For each random n-vector there is an entropy vector of length 2^n-1. A fundamental question in information theory is to characterize the region formed by these entropic vectors. The region is bounded by Shannon's inequalities, but not tightly bounded for n>3. Chan and Yeung discovered that random vectors constructed from groups fill out the entropic region, so that information theoretic properties may be interpreted to give properties of groups and combinatorial results for groups may be used to better understand the entropic region. I will elaborate on these connections and present some simple and interesting questions about groups that arise.

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## Upcoming Seminars

### Mon Apr 24, 2017

I will describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth- order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. I first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. I will then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. I will demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

This is joint work with W. Feng (UTK), A. Salgado (UTK), and C. Wang (UMassD).

We develop basic properties of generators of J_{\lambda^+}(a). Then we present the theorem of Shelah that under certain circumstances, max(pcf(a)) has the largest possible cardinality.

I will present the Strichartz estimate for non-endpoint and endpoint based on the dispersive estimate.

### Tue Apr 25, 2017

We discuss an inverse theorem on the structure of pairs of discrete probability measures which has small amount of growth under convolution, and apply this result to self-similar sets with overlaps to show that if the dimension is less than the generic bound, then there are superexponentially close cylinders at all small enough scales. The results are by M.Hochman.

A special case of the GRS Conjecture predicts a surprising link between values of derivatives of p-adic and global L-functions. Recently, Dasgupta-Kakde-Ventullo have used Hida families of modular forms to make progress towards the proof of a rational form of this special case. In this lecture I will report on an independent approach and progress towards the integral GRS conjecture, building upon my joint work with Greither in equivariant Iwasawa theory.

The problem of understanding CR geometries embedded as submanifolds in

higher dimensional CR manifolds arises in higher dimensional complex

analysis, including the study of singularities of analytic

varieties. It has also been studied intensively in connection with

rigidity questions. Despite considerable earlier work the local theory

has not been fully understood.

We develop from scratch a CR invariant local theory based on CR

tractor calculus (i.e. the associated bundle). This produces the tools

for constructing local invariants and invariant operators in a way

parallel to the classical Gauss-Codazzi-Ricci calculus for Riemannian

submanifolds. It also enables a practical and conceptual approach to

a Bonnet Theorem and potentially the rigidity questions.

This is joint work with Rod Gover.

### Thu Apr 27, 2017

Abstract: When numbers are added in the usual way, "carries" occur along

the way. Making math sense of the carries leads to all sorts of

corners, in particular to the mathematics of shuffling cards. I will

show that it takes seven ordinary riffle shuffles to mix up 52 cards and

explain connections to fractals and other lovely mathematical objects.

This is a talk for a general audience, no specialist knowledge needed.

### Fri Apr 28, 2017

Abstract: Szegö's theorem and the Kac-Murdoch-Szegö theorems are

classical asymptotic results about the distribution of the eigenvalues

of structured matrices. I will explain how these are useful in a

variety of applications (in particular analysis on Heisenberg groups)

_and_ show how they are equivalent to lovely theorems in random matrix

theory.

Let us take a couple of 2x2 matrices A and B, and consider a long product of matrices, where each multiplier is either A or B, chosen randomly. What should we expect as a typical norm of such a product? This simple question leads to a rich theory of random matrix products. We will discuss some of the classical theorems (e.g. Furstenberg Theorem), as well as the very recent results.