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: Twisted toric geometry October 11

Presenter: Maxim Kontsevich, IHES

Abstract: Toric geometry studies a simple class of algebraic varieties encoded by convex integral polytopes. For example, the standard simplex corresponds to the projective space. I will describe a generalization of toric geometry which gives combinatorial counterparts of more complicated algebraic varieties, like curves, K3 surfaces etc. Natural examples come from Mirror Symmetry. The subject is related to non-archimedean geometry and motivic integration.

Topic: TBA October 12

Presenter: Michael Herman, University of Paris, VII

Topic: Mirror partners arising from integrable systems October 12

Presenter: Michael Thaddeus, Columbia University

Presenter: Dihua Jiang, Institute for Advance Study

Topic: Minuscule Representations of Lie Algebras October 13

Presenter: Jacob A. Lurie, Princeton University

Abstract: I will present a nice way of constructing simply-laced Lie algebras and some of their representations. If time permits, I'll give some applications which can help to understand the exceptional group E_6.

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 existence for a class of systems of nonlinear wave equations October 16

in three space dimensions

Presenter: Soichiro Katayama, Wakayama University and Princeton 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: Applied string/M theory October 17

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: Some algebraic applications of multiplier ideals October 17

Presenter: Robert K. Lazarsfeld, University of Michigan, Ann Arbor

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

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: Counting congruence subgroups of arithmetic groups and applications October 23

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

Topic: TBA October 23

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: 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: Jeremiah P. Ostriker, AST, Princeton University

Topic: TBA November 15

Presenter: Oded Schramm, Microsoft Research and Wiezmann Institute

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 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: 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