We will review the inventory of open problems related to hyperbolic and partially hyperbolic dynamics (including the trace map dynamics), conservative dynamics, complex dynamics, piecewise translations, and convolutions of singular measures that are in a focus of our seminar interests (or are natural candidates for this status). Many of the problems are suitable for beginning graduate students. 
 
 

The CMV matrix is a unitary operator on $\ell^2(\mathbb N)$ that is a central tool in the study of orthogonal polynomials on the unit circle. One may view it as a unitary analogue of the Jacobi matrix. We may extend the CMV matrix to be a unitary operator on $\ell^2(\mathbb Z)$. It is more natural to consider the extended CMV matrix in certain contexts: for example, if we wish to generate CMV matrices dynamically. The extended CMV matrix also plays an important role in the study of quantum random walks.

In this talk, we will discuss a Gordon lemma for the CMV matrix (The Gordon lemma is an important tool in the study of Jacobi matrices, used to rule out the possibility pure point spectrum). We will also discuss some results pertaining to the H\"older-continuity of the spectrum of the extended CMV matrix.

In this expository talk, we discuss some inequalities holding
for certain solutions of Ricci flow. Ricci flow is a form of the heat
equation for Riemannian metrics. So techniques from the study of the
heat equation apply. Examples of basic inequalities include the
Li-Yau inequality for positive solutions of the heat equation, which
motivated the Harnack inequalities of Hamilton and Perelman for Ricci
flow. Fundamental inequalities of Perelman are for the entropy and for
the reduced volume. Moreover, there are many other inequalities which
hold for certain classes of solutions, such as those proved by
Hamilton, Perelman, and others.

The Ricci flow is an important tool in geometry, and a main
problem is to understand the stability of fixed points of the flow and the
convergence of solutions to those fixed points. There are many approaches
to this, but one involves maximal regularity theory and a theorem of
Simonett. I will describe this technique and its application to certain
extended Ricci flow systems. These systems arise from manifolds with
extra structure, such as fibration or warped product structures, or Lie
group structures.