Mathematics Colloquium Spring 2012

April 3, 2012

The evolution of vortices in planar flows

Professor Matania Ben-Artzi, Einstein Institute of Mathematics, Hebrew University of Jerusalem

The celebrated work of Leray (eighty years ago) settled the basic questions concerning the well-posedness of the Navier-Stokes equations in the plane (two dimensions). However, the basic premise of this work was the assumption that the initial velocity field is (at least) locally square integrable (namely, locally finite kinetic energy). Thus, some primary fluid dynamical objects (notably point vortices) have been excluded (as they generate non square integrable velocities). The topic of well posedness of such flows has been taken up only some twenty five years ago, and the final uniqueness result was obtained by Gallagher and Gallay five years ago. In this talk we review the story of such flows, including basic open problems concerning the behavior of steady state solutions in bounded domains.

April 9, 2012 ***This colloquium is on Monday. Refreshments - 3:30 pm McHenry 1270 Talk - 4:00 pm McHenry 1240

Mathematical Methods in Origami Design

Robert J. Lang 

Robert J. Lang is recognized as one of the foremost origami artists in the world as well as a pioneer in computational origami and the development of formal design algorithms for folding. With a Ph.D. in Applied Physics from Caltech, he has, during the course of work at NASA/Jet Propulsion Laboratory, Spectra Diode Laboratories, and JDS Uniphase, authored or co-authored over 80 papers and 45 patents in lasers and optoelectronics as well as authoring, co-authoring, or editing 12 books and a CD-ROM on origami. He is a full-time artist and consultant on origami and its applications to engineering problems but keeps his toes in the world of lasers, most recently as the Editor-in-Chief of the IEEE Journal of Quantum Electronics from 2007–2010. He received Caltech’s highest honor, the Distiguished Alumni Award, in 2009.

ABSTRACT - The last decade of this past century has been witness to a revolution in the development and application of mathematical techniques to origami, the centuries-old Japanese art of paper-folding. Much of that revolution was enabled by the application of a variety of mathematical techniques to the problem of design: “how do you fold a —?” In this talk, I will discuss some of the underlying mathematical laws of origami, how those laws can be applied to real-world origami problems, and several examples of how the algorithms and theorems of origami design have shed light on long-standing mathematical questions and have solved practical engineering problems. I will discuss examples of how origami has enabled safer airbags, space telescopes, medical devices, and more.

April 17, 2012

Well posedness for swarming models

Professor Jesus Rosado, UCLA 

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and selfpropulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models

April 24, 2012

Categorifying Quantum Groups

Professor Monica Vazirani, UC Davis 

What is categorification? If you de-categorify Vector-Spaces, you replace isomorphism classes of objects with natural numbers (their dimensions), replace direct sum with addition of those numbers, replace tensor product with multiplication. To categorify is to undo this process. For instance, one might start with the ring of symmetric functions and realize it has replaced the representation theory of the symmetric group.

In this talk, I will discuss how Khovanov-Lauda-Rouquier (KLR) algebras categorify quantum groups. I will discuss their simple modules, and in particular that they carry the structure of a crystal graph. This is joint work with Aaron Lauda.

May 1, 2012

Representations of SL(2,Z) of a modular category

Siu-Hung Ng, Iowa State University 

Associated to a modular category is a canonical projective representation of SL(2,Z). This projective representation can be lifted to an ordinary representation but not uniquely. These liftings manifest the congruence property and Galois symmetry which has been conjectured for the modular data of certain rational conformal field theories. In this talk, we will discuss this result and some applications on central charges and global dimensions of modular categories.

May 8, 2012

Active Scalar Equations and a Geodynamo Model

Professor Susan Friedlander, University of Southern California 

We discuss an advection-diffusion equation that has been proposed by Keith Moffatt as a model for the Geodynamo. Even though the drift velocity can be strongly singular, we prove that the critically diffusive PDE is globally well-posed. We examine the nonlinear instability of a particular steady state and use continued fractions to construct a lower bound on the growth rate of a solution. This lower bound grows as the inverse of the diffusivity coefficient. In the Earth's fluid core this coefficient is expected to be very small. Thus the model does indeed produce very strong Geodynamo action. This work is joint with Vlad Vicol.

May 22, 2012

Convex billiards and non-holonomic systems

Robert Bryant, Director of MSRI 

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For 'most' convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other curves, besides ellipses, for which this is true. In this talk, I'll discuss the above-mentioned phenomenon and show how it is related to the geometry of nonholonomic plane fields (which will be defined and described). This leads to some surprisingly beautiful geometry, which will require nothing beyond multivariable calculus from the audience.

May 29, 2012

Frankel conjecture and Sasaki geometry

Weiyong He, University of Oregon 

This is joint work with Song Sun at Imperial college. We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the K\"ahler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the K\"ahler cone. First, we deform the metric along the Sasaki-Ricci flow and obtain a limit Sasaki-Ricci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field along the negative gradient of the volume functional, we deform the Sasaki-Ricci soliton to a Sasaki-Einstein metric with positive transverse bisectional curvature, i.e. a round sphere. The second deformation is only possible when one treats simultaneously regular and irregular Sasaki manifolds, even if the manifold one starts with is regular(quasi-regular), i.e. K\"ahler manifolds(orbifolds).

June 5, 2012

Complex dynamics and adelic potential theory

Professor Matt Baker, University Of California Berkeley 

We will discuss the following theorem (joint work with Laura DeMarco): For any fixed complex numbers a and b, the set of complex numbers c for which a and b both have finite orbit under the map z --> z^2 + c is infinite if and only if a^2 = b^2. I will explain the motivation for this result and give an outline of the proof, which involves both complex analysis and number theory (with non-archimedean Berkovich spaces playing an essential role).