Topic: Arithmetic progressions of length four November 8

Presenter: T. Gowers, Cambridge University and Princeton University

Abstract: The famous theorem of Szemer\'edi, proving a conjecture of Erd\"os and Tur\'an from 1936, asserts that for every $\delta>0$ and every natural number $k$ there exists an $N$ such that every subset $A\subset\{1,2,\dots,N\}$ of cardinality at least $\delta N$ contains an arithmetic progression of length $k$. When $k=2$ this assertion is trivial. The case $k=3$ was proved by Roth in 1953 using the circle method. The case $k=4$ is much harder (for reasons that can be made quite explicit) and was not solved until 1969, when Szemer\'edi found a highly ingenious combinatorial argument, which over the next few years he was able to extend to progressions of arbitrary length. In 1977, Furstenberg discovered a completely different and more conceptual argument using ergodic theory, which led to many extensions of the original theorem. Both these proofs ignored Roth's method, and indeed there are very serious obstacles to extending this method to progressions of length greater than three, as I shall demonstrate. However, it can be done, and this will be the main topic of the talk. One of the main advantages of the new approach to the theorem is that it gives very greatly improved bounds for the dependence of $N$ on $k$ and $\delta$. Another is that the proof is much more closely related to results in additive number theory and may eventually lead to the solution of problems in that area.

Topic: Bernoulli Convolutions and K-Partitions with Intermediate Values of November 9

Conditional Entropies

Presenter: Elon Lindenstrauss, Institute for Advanced Study (joint work with Yuval Peres and Wilhelm Schlag)

Abstract: In my talk I will describe how one can use the theory of Bernoulli convolutions to study certain linear realizations of a Bernoulli process on two symbols as a real valued stationery process, which we show have trivial left and right tails. This work was motivated by a question suggested to us by Ya. Sinai regarding the possible values of the entropy of $ K $-partitions, which we answer for the case of Bernoulli measure preserving systems. Time permitting, I will also discuss related new results regarding Bernoulli convolutions and more general projected measures in the symbolic context.

Topic: Local well-posedness results for the quasilinear wave equations November 9

Presenter: Igor Rodnianski, Princeton University

Topic: Monochromatic Unit Fractions November 9

Presenter: E. Croot, Berkeley University

Abstract: We will outline the proof of a general theorem on unit fractions, which has the following corollary:

There exists a constant $b>0$ (effective and computable) such that for any $r$-coloring of the natural numbers $\geq 2$, there exits a monochromatic set $S \subset [2,b^r]$ such that $$\sum_{n \in S} {1 \over n} = 1. $$ In fact, we will show that $b$ may be taken to be $e^{167000}$, for $r$ sufficiently large, and we note that $b$ cannot be taken to be smaller than $e$, since the integers in $[2,e^{(1-\epsilon)r}]$ can be partitioned into $r$ subsets such that the sum of the reciprocals of the numbers in each subset is just under $1$.

Topic: P. A. Smith theory and duality November 9

Presenter: Bernhard Hanke, University of Munich/Notre Dame

Topic: Sidon Sequences November 10

Presenter: Ben Green, Trinity College, Cambridge

Abstract: A Sidon Sequence is a sequence of integers containing no solutions to the equation a + b = c + d apart from the obvious ones. I shall attempt to do justice to an extremely nice paper of Imre Ruzsa in which progress is made towards an old conjecture of Erdos. The conjecture is that there are Sidon sequences which grow about as quickly as certain reasonably obvious upper bounds allow. Ruzsa's remarkable and implausible construction involves rearranging the binary expansions of the numbers clog(p) for p a prime, where c is a random real number. I then hope to present a small selection of other interesting open problems in the area. For the benefit of any harmonic analysts that might attend I may briefly mention the (different) objects that they call Sidon Sequences. Confusingly enough the Sidon sequences of interest to me have a role to play in certain problems connected with the other type of Sidon sequence, as was discovered by Rudin.

Topic: Minimal graphs in R^3 over unbounded domains November 10

Presenter: Joel Spruck, Johns Hopkins University

Topic: Solvability of differential operators on the Heisenberg group November 13

Presenter: Fulvio Riccio, Scuola Normale, Sup. Di Pisa

Topic: Knotted Phase Singularities in Motionless Media November 13

Presenter: Arthur T. Winfree, University of Arizona

Absract: Idealized models of such excitability as found in a wide variety of biological media and some chemical media in gas and liquid states facilitate thinking about their possible modes of activity. These include periodic waves radiating from space curves defined by a phase singularity. In the laboratory these form closed rings that shrink and vanish. In numerical experiments with the corresponding parabolic PDE of reaction and diffusion the rings can also link and knot, making the rings topologically stable. Activity then persists indefinitely.

Topic: Dependent Random Selections and the Balog-Szemeredi Theorem November 14

Presenter: Tim Gowers, University of Cambridge

Topic: Galois groups of function fields November 14

Presenter: Yuri Tschinkel, Princeton University

Topic: Correlation functions in the 1D Holstein-Hubbard quasi-crystal model November 15

Presenter: Vieri Mastropietro, Rome University

Topic: TBA November 15

Presenter: Oded Schramm, Microsoft Research and Wiezmann Institute

Topic: TBA November 16

Presenter: Alex Eskin, University of Chicago

Topic: Induction theorems in algebra and topology November 16

Presenter: J. Grodal, Institute for Advanced Study

Topic: TBA November 16

Presenter: Richard Wentworth, Johns Hopkins University

Topic: Sharp Sobolev-Poincare inequalities on compact Riemannian manifold 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: The PCT theorem and local observables November 21

Presenter: Jakob Yngvason, University of Vienna

Topic: The monodromy matrix, the adiabatic WKB method and the spectrum November 22

of quasi-periodic operators on the real line

Presenter: Frederic Klopp, University of Paris

Topic: Billiards in Rational Polygons and Moduli Spaces of Holomorphic Differentials November 22

Presenter: A. Eskin, University of Chicago

Abstract: A polygon is called rational if all angles are rational multiples of pi. It turns out that the problem of counting periodic billiard trajectories on such a polygon can be reduced to a certain dynamical problem of the moduli space of pairs (M,w) where M is a Riemann surface, and w is a holomorphic 1-form on M. Even though it is not locally homogeneous, this moduli space is analogous in many ways to the moduli space of Euclidean lattices SL(n,R)/SL(n,Z). I will discuss these constructions and some associated problems in combinatorial enumeration.

Topic: Harmonic analysis on the infinite symmetric group November 27

Presenter: Alexei Borodin, University of Pennsylvania

Topic: Non-uniform structures in granular and gas-solid flows November 27

Presenter: Sankaran Sundaresan, Chemical Engineering, Princeton University

Topic: Nonperturbative analysis of a Model Quantum System under Time November 29

Periodic Forcing: Generic and Exceptional Cases

Presenter: Ovidiu Costin, Rutgers University

Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation $\eta(t)$. The amplitude of $\eta$ is unrestricted. We show that for generic $\eta(t)$, which includes the sum of any finite number of harmonics, the system, started in a localized state, gets fully ionized (the particle leaves the bound state) as time grows indefinitely, irrespective of the magnitude or frequency (resonant or not) of $\eta(t)$. There are however exceptions to full ionization which while very non-generic include rather simple explicit functions. For these $\eta(t)$ the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.

Topic: TBA November 29

Presenter: H. Hofer, Courant Institute and Princeton University

Topic: TBA November 30

Presenter: V. Shokurov, Johns Hopkins University

Topic: TBA December 1

Presenter: Wang Guo-Fang, Max Planck Institute

Topic: d-bar-regularity for weakly pseudoconvex domains in compact Hermitian December 4

symmetric spaces with respect to invariant metrics

Presenter: Yum-Tong Siu, Harvard University

Topic: TBA December 4

Presenter: Salvatore Torquato, Chemistry, Princeton University

Topic: The Theta Body and Partitionable Graphs December 5

Presenter: Bruce Shepherd, Lucent Technologies

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

Topic: Brownian Dynamics Methods for the Solution of Complex Polymeric December 11

Flows Based on Kinetic Theory Models: Early (CONNFFESSIT) and

More Recent (Configuration Field) Approaches

Presenter: Antony N. Beris, University of Delaware

Abstract: In the past, closed-form continuous models have been used for the solution of complex (i.e. multidimensional and/or time dependent) flow problems involving polymer solutions or melts. However, those models involve closure approximations that result in unpredictable errors in the complex flows that they are employed. Nevertheless, the presence of one or more internal variables makes the dimensionality of the microscopic problem prohibitively large to allow for a direct solution of the microscopic equations (such as those arising from kinetic theory) in even the simplest multidimensional flow problems. Fortunately, in the last decade new methodologies has emerged that allow for such solutions to be obtained at significantly less computational cost through the coupling of a stochastic solution for the polymer chain configuration to a more traditional macroscopic finite-element or spectral flow approximation of the momentum and continuity equations. In this presentation, after reviewing the first of these approaches (called CONNFFESSIT) which has first been developed by Laso and Oettinger, we will discuss the more recently developed (by Hulsen and co-workers) configuration fields approach that result in even more reduced computational requirements.

Topic: Coherent and Dissipative Electronic Transport in Aperiodic Media December 12

Presenter: Jean V. Bellissard, Universite Paul Sabatier, Toulouse, France and Institut Universitaire de France