Topic: Universality and the two dimensional Ising Model September 27

Presenter: Thomas Spencer, Institute for Advanced Study

Topic: Ricci flow on K\"ahler Einstein manifolds September 29

Presenter: Xiuxiong Chen, Princeton University

Topic: Hypoellipticity and nonhypoellipticity of the Kohn Laplacian for very flat October 2

Cauchy-Riemann structures

Presenter: Michael Christ, University of California-Berkeley

Topic: Long-wave unstable thin film equations - singularities, steady states, October 2

and heteroclinic orbits

Presenter: Mary Pugh, University of Pennsylvania

Abstract: I consider long-wave unstable interface models of the type $$ h_t = - (h^n h_{xxx})_x - B (h^m h_x)_x $$ where $B 0$, and $n$ and $m$ are constants. Equations of this type appear in fluid dynamics and in population biology. I will discuss the models as well as some of the mathematical difficulties one confronts in working on fourth-order PDEs. One interesting aspect of these equations is that there is a range of long-time behavior. There is a competition between a stabilizing fourth-order term and a destabilizing second-order term. Is it possible for the second-order term to win and cause finite-time singularities? I will discuss when this may be possible (joint work with Andrea Bertozzi of Duke University). Also, I will discuss the steady states, their linear stability

properties, and will present numerical simulations of evolving solutions (joint work with Richard Laugesen of the University of Illinois of Champaign-Urbana). At least a third of the talk will be within the reach of first and second-year graduate students and I hope some will attend.

Topic: Applied string/M theory October 3

Presenter: Nikita Nekrassov, Princeton University Physics and IHES

Abstract: The four lecture mini-course will give a brief introduction to modern string theory, aimed at the mathematical audience. The first lecture will be devoted to the basics: from quantum field theory to first quantized string theory, bosonic string amplitudes, Deligne-Mumford moduli space of curves and Mumford measure on it. The second lecture will deal exclusively with closed topological strings, Gromov-Witten theory, mirror symmetry, applications to singularity theory. The third lecture will deal with open topological strings and D-branes, deformation quantization, Chern-Simons theory. The last lecture will introduce to M-theory, and noncommutative geometry as seen by string theory.

Topic: Factorization of birational maps October 3

Presenter: Kalle Karu, Harvard Univ and IAS

Topic: On discrete Schroedinger equations with potentials given by deterministic October 4

dynamics

Presenter: Wilhelm Schlag, Princeton University

Topic: Hard Constraints and the Bethe Lattice October 4

Presenter: P. Winkler, Bell Labs

Abstract: In recent years an explosion of work in the intersection of combinatorics and statistical mechanics has contributed lively new ideas to both areas. Of particular interest to combinatorialists are physical systems with hard constraints, such as the hard-core gas model (a.k.a. random independent sets in a graph). In work with Graham Brightwell of the London School of Economics, we model hard-constraint systems by the space Hom(G,H) of homomorphisms from an infinite graph G to a fixed finite constraint graph H. These spaces become tractable when G is a regular tree (often called a Cayley tree or Bethe lattice), because the simple, invariant Gibbs measures on Hom(G,H) then correspond to node-weighted branching random walks on H. With this approach we can characterize the constraint graphs H which, by admitting more than one such measure, exhibit phase transitions. Applications to a physics problem (multiple critical points for symmetry-breaking) and a combinatorics problem (random coloring) will be mentioned.

Topic: The {3x+1} conjecture October 5

Presenter: Toufic M. Suidan, Princeton University

Abstract: This will be an expository talk on the {3x+1} conjecture. Let T be the function on the natural numbers which assigns the value x/2 for even x and 3x+1 for odd x. The conjecture is: For every x there is a k such that T^k x =1. I will discuss several theorems related to this conjecture. The talk will be self contained.

Topic: Applied string/M theory October 5

Presenter: Nikita Nekrassov, Princeton University Physics and IHES

Abstract: The four lecture mini-course will give a brief introduction to modern string theory, aimed at the mathematical audience. The first lecture will be devoted to the basics: from quantum field theory to first quantized string theory, bosonic string amplitudes, Deligne-Mumford moduli space of curves and Mumford measure on it. The second lecture will deal exclusively with closed topological strings, Gromov-Witten theory, mirror symmetry, applications to singularity theory. The third lecture will deal with open topological strings and D-branes, deformation quantization, Chern-Simons theory. The last lecture will introduce to M-theory, and noncommutative geometry as seen by string theory.

Topic: TBA October 9

Presenter: Vladimir Sverak, University of Minnesota

Topic: Edge channels, Chern numbers and Bott periodicity in the integer October 11

quantum Hall effect

Presenter: Hermann Schulz-Baldes, University of California @ Irvine

Topic: TBA October 11

Presenter: Jinho Baik, Princeton University

Topic: Applied string/M theory October 18

Presenter: Nikita Nekrassov, Princeton University Physics and IHES

Abstract: The four lecture mini-course will give a brief introduction to modern string theory, aimed at the mathematical audience. The first lecture will be devoted to the basics: from quantum field theory to first quantized string theory, bosonic string amplitudes, Deligne-Mumford moduli space of curves and Mumford measure on it. The second lecture will deal exclusively with closed topological strings, Gromov-Witten theory, mirror symmetry, applications to singularity theory. The third lecture will deal with open topological strings and D-branes, deformation quantization, Chern-Simons theory. The last lecture will introduce to M-theory, and noncommutative geometry as seen by string theory.

Topic: Interacting Fermi liquid in two dimensions at finite temperature October 25

Presenter: Margherita Disertori, Institute for Advanced Study