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

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.

Presenter: Nadya Gurevich, Princeton University and the Institute for Advance Study

Topic: Homomorphic disks and topological invariants for three-manifolds October 5

Presenter: Peter Ozsvath, Princeton University

Topic: The Creation of Spectral Gaps by Graph Decoration October 6

Presenter: Jeffrey H. Schenker, Princeton University

Abstract: We present a mechanism for the creation of gaps in the spectra of self-adjoint operators defined over a Hilbert space of functions on a graph. The mechanism is based on the process of "graph decoration" which we define. The resulting operators can be viewed as associated with discrete models exhibiting a repeated local structure and a certain bottleneck in hopping amplitudes.

Presenter: Harold Donnelly, Institute for Advance Study and Purdue University

Topic: On self-similar singular solutions of the complex Ginzburg-Landau equations October 9

Presenter: Vladimir Sverak, University of Minnesota

Topic: "Coarse" stability and bifurcation analysis using time steppers October 9

Presenter: Yannis Kevrekidis, Chemical Engineering, Princeton University

Abstract: Evolutionary, pattern forming partial differential equations are often derived as limiting descriptions of microscopic, kinetic theory based models of molecular processes (e.g. diffusion and reaction). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper based approaches, like the recursive projection method (Shroff and Keller, 1993) provide an attractive framework for the latter. We demonstrate an adaptation of this approach that permits a direct, effective ("coarse") bifurcation analysis of certain microscopic, kinetic based models. This is illustrated through a comparative study of the Fitzhugh-Nagumo PDE and of a corresponding Lattice-Boltzmann model.

Topic: Hodge theoretic obstruction to existence of quaternion algebras October 10

Presenter: A. Kresch, University of Pennsylvania

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

Hall Effect

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

Topic: Limiting fluctuations in random growth models October 11

Presenter: Jinho Baik, Princeton University

Topic: Mirror partners arising from integrable systems October 12

Presenter: Michael Thaddeus, Columbia University

Presenter: Dihua Jiang, Institute for Advance Study

Topic: Vertex algebras in differential geometry October 13

Presenter: Zhou Jiang, MIT

Topic: Minimal disks and two-convex hypersurfaces October 13

Presenter: Aliana Fraser, Brown University

Topic: Global Regularity of 3D Navier-Stokes Equations for 3D Flows with October 16

Uniformly Large Vorticity

Presenter: Alex Mahalov, Arizona State University

Abstract: We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for three-dimensional flows having uniformly large vorticity at an initial time t=0. This global regularity is proven for periodic or stress-free boundary conditions for all domain aspect ratios; smoothness assumptions are the same as for local existence theorems. The global regularity is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity is obtained by bootstrapping from global regularity of the limit equations and convergence theorems. In generic cases, sharper regularity results are derived from the algebraic geometry of resonant Poincare curves.

Topic: TBA October 16

Presenter: Soichiro Katayama, Wakayama University and Princeton University

Topic: TBA October 17

Presenter: R. Lazarsfeld, University of Michigan, Ann Arbor

Topic: The lace expansion for self-avoiding walks October 18

Presenter: Daniel Ueltschi, 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: TBA October 18

Presenter: R. Lazarsfeld, University of Michigan, Ann Arbor

Topic: Visibility of Shafarevich-Tate groups October 19

Presenter: William Stein, Harvard University

Abstract: I will describe how to unconditionally compute examples of nontrivial visible elements of Shafarevich-Tate groups of certain modular abelian varieties. Then I will discuss some preliminary data on possible analogues of these computations in the case of modular motives. Finally, I will touch on how visibility of Shafarevich-Tate groups might be connected with the problem of constructing points on certain elliptic curves of analytic rank greater than one.

Topic: Symplectic canonical class, surface cone and symplectic cone of 4-manifolds October 19

Presenter: Tian-Jun Li, Princeton University

Topic: TBA October 20

Presenter: Joel Spruck, Johns Hopkins University

Topic: TBA October 23

Presenter: Carlos Kenig, University of Chicago

Topic: Degrees of Algebraic Approximations October 24

Presenter: Harvey Friedman, Ohio State University

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

Presenter: Margherita Disertori, Institute for Advanced Study

Topic: TBA October 25

Presenter: H. Friedman, Ohio State University

Topic: TBA October 30

Presenter: Jeremiah P. Ostriker, AST, Princeton University

Topic: TBA November 17

Presenter: Emmanuel Hebey, Universite de Cergy-Pontoise

Topic: TBA November 20

Presenter: Haim Brezis, Université de Paris VI and Rutgers University

Topic: Is M_{g,n} a Mori dream space (mod p)? November 21

Presenter: Sean Keel, University of Texas

Topic: TBA December 1

Presenter: Wang Guo-Fang, Max Planck Institute

Topic: The moduli space of cubic surfaces is complex hyperbolic December 5

Presenter: Jim Carlson, University of Utah

Topic: TBA December 8

Presenter: Claude Le Brun, SUNY Stony Brook