite general final result for subsolutions of a class of elliptic variational equations.'

Colloquium: Friday, January 21, 2011

Productsets of large sets in amenable groups

 Alexander Fish (University of Wisconsin)
2:00 pm / CH 445

Abstract. 'We will review recent developments in the study of products of sets of positive upper Banach density in countable amenable groups. The methods are of dynamical origin. In particular, Kronecker factor of an appropriate ergodic \Gamma-system (where \Gamma is the underlying amenable group) will play a crucial role.

Colloquium: Friday, January 14, 2011

Manifolds with pinched flag curvature

Lei Ni (University of California, San Diego)
2:00 pm / CH 445 

Abstract. 'Manifolds with positive sectional curvature have been extensively studied. Ricci flow has been used to prove classification result for manifolds with pinched sectional curvature. This talk is about how the method can be used to classify manifolds with pinched flag curvature, a condition which is weaker than the pinched sectional curvature. '

Colloquium: Friday, January 13, 2011

A shape-based method for determining protein binding sites in a genome

Valerie Hower (University of California - Berkeley)
9:30 am / CH 458

Abstract: We present a new algorithm for the identification of bound regions from ChIP-Seq experiments. ChIP-Seq is a relatively new assay for measuring the interactions of proteins with DNA. The binding sites for a given protein in a genome are "peaks" in the data, which is given by an integer-valued height function defined on the genome. Our method for identifying statistically significant peaks is inspired by the notion of persistence in topological data analysis and provides a non-parametric approach that is robust to noise in experiments. Specifically, our method reduces the peak calling problem to the study of tree-based statistics derived from the data. The software T-PIC (Tree shape Peak Identification for ChIP-Seq) is available at http://math.berkeley.edu/~vhower/tpic.html and provides a fast and accurate solution for ChIP-Seq peak finding.

Colloquium: Monday, January 10, 2011

Scaling limits of random walks in random scenery: local time and indicator fractional stable motions

Paul Jung (Sogang University)
10:00 am / CH 458 

Abstract. 'Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric α-stable motions called local time fractional stable motions. When α = 2, these processes are precisely fractional Brownian motions with 1/2 < H < 1. Motivated by random walks in alternating scenery, we find a “complementary” family of symmetric α-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when α = 2, one gets fractional Brownian motions with 0 < H < 1/2. After a short introduction/review of stable integrals, we will introduce these two complementary families, and then go on to discuss their ergodic-theoretic properties. '

Colloquium: Friday, January 7, 2011

Novel techniques for acoustic and electromagnetic field manipulations and their applications

Daniel Onofrei (University of Utah)
10:00 am / CH 458 

Abstract. 'During the recent years, there has been, in the mathematical and physical communities as well, a large interest in the theoretical and practical design of structures and antennas offering new possibilities in the manipulation of acoustic or electromagnetic fields with a wide span of applications. In the first part of this talk I will introduce the ideas behind transformation acoustic/optics for passive acoustic/electromagnetic field manipulations, and discuss our recent analytical and numerical results on the mathematical and physical feasibility of this technique for cloaking. In the second part of the talk, I will describe our novel alternative strategy for active acoustic/electromagnetic field manipulations, and discuss its feasibility, advantages and limitations. Applications of this strategy to sonar or radar cloaking, to the realization of static or moving illusions or to the remote control of targets will be presented as well.'E techniques that in two dimensions, the only simply connected domains for which the expected exit time is maximized by zero drift are the discs.'

Colloquium: Friday, November 19, 2010

Complete Hypersurfaces of Constant Curvature in Hyperbolic Space
Bo Guan (Ohio State University)
2:00 pm / CH 445

Abstract. 'We discuss the problem of finding complete hypersurfaces in hyperbolic space with asymptotic boundary at infinity determined by a symmetric function of principal curvatures. We shall use the upper half space model and method of partial differential equations to prove exsitence results. Results described in this talk are joint work with Joel Spruck and Marek Szapiel.'

Colloquium: Friday, November 12, 2010

Gluing constructions for minimal surfaces
Nicos Kapouleas (Brown University)
2:00 pm / CH 445

Abstract. 'I will concentrate on doubling and desingularization constructions: I will first discuss doubling constructions for the Clifford torus and the equatorial two-sphere in the round three-sphere. In doubling constructions minimal surfaces are constructed resembling two copies of the given minimal surface joined by many small catenoidal bridges. I will then discuss desingularization constructions where minimal surfaces are constructed by replacing the intersection curves of minimal two-surfaces in a Riemannian three-manifold with handles modeled after the singly periodic Scherk surfaces and then perturbing to minimality. Finally I will discuss some applications and open questions for closed embedded minimal surfaces in the three-sphere.'

Colloquium: Friday, November 5, 2010

Some applications of heat equation method to the Ricci flow
Qi Zhang (University of California, Riverside)
2:00 pm / CH 445

Abstract. 'We start with some old ideas involving heat equation such as monotonicity of entropy and energy. Then we discuss how these are applied by Perelman and others to study 3 dimensional Ricci flow. Some concrete and potential applications to high dimensional Ricci will also be addressed.'

Colloquium: Friday, October 29, 2010

The Kahler Ricci flow on Toric Fano surface

Bing Wang (Princeton University)
2:00 pm / CH 445

Abstract. 'We will discuss the convergence of the K\"ahler Ricci flows on Toric Fano surfaces, which are complex surfaces with positive first Chern Classes and toric structures. If we assume the initial metric has a toric symmetry, then we can show the convergence by some topological method. If the initial metric is arbitrary, then the convergence is much more complicated. In some special cases, we can also prove the convergence by the classification of complex surfaces and weak compactness theorem along the K\"ahler Ricci flow.'

Colloquium: Friday, October 22, 2010

Some new geometric evolution equations

Jeff Streets (Princeton University)
2:00 pm / CH 445

Abstract. 'After giving a brief history of the application of geometric flows in geometry and topology, I will discuss two new such flows. The first is the gradient flow of the L^2 norm of curvature, which can be used to show a weak type of "sphere theorem." The second is a geometric flow on complex, non-Kahler manifolds. After discussing a recently discovered connection between this flow and mathematical physics, I will discuss some regularity theorems and the potential application to understanding the classification of nonKahler surfaces.'

Colloquium: Friday, October 8, 2010

Diffusion of waves in a random environment: problems and results

Jeffrey Schenker (Michigan State University)
2:00 pm / CH 445

Abstract. 'I will discuss the problem of proving diffusion of waves in a random environment in the context of the lattice Schroedinger equation. A major difficulty that arises is recurrence -- return of portions of the wave packet to regions previously visited. I will show that, if recurrence is eliminated by making the environment evolve randomly in time, then diffusion results in an elementary way.'

Colloquium: Friday, October 1, 2010

Circle Decompositions of Surfaces

Nandor Simanyi (UAB)
2:00 pm / CH 445

Abstract. 'We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.'

Colloquium: Friday, September 24, 2010

Measure of Buried Points in Julia Sets

John Mayer (UAB)
2:00 pm / CH 445

Abstract. ' I'll give an introduction to Julia sets (complex analytic dynamical systems). The talk will be completely accessible to anyone who has taken introductory topology and understands the measure theory in the first half of the first semester of our real analysis course.'

Colloquium: Friday, September 17, 2010

Time reversal method in Thermoacoustic tomography

Yulia Hristova (IMA and Texas A&M)
2:00 pm / CH 445

Abstract. 'Thermoacoustic tomography (TAT) is a novel biomedical imaging modality characterized by high resolution and contrast. Reconstruction of images in this type of computerized tomography requires solving an inverse source problem for the wave equation. In this talk, I will consider time reversal as a versatile technique for image reconstruction in TAT.'

Colloquium: Friday, September 10, 2010

Modeling Risk and Uncertainty – Applications in Power Markets and Trading
Jeff Baker (Southern Company)

2:00 pm / CH 445

Abstract. 'Economically running a reliable power system depends on a variety of highly volatile variables. These include weather, fuel costs, generator availability, legislation, and many others. Volatility introduces risk and uncertainty when planning for the short term (tomorrow) as well as the long term (next 5-20 years). The physical characteristics of power also present a number of challenges when valuing power as a tradable commodity, as opposed to other commodities such as wheat, corn, oil, etc. In this talk I will present some of the models and tools Southern Company’s Fleet Operations and Trading use to approach these problems. I will also give some insight about what it is like to be a mathematician in industry.'

Colloquium: Friday, August 20, 2010

Exit times of diffusions with incompressible drifts

Andrej Zlatos (University of Wisconsin)
2:00 pm / CH 445
 
Abstract. 'We consider the influence of an incompressible drift on the expected exit time of a diffusing particle from a bounded domain. Mixing resulting from an incompressible drift typically enhances diffusion so one might think it always decreases the expected exit time. Nevertheless, we show using PDE techniques that in two dimensions, the only simply connected domains for which the expected exit time is maximized by zero drift are the discs.'