In this talk I will explain some of the reasons why it is important to study rational curves on complex algebraic varieties. The talk will have two parts. First I will briefly introduce the Minimal Model Program, which aims at classification of projective algebraic varieties. As I shall explain, rational curves play a very important role in this theory. In the second part of the talk, I will concentrate on varieties that contain many rational curves. I will explain how we can recover important properties of such varieties by studying the geometry of certain families of rational curves on them. This will be an introductory talk, and I will not assume any background in algebraic geometry.

Paul Horja, University of Michigan

Jinhoo Lee , PACM, Princeton University

Zheng-Chao Han, Rutgers University

For the one-dimensional lattice ("integers") there are only two natural subsets: the intervals and the arithmetic progressions. The case of intervals is usually trivial, and the case of arthmetic progressions is extremely hard. Note that very recently Gowers proved some remarkable upper bounds for the van der Waerden threshold number concerning monochromaric arithmetic progressions. If we restrict ourselves to arithmetic progressions starting from zero, then even the "unbounded discrepancy problem" is unsloved (this is a famous open problem raised by Erdos in the 1930's).
For the 2-dimensional lattice one type of natural subset it the "sublattice," but there are many other natural subsets as well, for example the "set of lattice points contained by a rectangle," or simply the "lattice-set of a rectangle." Instead of a rectangle one can take a circle, or other natural geometric shapes. Here the analogue of Erdoos's "unbounded discrepancy problem" is solved. In my talk I focus on three shapes: rectangles, circles, and hyperbola segments. The proofs combine analytic tools like "Riesz product" with combinatorial considerations.

Paul Seidel, University of Chicago

Andrzej Hulanicki , Wroclaw University and CUNY

New developments of nonparametric methods in financial econometrics

Jianqing Fan, Operations Research and Financial Engineering, Princeton University

Christopher Woodward, Rutgers University

Michael Keene, Wesleyan University

One of the distinguishing properties of the present scientific method is reproducibility.

In one of its guises, probability theory is based on statistical reproduction, near certainty being obtained of truth of statements  by averaging over long term to remove randomness occurring in individual experiments.

When one assumes, as is often the case, that events farther and farther in the past have less and less influence on the present, the probabilistic paradigm is currently well understood and is successful in many scientific and technological applications.

Recently, however, we have come to realize that precisely in these applications important stocahstic processes occur whose present
outcomes are significantly influenced by events in the remote past.

This behaviour is not at all well understood and some of the simplest questions remain today irritatingly beyond reach.

A salient example occurs in the theory of random walks, where there  is a dichotomy between recurrent and transient behaviour.

After explaining this classical dichotomy, we present a very simple example with infinite memory which is neither known to be transient  nor recurrent.

Then, using a reinforcement mechanism due to POLYA, we explain the nature of a particular infinite memory process in terms of spontaneous emergence of opinions.

Finally we would like to discuss briefly some of our recent results towards understanding the recurrence-transience dichotomy for
reinforced random walks, and indicate an application to universal coding used in optical CD technology.

Signals received by an array, time reversed and re-emitted into the environment will pack-propagate to the vicinity of the sources that produced them. It is remarkable that the focusing resolution in time reversal is much better in a strongly scattering medium than in a homogeneous one, assuming dissipation is negligible. This interesting phenomenon has many surprising applications in imaging and communications through clutter. I will describe time reversal and its properties, explain mathematically how super-resolution occurs in random media and introduce some imaging methods that deal effectively with clutter. I will also describe how time reversal can be used in communications.

Thomas Nevins , University of Michigan

Michael Shapiro, Michigan State University

Upper bounds on coarsening rates

Robert Kohn, Courant Institute of Mathematics, New York University

C. C. Liu, Harvard University

Vorticity and Wave Confinement A Rotationally Invariant Limiter Approach to Modeling Small Scales

Alexander Braverman, Brown University

TBA

Nicholas J. Pippenger, Computer Science, Princeton University