**Friday, April 23, 2004Plamen Stefanov (Purdue University)**

Boundary rigidity of Riemannian manifolds

2:30 pm / CH 445

**Abstract.**In this talk, I will present the recent progress in solving the boundary rigidity problem. Let (M,g) be a compact Riemannian manifold with boundary. Assume that for each pair of boundary points (x,y), we know the distance function d(x,y) . The manifold M is called boundary rigid, if it is determined uniquely by d(x,y) (known on the boundary). At present, boundary rigidity is known only for some classes of manifolds. The distance between boundary points measures also the travel times of light rays through the domain. This information can be extracted from the scattering kernel or from the associated hyperbolic Dirichlet-to-Neumann map.

We study this problem for simple manifolds, i.e., strictly convex ones with no caustics inside. The associated linearized problem is recovery of the so-called solenoidal part of a tensor from its X-ray transform If along geodesics. We show that N=I^*I is a pseudodifferential operator, analyze its principal symbol and invert it microlocally. We prove an a priori stability estimates for N. Using analytic pseudodifferential calculus, we show that for analytic metrics, the linearized problem is invertible. For the non-linear one, we prove generic local uniqueness and stability.

**Friday, April 16, 2004** **Yanyan Li (Rutgers University)**

On a fully nonlinear version of the Yamabe problem

2:30 pm / CH 445**Abstract.** We present some recent results on a fully nonlinear version of the Yamabe problem which includes existence and compactness of solutions and some Liouville type theorems.

**Friday, April 2, 2004** **Robert Sims (Princeton University)**

A variational principle for the Sherrington-Kirkpatrick model

2:30 pm / CH 445**Abstract.** Ising systems are models of "simple" magnetic structures composed of particles (called spins) whose values are restricted to be +1 or -1. For a finite region of such a system, one may ask: given a particular energy function, describing the interaction between these particles, is there a well-defined "infinite volume limit". For extensive systems, (roughly speaking: systems where the range of the interactions is assumed to be finite) such an infinite volume limit was known to exist. For the Sherrington-Kirkpatrick model, in which the range of the interactions is not restricted, the existence of such a well-defined limit was only recently proven. To make matters slightly more interesting, not only is the Sherrington-Kirkpatrick model one of infinite range, the couplings between spins (i.e. the interactions) are also random variables. I will discuss a recent result which characterizes this infinite volume limit in terms of a variational principle over "random overlap structures". I will also begin with a basic introduction to the methods of statistical mechanics.TBA

**Friday, March 19, 2004** **Tomio Umeda (Himeji Institute of Technology)**

Generalized eigenfunctions of relativistic Schroedinger operators, the massless case

2:30 pm / CH 445**Abstract.** It is well-known that generalized eigenfunctions of Schroedinger operators are superposition of plane waves and spherical waves satisfying the Sommerfeld radiation conditions. This fact is very natural from the view points of physics as well as mathematics. Unlike the Schroedinger operators, it is highly nontrivial from the view point of mathematics that generalized eigenfunctions of relativistic Schroedinger operators are superposition of plane waves and spherical waves satisfying the Sommerfeld radiation conditions, even though it is natural from the physics view point. In this talk, I will describe how to prove that generalized eigenfunctions of relativistic Schroedinger operators are superposition of plane waves and spherical waves, how different it is from that for the Schroedinger operators, and why it must be different.

**Friday, March 12, 2004** **Marco Merkli (McGill University)**

Some recent developments in the theory of open quantum systems

2:30 pm / CH 445**Abstract.** Over the last ten years, the mathematical theory of open quantum systems has experienced significant progress. The aim of this talk is to give an overview of phenomena, results and techniques involved in this branch of rigorous non-equilibrium quantum statistical mechanics. Systems considered consist either of a single heat reservoir (modelled by free Bosons or Fermions) interacting with a "small" quantum system possessing finitely many degrees of freedom (e.g. a spin), or several heat reservoirs interacting with each other, possibly via small systems. For the former class of systems, one is interested in the existence and stability of equilibria ("return to equilibrium", "thermal ionization"). For models where several reservoirs interact with each other, the goal is to show that the system approaches, in the large time limit, a "non-equilibrium stationary state", and to characterize that state by its thermodynamic properties, such as the strict positivity of entropy production, or the presence of energy or matter flows between the reservoirs. We formulate the theory of open quantum systems in the language of C* or W* dynamical systems. The dynamics can be described by a so-called Liouville operator (positive temperature Hamiltonian), whose spectral properties reflect the asymptotic behavior of the system. Many recent investigations focus on developing tools for spectral analysis of Liouville-type operators (complex deformation, Mourre theory). Another approach, based on algebraic scattering techniques, allows for an explicit and detailed description of limiting states for relatively simple, but physically relevant models.

**Friday, March 5, 2004** **Ben Chow (University of California at San Diego)**

Monotonicity formulas for geometric flows

2:30 pm / CH 445**Abstract.** Geometric flows deform geometric structures on manifolds such as metrics, submanifolds, and maps. The processes we will discuss are like heat equations where a smoothing process occurs. Monotonicity formulas tell us that the geometric structures are improving and are used to understand how the structures evolve. We shall focus on the heat equation on manifolds and the Ricci flow of metrics where a metric is evolved in the direction of its Ricci curvature. This talk will be aimed at graduate students and nonspecialists.

**Friday, February 27, 2004** **Dirk Hundertmark (University of Illinois at Urbana-Champaign)**

Some results on Lieb-Thirring inequalities

2:30 pm / CH 445**Abstract.** Non-relativistic quantum mechanics, or the spectral theory of Schr"odinger operators, is a fascinating theory for real matter. Up to date it is the best theory physicists came up with and, as a bonus, it also makes sense for mathematicians! However, it is a fairly abstract theory and getting an intuition for it is hard. This has mainly to do with the fact that we live in a classical world and, from our day to day life, are used to classical concepts, but certainly are not at all familiar with the quantum world.

So how does one bridge the gap between real matter and our classical intuition? Often one takes limits, for example, noticing that the Planck constant is really small, one takes it to zero, thereby ending up with the so-called semi-classical theories. Another one is trying to bound quantum mechanical expressions by the corresponding classical ones, not necessarily for a small Planck constant. A particularly useful type of inequalities, which does exactly this, are the so-called Lieb-Thirring inequalities. We will discuss the motivation behind these inequalities and present some of the recent results. The talk will be mainly non-technical, some understanding of convexity is helpful.

**Friday, February 6, 2004** **Robert L. Devaney (Boston University)**

The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence

2:30 pm / CH 445**Abstract.** In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.

**Friday, January 30, 2004** **Robert Kauffman (University of Alabama at Birmingham)**

Discovery Teaching at the Graduate Level

2:30 pm / CH 445**Abstract.** Many people in this department have used a method of teaching where students prove theorems at the board, in class, instead of listening to lectures. This is standard in advanced calculus here, and has been used in other courses as well. Could this method of teaching be used in more graduate courses? Is it possible to cover a reasonable syllabus this way? What are the virtues and flaws of such a method?

This colloquium will feature a short presentation by Bob Kauffman, followed by comments from a panel consisting of Gunter Stolz, Lee Goswick, Gary Russell and Mitch Wyatt, and then by an open discussion between the audience and the panel and Bob.

**Friday, January 23, 2004** **Raul Manasevich (Universidad de Chile)**

A homotopy along $p$ for a system of PDE with a vector $p$-Laplace operator, the radial case.

2:30 pm / CH 445**Abstract.** We consider the system of PDE containing a vector p-Laplace operator {\rm div} ( | \nabla u |^{p-2} \nabla u)+ \lambda |u|^{p-2}u=0,\quad x\in B(0,R) u(x)=0,\quad x\in \partial B(0,R),\\ p>1, under radial symmetry. Here B(0,R) denotes the ball center 0 and radius R in R^N. Also u(x)=(u_1,\cdots,u_n)(x) in R^n for each x and |\cdot| denotes the Euclidean norm in R^{nN}. For this problem we extend a method that uses a suitable homotopy along p to evaluate an associated Leray-Schauder degree. We apply this result to prove existence of nontrivial radial solutions to a system of the form {\rm div} ( a(| \nabla u |) \nabla u)+\lambda a(| u |) u=0,\quad x\in B(0,R)\\ u(x)=0,\quad x\in \partial B(0,R).

**Friday, January 16, 2003** **Jan Dijkstra (the Free University of Amsterdam)**

The (in)stability of (complete) Erd\H os space

2:30 pm / CH 445**Abstract.** It is proved that the countable infinite power of complete Erd\H os space E_c is not homeomorphic to E_c. The method by which this result is obtained consists of showing that E_c does not contain arbitrarily small closed subsets that are one-dimensional at every point. This observation also produces solutions to several problems that were posed by Aarts, Kawamura, Oversteegen, and Tymchatyn. In contrast, we can also show that the original (rational) Erd\H os space is homeomorphic to its infinite power. We obtain this last result by developing topological characterizations of Erd\H os space.

**Friday, December 5, 2003** **Avy Soffer (Rutgers University)**

Multichannel Nonlinear Dynamics and Scattering

2:30 pm / CH 445**Abstract.** The Nonlinear Schrodinger equation, which results from the Hartree Fock approximation in Condensed Matter theory ,and in nonlinear optics, is an example of a dispersive wave equation which has many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many recent scientific advances,such as Bose-Einstein condensates and optical devices . I will discuss the solutions of such equations,including the large time behavior. Asymptotic Stability of N-Soliton solutions will be described. Furthermore, for nonhomogenous equations, rigorous results have shown, for the first time, the phenomena of ground state selection,asymptotic instability of the excited states and more. These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties. The talk will be introductory for general audience.

**Friday, November 21, 2003** **Alexander Blokh and Lex Oversteegen (University of Alabama at Birmingham)**

Wandering triangles exist

2:30 pm / CH 445

**Abstract.**In his preprint

*On the combinatorics and dynamics of iterated rational maps,*Princeton, 1985, W. P. Thurston developed the theory of quadratic invariant laminations. He studied connections between the dynamics of complex polynomials on their Julia sets and the dynamics of the map P

_{d}(z) = z

^{d}on the unit circle in the complex plane. The following notion was introduced in the preprint: a set A

_{0}= {x

_{0},y

_{0},z

_{0}} on the unit circle is said to be a

*wandering triangle*(for f = P

_{d}) if for the sets f

^{n}(A

_{0}) = A

_{n}, n = 0, 1, ... we have that:

(1) each A

_{n}consists of three distinct points for all n = 0, 1, ...;

(2) the convex hulls of all sets A

_{n}in the unit disk are pairwise disjoint.

The theorem below was proven by Thurston in his preprint.

**No Wandering Triangle Theorem.**

*There exist no wandering triangles for P*

_{2}.Recently we constructed an example which shows that wandering triangles exist for P

_{3}and actually for all P

_{d}with d greater than 2. Thus, Thurston's No Wandering Triangle Theorem fails for d greater than 2.

In this talk we will describe our example. We will try to make the presentation accessible to all members of the department, including all students.

**Friday, November 14, 2003** **Lennie Friedlander (University of Arizona)**

Witten deformation of the deRham complex

2:30 pm / CH 445**Abstract.** Twenty years ago, Witten found an analytical proof of Morse inequalities. This proof is based on the study of the spectrum of the Laplacians that are accosiated to the family of deformed deRham differentials $\exp{-th}d\exp{th}$ where $h(x)$ is a Morse function on a manifold. Later, Helffer and Sj"ostrand applied methods of micro-local analysis to study in details the asymptotic behavior of the deRham differential when $t\to\infty$. This analysis made possible to apply the Witten deformation to the study of analytic torsion. I am planning the talk to be as elemetary as possible, and most of it will be accessible to graduate students. Part of the talk is based on the work done by me in collaboration with Dan Burghelea and Thomas Kappeler.

**Friday, November 7, 2003** **Marcel Griesemer (University of Alabama at Birmingham)**

Electron-Photon Scattering

2:30 pm / CH 445**Abstract.** An atom or molecule in an excited state with energy below the ionization threshold will eventually relax to its ground state by dissipating excess energy in the form of radiation. This process of *relaxation to the ground state* is one of the basic phenomena responsible for the production of visible light.

I will introduce a simple mathematical model of atoms interacting with quantized radiation (photons) and give a mathematical formulation of the phenomenon of relaxation to the ground state. The proof is based on methods and ideas from the scattering theory of N-particle quantum systems; in particular the positivity of certain commutators and propagation estimates play an important role.

**Friday, October 31, 2003** **Sumio Yamada (University of Alabama at Birmingham)**

On Penrose Inequality with Electric Charge.

2:30 pm / CH 445**Abstract.** There has been much progress in the subject of Penrose inequality which provides a lower bound on the total mass of asymptotically flat space-time in terms of the geometry of the black holes it contains. In this collaborative work with Gilbert Weinstein, we seek a larger bound for the total mass, by taking into account the electric charges inside the black holes. In particular we provide a counter example to a version of generalized Penrose inequalities, and discuss the implications of this counter example in further understanding of the subject.

**Friday, October 24, 2003** **Robert Gulliver (University of Minnesota)**

Curves of Small Curvature and Embedded Minimal Surfaces

2:30 pm / CH 445**Abstract.** Every closed curve in Euclidean space is the boundary of a minimal surface: think of a soap film which spans an arbitrarily curvy wire. Some mathematical formulations of this phenomenon produce a surface of the topological type of the disk, but allowing the surface to intersect itself (Rado-Douglas, 1930). On the other hand, it is known that a curve with total curvature less than 4 pi is unknotted (Fary-Milnor, 1949). A new result (Ekholm-White-Wienholtz 2002) proves this unknottedness by showing that any minimal surface of the type of the disk is embedded. Their result is extended to the case where the ambient space has nonpositive curvatures. There are some fascinating open questions, even for Euclidean space. This is joint work with Jaigyoung Choe.

**Friday, September 26, 2003** **Jan Lang (Ohio State University)**

p-Laplacian in dimension one and Sobolev embeddings

2:30 pm / CH 445**Abstract.** We will show the relation between eigenvalues of the p-Laplacian and exact approximative numbers of the Sobolev embedding and also relation between eigenfunctions of the p-Laplacian and extreme functions for the best n-dimensional approximation of the Sobolev embedding. Exact value of approximative numbers and the corresponding best n-dimensional linear approximation of the Sobolev Embedding are going to be shown. Difference between L^2 case (Hilbert space) and L^p, p not equal to 2 (non-Hilbert spaces) will be discussed.

**Friday, September 19, 2003** **Anthony Skjellum (University of Alabama at Birmingham)**

Polyalgorithms for Matrix Multiplication: Parallel and Cache-Friendly Methodologies

2:30 pm / CH 445**Abstract.** Although dense matrix multiplication is a straightforward mathematical operation, both cache-memory hierarchy and related distributed memory parallel computing create a large space of potential algorithms, data structures, and performance differentials that vary with problem size, matrix shape, and data layout. This richness can be described in terms of polyalgorithms for the O(N^3) algorithm, and some of the methods that evolve help with Strassen-type algorithms as well. A taxonomy for 2D decompositions of matrix algorithms is given, based on the work of Li, Falgout, and the speaker; a discussion of efficient 2D decompositions inside a single memory space based on modified morton ordering, and due to several researchers, is also discussed, including recent results by Wu and the speaker. Among the goals of this research are to always use the highest performing mathematical operation, to be able to predictive decide what will be highest performance, and to be able to smooth performance as a function of problem size, memory hierarchy, and concurrency.

**Friday, September 12, 2003** **W.D. Evans (University of Wales, Cardiff) **

Inequalities associated with magnetic fields

2:30 pm / CH 445**Abstract.** Hardy and Sobolev-type inequalities in L^p(R^n), 1 < p < \infty, n > 1, associated with magnetic fields, will be discussed, with particular emphasis on the sharpness, or otherwise, of constants. Similar inequalities in the case p=2, n=2 for Aharonov-Bohm-type magnetic fields have already been determined by Laptev/Weidl and Balinsky/Evans/Lewis. The lecture will report on recent joint work with Christer Bennewitz.

**Friday, August 29, 2003** **Ken McLaughlin (University of North Carolina)**

Introduction to random matrix theory

2:30 pm / CH 445**Abstract.** There has been a number of developments of late connected to eigenvalues statistics of random matrices from the so-called "invariant ensembles". Most of these results have to do with the statistics of eigenvalues when the size of the matrices in question grows to infinity. This presentation will be accessible to graduate students and should serve as an introduction to some aspects of this field of research. Bring fruits! If at the end of the presentation viewers have not seen (1) basic questions about eigenvalue statistics connected directly to (2) explicit probability measures on spaces of matrices and (3) a discussion of the behavior of said statistics using orthogonal polynomials, then these fruits may be freely hurled at the speaker. In the time remaining, the speaker will either partake of the fruits of his labor, or discuss some more recent connections between random matrix theory and other combinatorial problems.