Math Colloquium

Math colloquium meets on Mondays at 12:15 in Schreiber 006, Tel Aviv University.

Spring 2008

25.02.2008, 12:15
Shai Dekel , GE Healthcare, Israel
Multivariate adaptive representations and anisotropic function spaces.

ABSTRACT: Anisotropic phenomena appear in various contexts in mathematical analysis and its applications: sharp edges in digital images and the formation jump discontinuities of solutions of hyperbolic conservation laws, to name just two examples. We show algorithms for sparse representations of the anisotropic phenomena that can outperform existing state-of-the-art adaptive methods that are based on isotropic building blocks such as wavelets. We also address one of the main challenges of modern approximation theory, which is: how to measure the (weak-type) smoothness of such phenomena. Existing classical function spaces such as Besov-Sobolev are optimal in the univariate case, but are inadequate for higher dimensions. We present a framework for anisotropic regularity that generalizes the classical function spaces and allows capturing of the low-dimensional structure of the anisotropic phenomena.

03.03.2008, 12:15
Boaz Tsaban ,Bar Ilan University, Israel
Mathematical Selection Principles: Past, present, future.

ABSTRACT: The mathematical field of Selection Principles deals with qualitative analysis of diagonalization procedures, where the diagonalized object carries a mathematical (topological or other) structure. This field aims at applications to classical and new mathematical objects and phenomena, in a variety of mathematical disciplines, including: Ramsey theory, game theory, function spaces and convergence, topological groups, dimension theory, covering properties, combinatorial set theory, hyperspaces, etc. The uniform treatment given by Selection Principles to this vast variety of fundamental mathematical properties is not only aesthetically pleasing. It brought new perspectives, focused attention on essential parts of classical problems, and in several cases lead to their solution. We will illustrate this by several fundamental examples, and discuss some of our future plans regarding this fascinating topic.

17.03.2008, 12:15
Albert Fathi , Ecole Normale Superieure de Lyon
On smooth critical subsolution of the Hamilton-Jacobi equation.
(In the framework of Blumental Lectures in geometry.)

ABSTRACT: click here

24.03.2008, 12:15
Felix Schlenk , ULB, Brussels
Complexity of Hamiltonian systems and modern symplectic topology.

ABSTRACT: One goal in the study of dynamical systems is understanding their complexity.Hamiltonian systems describe systems without friction, and so one can expect that these systems are complicated in some way. A quite intuitive measure for complexity is topological entropy. We explain how modern techniques in symplectic topology can be used to show that many interesting Hamiltonian systems have indeed positive topological entropy. The talk is based on joint works with Urs Frauenfelder and Leo Macarini.

31.03.2008, 12:15
Yuval Peres , Microsoft Research
Scaling limits for internal aggregation models.

ABSTRACT: Start with $n$ particles at each of $k$ points in the lattice $Z^d$, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed. We prove that if the distances between the starting points are scaled by $n^{1/d}$, then the set of occupied sites has a deterministic scaling limit. In two dimensions, the boundary of the limiting shape is an algebraic curve of degree $2k$. (For $k=1$ it is a circle, as proved in 1992 by Lawler, Bramson and Griffeath). The limiting shape can also be described in terms of a free-boundary problem for the Laplacian and quadrature identities for harmonic functions, revealing a connection with potential theory and fluid mechanics. I will show simulations of the process, that suggest several intriguing open problems. Joint work with Lionel Levine.

05.05.2008, 12:15
Shlomo Sternberg , Harvard University
The symplectic category and the split orthogonal category.

ABSTRACT: In 1979 Victor Guillemin and I introduced the linear symplectic category whose objects are symplectic vector spaces and whose morphisms are Lagrangian subspaces of the twisted direct product. In a recent remarkable paper, Alekseev, Bursztyn, and Meinrenkin used an analogue of this category with ``symplectic" replaced by "split orthogonal" (and an important subcategory thereof) to study Dirac structures on differentiable manifolds, unifying and extending much recent work in this area. In this lecture I will go over the elementary linear algebra involved in the construction of these categories.

12.05.2008, 12:15
Lenya Ryzhik , University of Chicago
Reaction-diffusion fronts in random and heterogeneous media.

ABSTRACT: It is well known that scalar reaction-diffusion equations in homogeneous media admit special planar front solutions which propagate with a constant speed. They are stable for a large class of nonlinearities which makes them relevant for many applications. Similar results have been established in periodic media by Berestycki and Hamel, and Weinberger. I will discuss generalizations of the notion of a traveling front to general heterogeneous media and, in particular, random media, as well as stability of those fronts. This is a joint work with J. Nolen, J.-M. Roquejoffe and A. Mellet.

19.05.2008, 12:15
Haim Brezis , Paris 6, Rutgers, Technion
New ideas about the topological degree.

ABSTRACT: I will present an unusual connection between Fourier series and the topological degree of maps from S^1 into itself. Also, recent estimates for the degree in S^n.

26.05.2008, 12:15
David Mumford , Brown University (Laureate of the 2008 Wolf Prize in Mathematics)
The Interplay of Different Cultures in the Progress of Mathematics.

ABSTRACT. It is striking how differently mathematics developed in Greece/Europe, India/Middle East and China with rare but significant exchanges. In the end, there is only one Mathematics to be discovered, but there are often many routes to the same understanding. I will illustrate this interplay with examples from "Pythagoras's" theorem and negative numbers to the differential equation for sine.

02.06.2008, 12:15
Amir Beck , Technion, Israel

09.06.2008, 12:15
No Colloquium (Shavuot)

16.06.2008, 12:15
Fedor Pakovich , BGU, Israel
The Ritt theorems and their generalizations.

ABSTRACT: In 1922 J. Ritt constructed the theory of decompositions of polynomials with respect to the composition operation. In particular, he proved that two decompositions of a polynomial into a composition of indecomposable polynomials have the same number of components, and described polynomial solutions of the equation A(B)=C(D). In the talk we discuss some recent results of the author and of the author and M. Muzychuk which generalize the Ritt theorems to the Laurent polynomials (rational functions with at most two poles). On the other hand, using the Belyi functions corresponding to the Platonic solids we construct explicit examples of rational functions which have decompositions of different lengths. Finally, we give a description of solutions of the equation A(B)=C(D), where A,C are polynomials while B,D are arbitrary entire functions.

14.07.2008, 12:15
Alex Nabutovsky (Pennstate and U Toronto)
Kolmogorov complexity and variational problems in Riemannian geometry.

ABSTRACT: First, I am going to explain some applications of recursion theory to combinatorics of the space of triangulations of a compact manifold. Then I will discuss R. Thom and S.T. Yau question ``What is the best Riemannian metric on a given smooth manifold?" Finally, I am going to talk about possible connections between these topics and Quantum Gravity.
Organizer: Semyon Alesker , e-mail: semyon AT post DOT