School of Mathematics

Reducibility for the Quasi-Periodic Liner Schrodinger and Wave Equations

Lars Hakan Eliasson
University of Paris VI; Institute for Advanced Study
February 21, 2012
We shall discuss reducibility of these equations on the torus with a small potential that depends quasi-periodically on time. Reducibility amounts to "reduce” the equation to a time-independent linear equation with pure point spectrum in which case all solutions will be of Floquet type.

For the Schrodinger equation, this has been proven in a joint work with S. Kuksin, and for the wave equation we shall report on a work in progress with B. Grebert and S. Kuksin.

Arithmetic Fake Compact Hermitian Symmetric Spaces

Gopal Prasad
University of Michigan
February 16, 2012
A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David Mumford in 1978 using p-adic uniformization. This construction is so indirect that it is hard to obtain geometric properties of the fake projective plane. A major problem in the theory of complex algebraic surfaces was to find all fake projective planes in a way which allows us to discover their geometric properties.

The Inner Equation for Generalized Standard Maps

Pau Martin
Universitat Poliecnica de Catalunya, Barcelona, Spain
February 15, 2012
We study particular solutions of the "inner equation" associated to the splitting of separatrices on "generalized standard maps". An exponentially small complete expression for their difference is obtained. We also provide numerical evidence that the inner equation provides quantitative information of the splitting of separatrices even in the case when the limit flow does not predict correctly its asymptotic behavior.

On the Colored Tverberg Problem

Benjamin Matschke
Institute for Advanced Study
February 14, 2012
In this talk I will present a colored version of Tverberg's theorem about partitioning finite point sets in R^d into rainbow groups whose convex hulls intersect. This settles the famous Bárány-Larman conjecture (1992) for primes minus one, and asymptotically in general. It implies some complexity estimates in computational geometry. I will also give some generalizations that connect to mass partition and center point theorems. The proofs use equivariant topology, which I will try to keep as elementary as possible. This is joint work with Pavle V.M.

High-Confidence Predictions under Adversarial Uncertainty

Andrew Drucker
Massachusetts Institute of Technology
February 13, 2012
We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x . Our main focus is the problem of successfully predicting a single 0 from among the bits of x . In our model we get just one chance to make a prediction, at a time of our choosing. This models a variety of situations in which we need to perform an action of fixed duration, and must predict a "safe" time-interval to perform it.