Consider the almost Mathieu operator in the case of the critical coupling.

For rational frequencies p/q we obtain power-law bounds of the form 1/q^C

for the widths of the gaps close to the center of the spectrum. It follows that

these gaps remain open for the spectrum in the case of a certain class of Diophantine irrational frequencies.

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.

In this talk I will introduce the moduli space of curves and a class of vector bundles on it.   I’ll discuss how these bundles, which have connections to algebraic geometry, representation theory, and mathematical physics, tell us about the moduli space of curves, and vice versa, focusing on just a few recent results and open problems.

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.

I will highlight some of the methods I've previously employed in undergraduate mathematics education. These methods were drawn from the research of others and some of my own inquires. I will give insight into how the methods are brought together so that they form a cohesive whole, while fitting the needs of the both the students and the department.

Central to the discussion is a particular enlightening experience I had with one of my students.

The experience has inspired my current methods. I will discuss this my vision for further developing curriculum and more so fostering a lively dynamic atmosphere around mathematics courses, both within and without the classroom walls.

Time permitting, I'll discuss some research that is underway with regards to these approaches.