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

Presenter: Daniel Ueltschi, Princeton University

Topic: Multiplier ideals and their applications October 18

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

Abstract: In recent years, multiplier ideals have come to play an increasingly important role in algebraic geometry. Originally introduced in the analytic side of the field, these are ideal sheaves that arise in generalizing the classical vanishing theorems. Applications of the theory have included questions involving the geometry of theta divisors in abelian varieties, the deformation invariance of plurigenera of varieties of general type (Siu's theorem), and most recently some problems in commutative algebra. This talk will present a survey of multiplier ideals and some of their applications.

Topic: Visibility of Shafarevich-Tate groups October 19

Presenter: William Stein, Harvard University

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

Presenter: Tian-Jun Li, Princeton University

Topic: Counting congruence subgroups of arithmetic groups and applications October 23

Presenter: Alex Lubotzky, Columbia University and Hebrew University at Jerusalem

Topic: Mean field theory and a small parameter for turbulence October 23

Presenter: Victor Yakhot, Institute for Advanced Study and Boston University

Abstract: Numerical and physical experiments on two-dimensional ($2d$) turbulence show that the differences of transverse components of velocity field are well described by Gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume the existence of a critical space-dimensionality $d=d_{c}$, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At $d<d_{c}$ the flow can be described by the ``mean-field theory'', leading to the observed gaussian statistics and Kolmogorov scaling of transverse velocity differences. It is shown that in the vicinity of $d=d_{c}$ the ratio of the relaxation and translation characteristic times decreases to zero, thus giving rise to a small parameter of the theory. The expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of $d=d_{c}$. The resulting equation describes experimental data on two-dimensional turbulence and demonstrate onset of intermittency as $d-d_{c}>0$ and $r/L\rightarrow 0$ in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity components in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.

Topic: A Liouville theorem for the viscous Burgers equation, with applications October 23

Presenter: Carlos Kenig, University of Chicago

Topic: TBA October 24

Presenter: Gil Kalai, Hebrew University

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

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

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.

Topic: Harmonic analysis on the infinite symmetric group November 6

Presenter: Alexei Borodin, University of Pennsylvania

Topic: TBA November 6

Presenter: Tom Powers, Brown University

Topic: The Theta Body and Partitionable Graphs November 7

Presenter: Bruce Shepherd, Lucent Technologies

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: 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: 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: Non-uniform structures in granular and gas-solid flows November 20

Presenter: Sankaran Sundaresan, Chemical Engineering, Princeton University

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

Presenter: Sean Keel, University of Texas

Topic: TBA November 22

Presenter: A. Eskin, University of Chicago

Topic: TBA November 29

Presenter: H. Hofer, Courant Institute and Princeton 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