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)

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).

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.

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.

For a compact Riemannian manifold, we can construct an action of the group of its volume-form-preserving diffeomorphisms on an infinite-dimensional Hilbert manifold with non-positive sectional curvatures. Exploiting this geometric property, we can study the injectivity of the Baum-Connes assembly map for discrete subgroups of the volume-form-preserving diffeomorphism group.

In 1987, the physicist Ed Witten gave a (physicist's) explanation showing that the Jones Polynomial of knots was in fact calculable from Chern-Simons Quantum Field Theory.  (For their separate contributions, both Ed and Vaughan were awarded the Fields Medal in 1990.)  Since these theories are topological in nature, in fact Jones/Witten had invented a whole new branch of Topology, called Quantum Topology, 25 years old today. Quantum Topology studies more generally the notion of a TQFT (topological quantum field theory) and its perturbative analogues.  In the early 90's the author and his collaborators showed that the Jones Polynomial extended to invariants of knots in 3-manifolds, and that all of the axioms of a TQFT were satisfied.  (Perturbative aspects of these theories involve what has become to be known as the theory of Finite Type Invariants, first explored for knots by Vassiliev.  The Kontsevich Integral is the Universal such knot invariant.) A TQFT can already be thought of as a Quantum Computer, since the Hilbert spaces involved are finite dimensional, at least if one defines a computer to be a finite set and operations thereupon, and defines a quantum computer to be the linear analogue (over the complex numbers).  If one wants to be a bit more restrictive, organizing a computer in terms of bits (a bit is a 2-pont set), then a quantum computer is just organized in terms of q-bits (a 2 dimensional complex Hilbert space or rather its projective analogue, the 2-sphere).  This is what one gets as the Hilbert space associated to a 2-sphere (not the one above) with 4 marked points, In theory then, 3-manifold/tangle pairs bounding such objects give (calculable) quantum computer operations. Problems for the 21st century: 1. (Mathematics) Use these observations to do effective and interesting (useful) computations. 2. (Computer Science)  Implement these calculations on an (ordinary) computer. 3. (Condensed Matter Physics) Design circuits on an atomic scale which will do the same.