The local properties of non-negative weak solutions to the singular parabolic equation $u_t-\Delta \ln u = 0$ are largely unclear though some research has been done for the Cauchy problem of such an equation.  In this talk, we address the local positivity of this equation in the form of a Harnack-type inequality. Under the assumption $\ln u$ is sufficiently integrable, we show if $u$ does not vanish identically in a space neighborhood of $x_0$ and on some time level $t_0$ then $u$ is positive in a neighborhood of $(x_0,t_0)$.

Over the last two decades our understanding of small index subfactors has improved substantially. We have discovered a slew of examples, some related to finite groups or quantum groups, and other `sporadic' examples. At present we have a complete classification of (hyperfinite) subfactors with index at most 5, and a few results that push past 5. I'll explain the main techniques behind these classification results, and also spend a little time describing how we construct the sporadic examples. (Joint work with many people!) Tea at 3:30 pm in SC 1425.

I'll introduce Khovanov homology, a `categorical' extension of the Jones polynomial. Since the discovery of Khovanov homology, representation theorists have opened up a new world of categorical quantum groups. In this talk, I'll head in a different direction, explaining how the 4-dimensional nature of Khovanov homology makes it ideally suited for building a new 4-manifold invariant. I'll explain the construction, then discuss its present limitations and how we hope to get past them. (Joint work with Kevin Walker)

There are only two known examples of positively curved compact (orientable) Einstein 4-manifolds: The round metric on the 4-sphere, and the Fubini-Study submersion metric on the complex projective plane. It is an open question whether or not this is the complete list. In this talk, we will prove that if we in addition assume that the metric is compatible with a complex structure on the manifold, then it has to be the Fubini-Study metric.

Compressed sensing is a recent development in the field of sampling Based on the notion of sparsity, it provides a theory and techniques for the recovery of images and signals from only a relatively small number of measurements. The key ingredients that permit this so-called subsampling are (i) sparsity of the signal in a particular basis and (ii) mutual incoherence between such basis and the sampling system. Provided the corresponding coherence parameter is sufficiently small, one can recover a sparse signal using a number of measurements that is, up to a log factor, on the order of the sparsity. Unfortunately, many problems that one encounters in practice are not incoherent. For example, Fourier sampling, the type of sampling encountered in Magnetic Resonance Imaging (MRI), is typically not incoherent with wavelet or polynomials bases. To overcome this `coherence barrier' we introduce a new theory of compressed sensing, based on so-called asymptotic incoherence and asymptotic sparsity. When combined with a semi-random sampling strategy, this allows for significant subsampling in problems for which standard compressed sensing tools are limited by the lack of incoherence. Moreover, we demonstrate how the amount of subsampling possible with this new approach actually increases with resolution. In other words, this technique is particularly well suited to higher resolution problems. This is joint work with Anders Hansen and Bogdan Roman (University of Cambridge).

One day in 1984, my friend from my grad school days in Geneva, Switzerland, Vaughan Jones, announced to me that he had discovered a polynomial for knots.  I first wondered what all the hubub surrounding his discovery was about, but all that changed for me in 1987, when a physicist named Ed Witten explained that the Jones Polynomial was best regarded in the light of Quantum Field Theory (For their separate contributions, Vaughan and Ed both received the Field's Medal). You should come to this lecture hoping to get an introduction to higher math, physics, and computer science, and how they are related (In fact, even biologists and organic chemists have gotten interested in Vaughan's polynomial, since strings of DNA can get entangled, but I won't have time to talk about that).  But don't be scared.  I will try to keep things as elementary as possible.  And then the fun starts.  You can go home and teach your little brother or sister to calculate (a version of) the Jones polynomial.  You can tell your Mom and Dad that the hydrogen atom is not like the moon and the earth, but more like a cloudy day all around the earth, and that even Einstein made mistakes. And you can tell your friends that you hope to beat Bill Gates (and maybe even become rich) by starting now to work on the Quantum Computer.

A "pictures mod relations'' presentation of the representation theory of SL(n). The representation category of SL(n) is a pivotal tensor category. This means that one can draw planar diagrams representing morphisms, with composition corresponding to vertical stacking, and tensor products corresponding to horizontal juxtaposition. Any planar isotopy of such a diagram gives equations between the corresponding morphisms. For any such category, we'd like to be able to give a presentation via certain generators modulo local relations. For Rep(SL(n)), we've had a conjectural presentation for several years, but no good tools for showing that we have all the relations. With Sabin Cautis and Joel Kamnitzer, we now have not only a proof that this presentation is correct, but also a clear conceptual explanation of how the relations arise. This explanation uses skew Howe duality.

I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity. This is  joint work with David G. Ebin.