

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 trianglefree 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 polynomialtime algorithm that decides whether an input graph of girth at least q drawn in S is kcolorable? 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 polynomialtime algorithm that computes the chromatic number of an input trianglefree graph G drawn in S. The new contribution here is deciding whether G is 3colorable, 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 3coloring plays an important role. The second ingredient is a theorem bounding the number and sizes of faces of size at least four in 4critical trianglefree 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: $S^3K \conj S^1 \times D^2$). 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 $(p1)(q1)/2$. 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 SH101 
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 Floertype homology invariants is intriguing and still poorlyunderstood. In this talk, I will describe a connection between Khovanov's categorification of the reduced ncolored 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 ColumbiaCourantPrinceton 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 nonpositive curvature. We will also discuss the relation with rigidity questions of group actions on these spaces. 


Joint ColumbiaCourantPrinceton 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 logconcavity 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 ChernSimons 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 Linfinity algebras together with some elementary Banach algebra techniques lead to a holomorphic function germ associated to every point in a moduli space of DonaldsonThomas type. This gives rise to the definition of a Milnor Fibre for every Schur object in the derived category of a CalabiYau threefold. This may lead to a categorification of DonaldsonThomas 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 deformationopen and satisfies the unique limit property, hence gives rise to the modular DeligneMumford 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 DonaldsonThomas theory 
Presenter: 
Kai Behrend, University of British Columbia 
Date: 
Wednesday, December 10, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
DonaldsonThomas invariants are algebraic analogues of Casson invariants. They are virtual counts of stable coherent sheaves on CalabiYau threefolds. Ideally, the moduli spaces giving rise to these invariants should be critical sets of "holomorphic ChernSimons functions". Currently, such holomorphic ChernSimons 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 DonaldsonThomas 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 nonzero 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 nonzero length modulo $m$ or there exists a set of at most $f(k)$ vertices intersecting every such cycle of nonzero 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 nonzero 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 SH101 
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 ladic representations attached to selfdual automrophic representations of GL(n). 


Topology Seminar 
Topic: 
Primitivestable 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 "primitivestable" representations containing discrete as well as dense representations, onwhich the action is properly discontinuous. 

