# Seminars & Events for 2008-2009

##### Multiscale Methods for Hydrodynamics of Polymer Chains in Solution

The hydrodynamics of complex fluids, such as polymer solutions and colloidal suspensions, has attracted great interest due to recent advances in fabrication of micro- and nano-fluidic devices. I will first review recent advances in mesoscopic numerical methods for simulating the interaction between complex fluid flow and suspended macro molecules and structures.

##### Equivalences from geometric $sl_2$ actions

We explain how $sl_2$ actions on derived categories of coherent sheaves can be used to construct new derived equivalences. The example I will describe in detail is an $sl_2$ action via correspondences on the cotangent bundles of Grassmannians which generalizes the basic Mukai flop.

##### Bosons in rapid rotation

One of the most remarkable manifestations of superfluidity in Bose-Einstein condensates is the way the condensate responds to rotation. In a superfluid the rotation of the container confining the fluid leads to the formation of vortices with quantized circulation. This phenomenon can be studied through the solutions of a nonlinear Schrodinger equation, the Gross-Pitaevskii equation.

##### An overview of the l-adic and p-adic monodromy theorems

I will introduce (focusing on the case of elliptic curves) two essential theorems of arithmetic geometry that bind the geometry of an algebraic variety over a local field (or number field) to its arithmetic, Galois-theoretic properties. The l-adic monodromy theorem is in fact entirely elementary (one needn't even know what 'monodromy' means!) but will motivate the subtler p-adic theorem.

##### Comparison isomorphisms for $p$-adic formal schemes and applications

##### Quasi-isometries, phase transitions, and other problems in additive number theory

This is a survey of recent work in combinatorial and additive number theory suggested by a problem of Richard Schwartz in metric geometry and geometric group theory. The central object is a group with an infinite set of generators, and the induced metric. Some results and many open problems will be discussed.

##### An Exotic Sphere with Positive Sectional Curvature

I'll discuss joint work with Peter Petersen that shows that the Gromoll-Meyer exotic 7-sphere admits a metric of positive sectional curvature. I'll discuss the history of the problem and give a coarse outline of the proof.

##### Bordered Heegaard Floer homology

I will describe a construction of invariants for three-manifolds with (parameterized) boundary. The invariant associates a differential graded algebra to an oriented surface, and a (suitably generalized) module to a three-manifold whose boundary is that surface. This invaraint also enjoys a "pairing theorem" relating it with HF-hat of closed three-manifolds.

##### Emissions Market Models

The main goal of the talk is to introduce a new cap-and-trade scheme design for the control and the reduction of atmospheric pollution. The tools developed for the purpose of the study are intended to help policy makers and regulators understand the pros and cons of the emissions markets at a quantitative level.

##### Fifth order KdV equations

We study the fifth-order KdV equations, which arise in the KdV hierarchy. In this talk, we discuss the initial value problem in Sobolev spaces with low regularity. In the linear part the fifth-order equation has stronger dispersion effect and so better smoothing than KdV equation. But it comes with stronger nonlinear parts compared to dispersion.

##### Finiteness theorems for algebraic groups over function fields

If $X$ is a smooth variety over a global field $k$, $G$ is an algebraic group over $k$ equipped with an action on $X$, and $x$ is a point in $X(k)$ then it is natural to ask how the property of $x'$ in $X(k)$ being in the $G(k)$-orbit of $x$ compares with being in the $G(k_v)$-orbit of $x$ for all places $v$ of $k$.

##### Local and Global Structure of Stationary States of Macroscopic Systems

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.

##### Coloring triangle-free graphs on surfaces

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.

##### A new proof of Gromov's theorem on groups of polynomial growth

##### On Khovanov homology and sutured Floer homology

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.

##### Mock modular forms

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.

##### Fibered Knots

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: $S^3-K\wedge S^1\times D^2$).

##### Morrison, Mori and Mumford: mirror symmetry, birational geometry, and moduli spaces

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar

##### Harmonic maps between singular spaces

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.

##### Convex bodies associated to linear series

Joint Columbia-Courant-Princeton Algebraic Geometry Seminar