SEMINARS
Updated: 12-3-2008
 DECEMBER 2008 Statistical Mechanics Seminar Topic: Local and Global Structure of Stationary States of Macroscopic Systems Presenter: Joel Lebowitz, Rutgers University Date: Wednesday, December 3, 2008, Time: 2:00 p.m., Location: Jadwin 343 Abstract: The microscopic structure of a macroscopic system in a steady state is described locally, i.e. at a suitably scaled macroscopic point $x$, by a time invariant measure of the corresponding infinite system with translation invariant dynamics. This measure may be extremal, with good decay of correlations, or a superposition of extremal measures, with weights depending on $x$ (and possibly even on the way one scales). I will illustrate the above by some exact results for 1D lattice systems with two types of particles (plus holes) evolving according to variants of the simple asymmetric exclusion process, in open or closed systems. Somewhat surprisingly, the spatially asymmetric local dynamics satisfy (in some cases) detailed balance with respect to a global Gibbs measure with long range pair interactions. Discrete Mathematics Seminar ***Please note special date Topic: Coloring triangle-free graphs on surfaces Presenter: Robin Thomas, Georgia Tech Date: Wednesday, December 3, 2008, Time: 2:15 p.m., Location: Fine Hall 224 Abstract: Let S be a fixed surface, and let k and q be fixed integers. Is there a polynomial-time algorithm that decides whether an input graph of girth at least q drawn in S is k-colorable? This question has been studied extensively during the last 15 years. We will briefly survey known results. Then we will describe a solution to one of the two cases left open (the prospects for the other one are not bright). For every surface S we give a polynomial-time algorithm that computes the chromatic number of an input triangle-free graph G drawn in S. The new contribution here is deciding whether G is 3-colorable, and has two main ingredients. The first is a coloring extension theorem that makes use of disjoint paths in order to construct a coloring. The notion of "winding number" of a 3-coloring plays an important role. The second ingredient is a theorem bounding the number and sizes of faces of size at least four in 4-critical triangle-free graphs on a fixed surface, a generalization of a theorem of Thomassen. By developing more structure theory and using the notion of bounded expansion of Nesetril and Ossona de Mendez we were able to implement the algorithm to run in linear time. This is joint work with Zdenek Dvorak and Daniel Kral. Department Colloquium Topic: A new proof of Gromov's theorem on groups of polynomial growth Presenter: Bruce Kleiner, Yale University Date: Wednesday, December 3, 2008, Time: 4:30 p.m., Location: Fine Hall 314 Graduate Student Seminar Topic: Fibered Knots Presenter: Margaret Doig, Princeton University Date: Thursday, December 4, 2008, Time: 12:30 p.m., Location: Fine Hall 314 Abstract: A fibered knot is a knot whose complement can be filled "nicely" by copies of an oriented surface bounded by the disk, i.e., is filled by $S^1$ copies of $D^2$ (in fact, this fibration is globally trivial: $S^3-K \conj S^1 \times D^2$). By the time the pizza is all eaten, we should even be able to understand Milnor's construction of a fibration of the $(p,q)$ torus knot by surfaces of genus $(p-1)(q-1)/2$. You may care about fibered knots if you have ever been or will ever be interested in any of the following: \begin{itemize} \item hyperbolic structures \item algebraic knots and links \item unbranched cyclic covers \item open book decompositions \end{itemize} Number Theory Seminar Topic: Mock modular forms Presenter: Sandors Zwegers, University College Dublin Date: Thursday, December 4, 2008, Time: 4:30 p.m., Location: IAS SH-101 Abstract: The main motivation for the theory of mock modular forms comes from the desire to provide a framework in which we can understand the mysterious and intriguing mock theta functions, as well as related functions, like Appell functions and theta functions associated to indefinite quadratic forms. In this talk, we will describe the nature of the modularity of the original mock theta functions, formulate a general definition of mock modular forms, and describe further examples. We will also consider a generalization to higher depth mock modular forms Topology Seminar ***Please note special time and place Topic: On Khovanov homology and sutured Floer homology Presenter: Elisenda Grigsby, Columbia University Date: Thursday, December 4, 2008, Time: 3:30 p.m., Location: Fine Hall 214 Abstract: The relationship between Khovanov- and Heegaard Floer-type homology invariants is intriguing and still poorly-understood. In this talk, I will describe a connection between Khovanov's categorification of the reduced n-colored Jones polynomial and sutured Floer homology, a relative version of Heegaard Floer homology developed by Andras Juhasz. As a corollary, we will prove that Khovanov's categorification detects the unknot when n > 1. This is joint work with Stephan Wehrli. Joint Columbia-Courant-Princeton Algebraic Geometry Seminar Topic: Morrison, Mori and Mumford: mirror symmetry, birational geometry, and moduli spaces Presenter: Seán Keel, University of Texas at Austin Date: Friday, December 5, 2008, Time: 2:30 p.m., Location: Columbia University, Mathematics Hall 312 Abstract: I'll explain how elementary ideas from mirror symmetry and birational geometry determine (conjecturally) a canonical toroidal compactification of the moduli space of polarized K3 surfaces. Joint work with Paul Hacking and Valery Alexeev. Differential Geometry and Geometric Analysis Seminar Topic: Harmonic maps between singular spaces Presenter: Georgios Daskalopoulos, Brown University Date: Friday, December 5, 2008, Time: 3:00 p.m., Location: Fine Hall 314 Abstract: We will discuss regularity questions of harmonic maps from a simplicial complex to metric spaces of non-positive curvature. We will also discuss the relation with rigidity questions of group actions on these spaces. Joint Columbia-Courant-Princeton Algebraic Geometry Seminar Topic: Convex bodies associated to linear series Presenter: Rob Lazarsfeld, University of Michigan Date: Friday, December 5, 2008, Time: 4:00 p.m., Location: Columbia University, Mathematics Hall 312 Abstract: In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, as well as opening the door to a number of extensions. I will explain the construction, and give a number of examples, applications and open questions. If time permits, I will also mention briefly how Yuan has carried over the construction to the arithmetic setting. Geometry, Representation Theory, and Moduli Seminar Topic: On the holomorphic Chern-Simons functional Presenter: Kai Behrend, University of British Columbia Date: Monday, December 8, 2008, Time: 4:00 p.m., Location: Fine Hall 314 Abstract: We explain how the transfer theorem for L-infinity algebras together with some elementary Banach algebra techniques lead to a holomorphic function germ associated to every point in a moduli space of Donaldson-Thomas type. This gives rise to the definition of a Milnor Fibre for every Schur object in the derived category of a Calabi-Yau threefold. This may lead to a categorification of Donaldson-Thomas theory. (This is joint work in progress with Getzler.) PACM Colloquium Topic: Computational Astrophysics and the Dynamics of Accretion Disks Presenter: James M. Stone, PACM & Astrophysical Sciences Date: Monday, December 8, 2008, Time: 4:00 p.m., Location: Fine Hall 214 Abstract: he ever increasing performance of computer hardware and improvements to the accuracy of numerical algorithms are revolutionizing scientific research in many disciplines, but perhaps none more so than astrophysics. I will begin by describing why computation is crucial for the solution of a variety of problems at the forefront of research in astronomy and astrophysics, with particular emphasis on understanding accretion flows onto black holes. I will outline the challenge of developing, testing, and implementing numerical algorithms for the investigation of these problems. Finally, I will present results that demonstrate how computation can help us understand the basic physics of magnetized accretion disks. Algebraic Geometry Seminar Topic: Towards a classification of modular compactifications of the moduli space of curves Presenter: David Smyth, Harvard University Date: Tuesday, December 9, 2008, Time: 4:30 p.m., Location: Fine Hall 322 Abstract: The class of stable curves is deformation-open and satisfies the unique limit property, hence gives rise to the modular Deligne-Mumford compactification of M_{g,n}. But the class of stable curves is not unique in this respect; one obtains alternate compactifications by considering, for example, a moduli problem in which elliptic tails are replaced by cusps or in which marked points are allowed to collide. In this talk, we will survey progress toward a systematic classification of these alternate compactifications. Department Colloquium Topic: The geometry underlying Donaldson-Thomas theory Presenter: Kai Behrend, University of British Columbia Date: Wednesday, December 10, 2008, Time: 4:30 p.m., Location: Fine Hall 314 Abstract: Donaldson-Thomas invariants are algebraic analogues of Casson invariants. They are virtual counts of stable coherent sheaves on Calabi-Yau threefolds. Ideally, the moduli spaces giving rise to these invariants should be critical sets of "holomorphic Chern-Simons functions". Currently, such holomorphic Chern-Simons functions exist at best locally (see my seminar talk on Monday), and it is unlikely that they exist globally. I will describe geometric structures on the moduli spaces (some conjectural) that exist globally and reflect the fact that the moduli spaces look as if they were the zero loci of holomorphic maps. These are: symmetric obstruction theories, which prove that Donaldson-Thomas invariants are weighted Euler characteristics of moduli spaces, and derived scheme structures, exhibiting the moduli spaces as the classical schemes underlying schemes of Gerstenhaber algebras. Discrete Mathematics Seminar Topic: Packing cycles with modularity Presenter: Paul Wollan, University of Hamburg Date: Thursday, December 11, 2008, Time: 2:15 p.m., Location: Fine Hall 224 Abstract: Erdős and Posa proved that a there exists a function $f$ such that any graph either has $k$ disjoint cycles or there exists a set of $f(k)$ vertices that intersects every cycle. The analogous statement is not true for odd cycles - there exist numerous examples of graphs that do not have two disjoint odd cycles, and yet no bounded number of vertices intersects every odd cycle. However, Reed has given a partial characterization of when there does not exist a bounded size set of vertices intersecting every odd cycle. We will discuss recent work on when a graph has many disjoint cycles of non-zero length modulo $m$ for arbitrary $m$. When $m$ is odd, we see that again there exists a function $f$ such that any graph either has $k$ disjoint cycles of non-zero length modulo $m$ or there exists a set of at most $f(k)$ vertices intersecting every such cycle of non-zero length. When $m$ is even, no such function $f(k)$ exists. However, the partial characterization of Reed can be extended to describe when a graph has neither many disjoint cycles of non-zero length modulo $m$ nor a small set of vertices intersecting every such cycle. Number Theory Seminar Topic: Langlands functoriality and the inverse problem in Galois theory Presenter: Gordon Savin, University of Utah Date: Thursday, December 11, 2008, Time: 4:30 p.m., Location: IAS SH-101 Abstract: In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type B_n, C_n and G_2 as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with l-adic representations attached to self-dual automrophic representations of GL(n). Topology Seminar Topic: Primitive-stable representations of the free group Presenter: Yair Minsky, Yale University Date: Thursday, December 11, 2008, Time: 4:30 p.m., Location: Fine Hall 314 Abstract: Automorphisms of the free group F_n act on its representations into a given group G. When G is a simple compact Lie group and n>2, Gelander showed that this action is ergodic. We consider the case G=PSL(2,C), where the variety of (conjugacy classes of) representations has a natural invariant decomposition, up to sets of measure 0, into discrete and dense representations. This turns out NOT to be the relevant decomposition for the dynamics of the outer automorphism group. Instead we describe a set called the "primitive-stable" representations containing discrete as well as dense representations, onwhich the action is properly discontinuous.