(September 14) Vadim Kaloshin: Examples of infinite dimensional Hamiltonian systems with growth of energy - During the talk we shall discuss problems of spreading of energy for infinite dimensional Hamiltonian systems such as coupled rotors, 1-dimensional crystal lattices, and periodic 1-dimensional NLS. Then we describe the main result which loosely speaking says that for a large class of infinite dimensional Hamiltonian systems for an appropriate "smooth" nonlinearity there are solutions with a polynomially fast growth of energy. This is related to one of the problems posed by Bourgain.

(September 21) Yulij Ilyashenko: Non-attracting attractors - One of the major problems in the theory of dynamical systems is the study of the limit behavior of solutions. It is a general belief that after a long time delay the observer will see the orbits that belong to this or that type of attractor. Therefore, knowledge of the attractor of the system predicts the long time behavior of solutions.

In the present talk we develop an opposite point of view. Namely, we describe dynamical systems whose attractors have a large part which is in a sense unobservable. This motivated a notion of ε-attractor. It is a set (not necessary uniquely defined) near which almost all the orbits spend in average more than (1-ε) part of the future time. We discuss the effect of drastic non-coincidence of actual attractor and ε-attractor. For ε sufficiently small, like 10^-30, the difference between actual attractors and ε attractors is unobservable in the computer and physical experiments. Therefore, ε-attractors with small ε have a chance to replace actual attractors in applications. Theorems and conjectures on the subject will be presented.

(September 28) Leonid Bunimovich: Escape from a circle and Riemann hypothesis - A fundamental issue in physical measurements is ensuring that the system is to be studied is little affected by the observation. Therefore the experimentalists often study systems by making a small hole and measuring the escape from a system of interest. Beautiful experiments on atomic optic billiards inspired us to suggest seemingly a new approach to the studies of closed systems: make several holes and compare a total escape rate with the sum of individual escapes through each hole. Already seemingly in the simplest case of two opposite holes in a circular billiard this problem occured to be equivalent to the Riemann hypotheses (RH) which once again reminds that one should watch out when entering reality. I'll describe a kind of new general problems in dynamical systems arising in this approach and explain how RH in particular examples comes about.

(October 12) Nira Dyn: Subdivision schemes for the refinement of geometric objects - Abstract

(October 19) Richard Wentworth: Harmonic maps and representations of surface groups - This lecture will give a broad survey of recent activity in the study of surface group representations into symmetric spaces and more general nonpositively curved metric spaces. Topics will include equivariant harmonic maps and Higgs bundles, Toledo invariants and the generalized Milnor-Wood inequality, harmonic maps to trees, and Teichmueller space and mapping class groups. The presentation should be accessible to a wide audience.

(October 26) Charles Weibel: The Norm Residue is an Isomorphism - Milnor conjectured in 1970 that the étale cohomology of a field (mod 2 coefficients) should have a presentation with units as generators and simple quadratic relations (the ring with this presentation is now called the "Milnor K-theory"). This was proven by Voevodsky, but the odd version (mod p coefficients for other primes) has been open until this year, and has been known as the Bloch-Kato Conjecture.

Using certain norm varieties, constructed by Rost, and techniques from motivic cohomology, we now know that this conjecture is true. This talk will be a non-technical overview of the ingredients that go into the proof, and why this conjecture matters to non-specialists.

Here is a fun consequence of all this. We now know the first 20,000 groups K_n(Z) of the integers, except when 4 divides n. The assertion that these groups are zero when 4 divides n (n>0) is equivalent to Vandiver's Conjecture (in number theory), and if it holds then we have fixed Kummer's 1849 proof of Fermat's Last Theorem. If any of them are nonzero, then the smallest prime dividing the order of this group is at least 16,000,000.

(November 16) Albert Cohen: Adaptive Approximation by Greedy Algorithms - This talk will discuss computational algorithms that deal with the following general task : given a function f and a dictionary of functions D in a Hilbert space, extract a linear combination of N functions of D which approximates f at best. We shall review the convergence properties of existing algorithms. This work is motivated by applications as various as data compression, adaptive numerical simulation of PDE's, statistical learning theory.

(November 30) Stephen DeBacker: Harmonic analysis on reductive p-adic groups - In recent years many old and cantankerous problems in p-adic representation theory have fallen. In this talk, I will discuss some problems in the area that have been solved via Bruhat-Tits theory. This beautiful theory was (re)introduced to the subject through the fundamental work of Allen Moy and Gopal Prasad. We shall attempt to dispel the notion that this is not good colloquium material by using Michigan's ``time-tested algorithm'' for giving a colloquium talk.

(December 7) Gilbert Strang: Minimum Cuts and Maximum Area - The oldest competition for an optimal shape (area-maximizing) was won by the circle. We propose a proof that uses the support function of the set -- a dual description that linearizes the isoperimetric problem.

Then we measure the perimeter in different ways, which changes the problem (and has applications in medical imaging). If we use the line integral of |dx| + |dy|, a square would win. Or if the boundary integral of max(|dx|,|dy|) is given, a diamond has maximum area. When the perimeter = integral of ||(dx,dy)|| around the boundary is given, the area inside is maximized by a ball in the dual norm.

The second part describes the **max flow-min cut theorem** for continuous flows. Usually it is for discrete flows on edges of graphs. The maximum flow out of a region equals the capacity of the minimum cut. This duality connects to the constrained isoperimetric problems that produce minimum cuts, and to the Cheeger constant.

(February 8) Carl Bender: Ghostbusting: Reviving quantum theories that were thought to be dead - The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p² +i x³, for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian H = p² +i x³ is not Dirac Hermitian, it is PT symmetric; that is, it is symmetric under combined space reflection P and time reversal T. In general, if a Hamiltonian $H$ is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for determining the adjoint operation under which H is Hermitian. (It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric g^μν in curved space is before solving Einstein's equations.) In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was the appearance of ghost states (states of negative norm). The cause of the disease is that the Hamiltonians for these models were inappropriately treated as if they were Dirac Hermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories.

(March 7) Rahul Pandharipande: Modern enumerative geometry - I will discuss counting questions in algebraic geometry. The classical problem is to count curves in a fixed space subject to basic geometric conditions. I will explain old and new perspectives and the connections to modern developments in Gromov-Witten theory.

(March 28) Richard D. James: New materials from mathematics: real and imagined - In this talk I will give two examples where mathematics played an important role for the discovery of new materials, and a third example where mathematics suggests a systematic way of searching for broad classes of yet undiscovered materials.

(March 28) Mario Bonk: Quasiconformal Geometry of Fractals - Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston's characterization of rational functions with finite post-critical set.

In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpinski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will report on some recent developments in this area.

(April 4) Freydoon Shahidi: Spectral Theory of Hyperbolic Laplacian and Functoriality Principle - Eigenvalues of the Laplacian on a hyperbolic Riemann surface play an important role in many problems in number theory, icluding counting lattice points within circles on these surfaces and estimating sums of Kloosterman sums. We will discuss these problems and how they can be fully resolved using transfers, in the sense of Langlands functoriality, of Maass forms to automorphic functions on certain larger groups. We also discuss some other transfers and their consequences in both local and global spectral analysis of corresponding groups. In particular, we state some very precise reducibility criterion for induced representations of these groups in terms of these transfers.

(April 18) Mark de Cataldo: The topology of algebraic maps - I will survey some results concerning the special place occupied in geometry by complex algebraic varieties and the maps between them.

(May 9) Kaushik Bhattacharya: Mathematical and numerical issues in computing material properties - We use disparate theories to describe materials at different scales: we use quantum mechanics at the smallest scales, atomistic theories at the intermediate scales and continuum mechanics at the macroscopic scale. However, how one theory emerges as a coarse-grained approximation to the other is not always clear, and this manifests itself as a challenge when we try to compute macroscopic properties of materials. We are now able to compute many macroscopic properties of defect-free solids starting from quantum mechanics. However, defects that are responsible for many macroscopic properties remain a challenge. This is because the different scales interact in a non- trivial manner and this raises difficult but interesting interesting mathematical and computational problems. This talk will introduce the theories and the difficulties, and then describe some mathematical results that enable the development of a new computational approach.