(September 26) Peter Li: The spectrum and the number of ends of a complete manifold — In this talk we will describe a general method of counting the number of ends for some classes of complete Riemannian manifolds. A typical class of manifolds being considered is that the greatest lower bound of the spectrum is of maximum value relative to a lower bound of the Ricci curvature. Another class are locally symmetric spaces of infinite volume and has its spectrum bounded by the maximum value. Finally, we will also address manifolds with finite volume.

(October 17) David Gabai: Volumes of hyperbolic 3-manifolds — We will outline a proof that the Weeks manifold is the unique lowest volume closed orientable hyperbolic 3-manifold and address the Thurston, Weeks, Matveev-Fomenko conjecture that among the low volume hyperbolic 3-manifolds, there is a close relation between volume and combinatorial/topological complexity. (Joint work with Peter Milley and Robert Meyerhoff.)

(October 31) Valery Alexeev: From stable curves to higher dimensions — Applications of stable curves and maps to enumerative geometry, physics, arithmetical geometry, symplectic geometry, etc., are very well known. Due to efforts by many people, this theory was partially extended to higher dimensions, when stable curves are replaced by stable surfaces, 3-folds, etc. I will attempt to describe the current state of the art in this field.

(November 07) Michael Renardy: Some new results on stability of inviscid shear flow — In this talk, I shall discuss two problem related to inviscid shear flow:

The first part of the talk will discuss the connection between long wave instabilities and ill-posedness of the hydrostatic approximation. In particular, the role of boundary conditions on well-posedness of the hydrostatic equations will be highlighted.

In the second part, I shall present a result on stability of short wave perturbations to inviscid shear flows.

(November 14) Gregoire Allaire: Topology Optimization of Structures — The typical problem of structural optimization is to find the "best" structure which is, at the same time, of minimal weight and of maximum strength or which performs a desired deformation. In this context I will present the combination of the classical shape derivative and of the level-set methods for front propagation. This approach has been implemented in two and three space dimensions for models of linear or non-linear elasticity and for various objective functions and constraints on the perimeter. It has also been coupled with the bubble or topological gradient method which is designed for introducing new holes in the optimization process. Since the level set method is known to easily handle boundary propagation with topological changes, the resulting numerical algorithm is very efficient for topology optimization. It can escape from local minima in a given topological class of shapes and the resulting optimal design is largely independent of the initial guess.

(January 30) Peter Sarnak: Unipotent orbits at prime exponents — Ratner's theorem gives the equidistribution properties of g.uⁿ, n = 1,2,3,..., in the homogeneous space Γ\G, where g is a fixed element in a connected Lie group G, Γ a lattice in G and u a fixed unipotent element in G. We discuss the apparently difficult problem of what can one say when n runs over primes. After a general introduction to the problem and approaches and difficulties associated with it, we report on progress by Adrian Ubis and myself for the case of SL(2,Z)\SL(2,R).

(February 20) Robert A. Fefferman: Some Generalizations of Classical Problems in Harmonic Analysis and Associated Open Problems — This will be a consideration of generalizations of some of the main themes from classical Fourier Analysis:

• The Hardy-Littlewood Maximal Function
• The Connection Between the Maximal Operator, Calderon-Zygmund Singular Integrals, and Classical Multiplier Operators
• Classical Marcinkiewicz Multipliers and the Double Hilbert Transform
• The Classical L^p Dirichlet Problem for the Unit Ball in Rⁿ

We shall also consider some open problems associated with these extensions and generalizations.

(March 27) Nimish Shah: Equidistribution of evolution of curves under geodesic flow — We let the parameter measure concentrated on a finite segment of a C² curve on the unit tangent bundle of a compact hyperbolic manifold evolve under the geodesic flow. We show that if the curve is transversal to the weak-stable leaves at almost all points on the curve then the limiting measure is the Liouville measure on the manifold, under a natural geometric condition on the curve.

Analogous phenomenon on SL(n,R)/SL(n,Z) gives interesting number theoretic consequences.

(April 3) A. Katok: Invariant geometric structures for group actions: measure rigidity and Zimmer program. — Zimmer program, very roughly, aims at showing that volume preserving smooth actions of sufficiently ''large'' and ''rigid'' groups, such as semisimple Lie groups of real rank greater than one, or lattices in such groups, are essentially algebraic, or, more precisely are built from algebraic blocks. Until recently successes were limited to very low dimensional situations, or to the local problems (perturbations of algebraic actions), or to the actions satisfying additional dynamical assumptions, such as presence of Anosov elements. I will discuss the first case of partial realization of this program without such limitations.

(April 10) Sergei Tabachnikov: Flavors of "bicycle mathematics": tire tracks monodromy, hatchet planimeter, Menzin's conjecture, and oscillation of unicycle tracks — The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel, fixed on the bicycle frame); the same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and hence its terminal position; the monodromy map sends the initial position to the terminal one. This circle mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. Moebius transformations are either elliptic, or parabolic, or hyperbolic. I shall outline a proof of a 100 years old conjecture: if the front wheel track is an oval with area at least pi then the respective monodromy is hyperbolic. I shall also discuss bicycle motions such that the rear wheel follows the trajectory of the front one. I shall explain why such ''unicycle" tracks become more and more oscillating in forward direction and cannot be infinitely extended backward.

(April 24) Carlos E. Kenig: The global behavior of solutions to critical nonlinear dispersive and wave equations — In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas.

(May 1) Leslie Greengard : The Fast Multipole Method and its Applications — In this lecture, we will describe the analytic and computational foundations of fast multipole methods (FMMs), as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a system of N particles from O(N²) to O(N) or O(N log N) operations. FMMs are equally useful, however, in solving partial differential equations by first recasting them as integral equations. We will present examples from electromagnetics, elasticity, and fluid mechanics.