Unipotent quantum groups are non-commutative versions of unipotent group schemes in algebraic geometry. Finite unipotent quantum groups only appear in positive characteristic. In this talk, we will provide a complete classification of such objects up to prime-cube dimension, which can be thought as a generalization of the well-known fact about the structure of p-groups of small orders. In contrast to the finiteness of isomorphism classes for each fixed order in group theory, we obtain nine infinite families among these prime-cube ones, which are all naturally parameterized by finite group quotients of the affine line. Further topics regarding representations, cohomology and invariant theory of small unipotent quantum groups will also be discussed during the talk.

During the past decade, much progress has made in refining the principle that high-dimensional statistical problems are tractable when they exhibit some form of low-dimensional structure. However, in practice, it is often unclear whether or not structural assumptions are justified by data, and the problem of validating such assumptions is unresolved in many contexts. In this talk, I will focus on the context of compressed sensing (CS) --- a signal processing framework that is built on the structural assumption of sparsity. Although the theory of CS offers strong guarantees for recovering sparse signals, many aspects of the recovery process depend on prior knowledge of the signal's sparsity level --- a parameter which is rarely known in practice. Towards a resolution of this issue, I will introduce a generalized family of sparsity parameters that can be estimated in a way that is free of structural assumptions. In connection with signal recovery, I will show that the error rate of the Basis Pursuit Denoising algorithm can be bounded tightly in terms of these parameters. Lastly, I will present consistency results for the proposed sparsity estimation procedure, including a CLT, which allows for the hypothesis of sparsity to be tested in a precise sense.

In this talk, it is shown that a rank decomposition of symmetric tensors must be its symmetric rank decomposition when the tensor's rank is less than its order. Furthermore, when the rank of symmetric tensors equals the order, the symmetric rank must be the rank. As a corollary, for symmetric tensors, rank and symmetric rank coincide when rank is at most order. This partially gives a positive answer to the Comon's conjecture. Finally, a sufficient condition under which a symmetric decomposition of symmetric tensors is a symmetric rank decomposition is presented. Some examples are presented to show the efficiency of the condition.

We will continue our talk about the Cyclic Sieving Phenomenon involving the relationship between q-rational catalan number and the set of non crossing partitions. By utilizing the injective mapping between rational deck paths and non-crossing partitions, we have constructed small cases of how to observe that the set $(NC(a,b),Z_{b-1},Cat_q(a,b))$ exhibits the Cyclic Sieving Phenomenon. We will also demonstrate a few of our MATLAB programs that helped aid us in the progress of trying to prove the Cyclic Sieving Phenomenon.

One of the many ways to represent juggling patterns is through so-called \emph{juggling cards}. These are templates which describe the spatial ordering of a set of balls at each point in a juggling pattern. In this talk we describe a number of new combinatorial and probabilistic results in the study of these objects, and state some related, unsolved problems. Attendees are welcome to bring their own chainsaws or lit torches for the interactive portion of the talk.

Last February, Mike Cranston spoke in this Seminar about a polymer model based on three-dimensional Brownian motion conditioned to hit (and keep returning to) the origin. I will discuss the construction and certain properties of this conditioned Brownian motion from two points of view (i) Dirichlet forms, and (ii) excursion theory. The latter gives a nice interpretation of the Johnson-Helms example from martingale theory. It turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional Brownian motion reflected at the origin.

Based on joint work with Liping Li of Fudan University.

The representation theory of subfactors generalizes the representation theory of quantum groups, and thus we think of subfactors as objects which encode quantum symmetries. In one sense, subfactors of small index are the simplest examples of subfactors, and we have a complete classification of their standard invariants to index 5. I will discuss recent joint work with Morrison which classifies certain examples at index $3+\sqrt{5}$. One important ingredient is a new variation of Bigelow's jellyfish algorithm which is universal for finite depth subfactors.

The inverse problem in spectral theory was proposed by Bochner and Weyl in the early 20th century. One formulation of the question is: how much of the dynamics of a classical dynamical system can be detected from the spectrum of its quantization? I will describe this question and review recent results for the case integrable dynamical system, where in certain fundamental cases going back to the work of Atiyah and Guillemin-Sternberg a full solution can be given. The talk will emphasize the interplay between symplectic geometry and semiclassical spectral theory and is intended for a general audience.

Numerically solving elliptic partial differential equations for a large number of degrees of freedom requires the parallel use of many computer processors. This in turn requires algorithms to partition domains into subdomains in order to distribute the work.

We present five novel algorithms for partitioning domains that utilize information from the underlying PDE. When a PDE has strong convection or anisotropic diffusion, a partition that favors this direction is desirable. Our schemes fall into two classes; one class creates rectangular shaped subdomains aligned in this direction and one class creates subdomains that increase in size as you move in this direction.

These schemes are mathematically described and analyzed in detail. Then they are tested on a variety of experiments which include solving the convection-diffusion equation for 1/4 billion unknowns on 512 processors using over 1 teraflop of computing power.

Theory and experiments demonstrate that these schemes improve the domain
decomposition convergence rate when the underlying PDE has directional
dependance. In our hundreds of experiments, the number of DD iterations required for convergence reduces by a factor between 0.25 and 0.75. In some cases, these methods improve the final finite element solution's accuracy also.