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

Presenter: Margherita Disertori, Institute for Advanced Study

Topic: Does normal mathematics need new axioms? October 25

Presenter: H. Friedman, Ohio State University

Abstract: According to conventional wisdom (CW), normal mathematics steers clear of foundational issues. Only a minimal fragment of the currently accepted axioms and rules for mathematics (ZFC) are used (in any remotely essential way) in current normal mathematics. The known set theoretic independence results from ZFC do not upset CW because they are known to involve abnormal subsets of uncountable sets. The known unprovability of consistency does not upset this conventional wisdom since normal mathematics is not concerned with properties of formal systems for mathematical reasoning. The study of Diophantine equations is highly normal, but the known impossibility of an algorithm does not upset CW since it does not lead to any need to reconsider the status of ZFC. This CW has been attacked inconclusively at the margins: every Borel subset of $R^2$ that is symmetric about y=x contains or is disjoint from the graph of a Borel function. It is necessary and sufficient to use uncountably many uncountable cardinalities to prove this Theorem. Standards are very high for the genuine overthrow of CW. The new Boolean relation theory (BRT) and its reduced forms, disjoint cover theory (DCT) and formal partition theory (FPT), promise to refute CW and ignite renewed interest in foundational issues. Initial indications are that in virtually any mathematical context (discrete or continuous), these thematic investigations are deep, open ended, varied, and explainable at the undergraduate level. BRT grew out of two examples, which indicate its flavor. The thinness theorem asserts that for F:N^k into N, there exists an infinite subset A of N such that F[A^k] is not N. The complementation theorem asserts that for any strictly dominating F:N^k into N, there exists a (unique) infinite subset A of N such that F[A^k]=N\A. We present statements of this kind involving two functions and three sets provable using large cardinal axioms but not ZFC. Restricting to rather concrete functions does not change matters. We conjecture that the general theory of such statements can be carried out with large cardinal axioms. Partial results have been obtained.

Topic: Eigenfunction Asymptotics and Rankin Triple L-functions October 26

Presenter: Thomas Watson, Princeton University

Abstract: I will explain a natural connection between the Quantum Unique Ergodicity conjecture of Rudnick-Sarnak and standard conjectures for automorphic L-functions. In particular, an optimal form of QUE is reduced to the Lindelof hypothesis for degree 6 L-functions. The full strength of Lindelof is not needed to establish QUE; any improvement over the trivial or 'convexity' bound for these L-functions will prove the conjecture. Such estimates are already known in some special cases, and these have other unconditional applications.

Topic: On the Bernstein problem for affine maximal surfaces October 26

Topic: Vanishing of L^2 cohomology and the Bergman metric October 26

Presenter: Jeff McNeal, Ohio State University

Topic: The computational complexity of some problems in geometry and topology October 26

Presenter: Joel Hass, UCDavis and the Institute for Advance Study

Topic: Triangulated Spheres and Systems of Disjoint Representatives for October 27

Families of Hypergraphs

Presenter: Maria Chudnovsky, Princeton University

Abstract: A hypergraph is a collection of subsets, called edges, of a given vertex set. It is a generalization of the notion of a graph, where all the edges are of size 2. We prove a generalization of Hall's theorem to families of hypergraphs, namely give sufficient conditions for a family of hypergraphs to have a system of disjoint representatives (a choice of edges from each hypergraph, such that two edges chosen from different hypergraphs are disjoint). In the course of the proof we show that a triangulation of S^n-1 can be extended it to a triangulation of B^n by adding points in the interior of B^n with some restrictions on the degrees of these points.

Topic: TBA October 30

Presenter: Kenji Nakanishi, Kobe University

Topic: Testing Cosmological Models October 30

Presenter: Jeremiah P. Ostriker, AST, Princeton University

Abstract: The study of cosmology, the origin, nature and future evolution of structure in the universe, has been totally transformed in the last decade, and computers have played a major role in the change. New theories have arisen which make the subject, formerly almost a branch of philosophy, into a quantitative science. Initial, semi-quantitative tests of these theories, either using data on galaxy distributions in the local universe or the cosmic background radiation fluctuations reaching us from the distant universe, indicate rough agreement with the simplest predictions of the theories. But now that fully three dimensional, time dependent numerical simulations can be made on modern, parallel architecture computers, we can examine (using good physical modelling) the detailed quantitative predictions of the various theories that have been proposed to see which, if any, can produce an output consistent with the real world being revealed to us by the latest ground and space based instruments. Simulations could address 32^3 = 10^4.5 independent volume elements a decade ago. Now 512^3 = 10^8.1 is the standard for hydro computations, with 1024^3 = 10^9.0 the current state-of-the art. Increasingly, unstructured, adaptive or moving mesh techniques are being used to improve the resolution in the highest density regions. In purely darkmatter (gravitation only) calculations, the ratio of volume to resolution element has reached 16,000^3 = 10^12.6. This has enabled detailed computation for phenomena, from gravitational lensing to X-ray clusters, to be made and compared with observations. Using these tools, we have been able to reduce to a small number the currently viable options for the correct cosmological theory.

race problem

Presenter: Kevin Ford, University of South Carolina

Topic: Twist, kinks, and drag: whirling elastica November 6

Presenter: Tom Powers, Brown University

Topic: The Theta Body and Partitionable Graphs November 7

Presenter: Bruce Shepherd, Lucent Technologies

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

Presenter: Joel Spruck, Johns Hopkins University

Topic: TBA November 13

Presenter: Peter Bunge, Geosciences, Princeton University

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

Presenter: Tim Gowers, University of Cambridge

Topic: TBA November 15

Presenter: Oded Schramm, Microsoft Research and Wiezmann Institute

Topic: Induction theorems in algebra and topology November 16

Presenter: J. Grodal, Institute for Advanced Study

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: TBA November 22

Presenter: A. Eskin, University of Chicago

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: 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: TBA December 4

Presenter: Salvatore Torquato, Chemistry, Princeton University

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