In this talk I will describe a few questions about intermediate subfactors motivated by the theory of finite groups, and report on some recent progress related to Hopf algebras, fusion categories, quantum groups and conformal field theory. I will also discuss similar problems in Vertex Algebras and recent results which are joint work with V. Kac, P. Moseneder and P. Papi.

In this talk, I will present a deterministic population model in the spread of an epidemic disease with intervention of isolation and quarantine methods. I will provide the epidemic background, the PDE model, the existence and uniqueness theorem, the computation strategy, some simulation results and explanations. We are recently trying to use the model to fit the data of SARS spread in Taiwan, 2003. I will briefly present some of the recent fitting results as well.

Let \Gamma be a discrete group acting properly and freely on a manifold M\tilde with compact quotient M = M\tilde / \Gamma. Given D any elliptic differential operator on the compact manifold M. Let D\tilde be the lifting of D to M\tilde. We may define the higher index of D\tilde, denoted by H-index(D\tilde), as an element of the K-group for the reduced C*-algebra of \Gamma K_0(C^*_r(\Gamma)). Let tr_* be the homomorphism on K-theory induced from the canonical trace on C^*_r(\Gamma). Then Atiyah's L^2-index theorem implies the following identity: tr_*(H-index(D\tilde)) = index(D). It follows that the Fredholm index of D is the 0-dimensional information of the higher index of D\tilde. I am going to sketch a proof of the Atiyah's L^2-index theorem via Geometric K-homology, as described in the joint paper by Willett and Yu in 2012.

To most Americans, voting is an infrequent, simple civic activity: you learn a little about the candidates, choose the one you like the most (or dislike the least!), mark a paper or electronic ballot, and move on with your life. Few of us reflect on how the electoral system might shape our public institutions, and still fewer on how the electoral system might be different, and how such changes could affect the power and workings of public institutions. We will discuss a few such ideas during this talk, including why the U.S. has a two-party system while other nations have several parties, and Arrow's Theorem stating (informally) that there is no perfect electoral system.

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.

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.

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.

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.

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.

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.

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.

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)

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.

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 $(x_0,t_0)$.

We propose a method for the 3D-rigid motion invariant texture discrimination for discrete 3D-textures that are spatially homogeneous. We model these textures as stationary Gaussian random fields. We formally develop the concept of 3D-texture rotations in the 3D-digital domain. We use this novel concept to define a `distance' between 3D-textures that remains invariant under all 3D-rigid motions of the texture. This concept of `distance' can be used for a monoscale or a multiscale setting to test the 3D-rigid motion invariant statistical similarity of stochastic 3D-textures. To extract this novel texture `distance' we use the Isotropic Mutliresolution Analysis. We also show how to construct wavelets associated with this structure by means of extension principles and we discuss some very recent results by Atreas, Melas and Stavropoulos on the geometric structure underlying the various extension principles. The 3D-texture `distance' is used to define a set of 3D-rigid motion invariant texture features. We experimentally establish that when they are combined with mean attenuation intensity differences the new augmented features are capable of discriminating normal from abnormal liver tissue in arterial phase contrast enhanced X-ray CT-scans with high sensitivity and specificity. To extract these features CT-scans are processed in their native dimensionality. We experimentally observe that the 3D-rotational invariance of the proposed features improves the clustering of the feature vectors extracted from normal liver tissue samples. This work is joint with R. Azencott, S. Jain, S. Upadhyay, I.A. Kakadiaris and G. Gladish, MD.

This is a joint work with Denis Osin. We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We will briefly discuss some applications to subgroup distortion, compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F\o lner functions, and elementary classes of amenable groups.

In 1973, Richard Thompson considered the question of whether his newly defined group $F$ was amenable. The motivation for this problem stemed from his observation --- later rediscovered by Brin and Squire --- that $F$ did not contain a free group on two generators, thus making it a candidate for a counterexample to the von Neumann-Day problem. While the von Neumann-Day problem was solved by Ol'shanskii in the class of finitely generated groups and Ol'shanskii and Sapir in the class of finitely presented groups, the question of $F$'s amenability was sufficiently basic so as to become of interest in its own right. In this talk, I will analyze this problem from a Ramsey-theoretic perspective. In particular, the problem is related to generalizations of Ellis's Lemma and Hindman's Theorem to the setting of nonassociative binary systems. The amenability of $F$ is itself equivalent to the existence of certain finite Ramsey numbers. I will also discuss the growth rate of the F\olner function for $F$ (if it exists). Tea at 3:30 pm in SC 1425.

The Einstein equations have been a source of many interesting problems in Physics, Analysis and Geometry. Despite the great deal of work which has been devoted to them, with many success stories, several important questions remain open. One of the them is a satisfactory theory of isolated systems, such as stars, both from a perspective of the time development of the space-time, as well as from the point of view of the geometry induced on a space-like three surface. This talk will focus on the former situation. More specifically, we shall discuss relativistic fluids with and without viscosity, and prove a well-posedness result for the Cauchy problem. The viscous case, in particular, is of significant interest in light of recent developments in astrophysics.