\indent We usually only justify time series estimators using asymptotic theory, but the sample size for time series, say those yearly macro series, is usually limited, not more than 100. Additionally, high dimensionality and serial dependence makes the asymptotics harder to be a good approximation for a finite sample. My works in high dimensional time series modeling tries to solve these problems, i.e. to quantify the interplay and strike a balance between degree of time dependence, high dimensionality and moderate sample size (relative to dimensionality). In this talk, I will talk about generalized dynamic factor models (briefly), large vector autoregressions for modeling expectation (in detail), and also dynamic volatility matrix estimation (briefly) if time permits.

\indent A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons and I will discuss several applications. I will then discuss some theoretical aspects of sums of squares representations of nonnegative polynomials, in particular, some underlying fundamental reasons that there exist nonnegative polynomials that are not sums of squares.

Tutte founded the enumeration theory of planar maps in a
series of papers in the 1960s. We are interested in rooted planar maps
which can be thought as connected planar graphs embedded in the sphere
with a directed edge distinguished as the root. A planar map is
non-separable if it has no loops and no cut-vertices. Non-separable
planar maps are also called 2-connected maps. Another class of maps of
our interest is bicubic maps, which after removing the root orientation
are connected regular bipartite graphs with vertex degree 3.
Cori, Jacquard and Schaeffer introduced description trees in 1997, to
give a general framework for the recursive decompositions of several
families of planar maps studied by Tutte. These trees are not only
interesting in their own right, but also they proved to be a useful tool
in obtaining non-trivial equidistribution results on planar maps,
certain pattern avoiding permutations, and objects counted by Catalan
numbers.

In this talk, I will provide an overview of several recent results and
research trends related to planar maps and description trees. Most of
the results are ``work in progress'' of several research teams.

The famous Carleson corona theorem asserts that the open
unit disk is dense in the maximal ideal space of the algebra of
bounded holomorphic functions on it (denoted $H^\infty$). Similar
statements for the algebra of bounded holomorphic functions on a
polydisk and for slice algebras for $H^\infty$ remain the major open
problems of multivariate complex analysis. For instance, the answer to
the last problem would be obtained if one were able to show that
$H^\infty$ has the Grothendieck approximation property. This problem
posed by Lindenstrauss in the early 1970th is also unsolved. The
strongest result in this direction was proved by Bourgain and Reinov
in 1983 and asserts that $H^\infty$ has the approximation property up to
logarithm. In the talk I will present a proof of the corona theorem
for slice algebras for $H^\infty$, describe topological structure of the
maximal ideal space of $H^\infty$ and as a corollary present some
results on $Sz$. Nagy operator-valued corona problem for $H^\infty$.

\indent Function theory of several complex variables is much less well understood than function theory of functions of one variable. One approach to attempting to bridge this divide is to study an analytic function $f$ on $D^n$ as follows. Fix a 1-dimensional algebraic variety $V\subset C^n$ and let $F$ denote the restriction of $f$ to $V$. Since $V$ is 1-dimensional, $F$ behaves somewhat like a function of one complex variable and we could potentially apply the theory of functions of one variable to understanding $F$. If we can use this approach to prove facts about $F$, then we could potentially extend some of these results to $f$. In particular, we take this approach to generalize to $D^n$ the classic Schwarz Lemma on the disc $D$. Familiarity with the definition of an analytic function of one variable is the only thing that will be assumed.

Function theory of several complex variables is much less well understood than function theory of functions of one variable. One approach to attempting to bridge this divide is to study an analytic function f on the polydisc as follows. Fix a $1$-dimensional algebraic variety $V$ in $C^n$ and let $F$ denote the restriction of $f$ to $V$. Since $V$ is $1$-dimensional, $F$ behaves somewhat like a function of one complex variable and we apply the theory of functions of one variable to $F$. We use this approach to prove facts about $F$ and then we extend certain results about $F$ to results about $f$. In particular, we take this approach to generalize to $D^n$ the classic Schwarz Lemma on the disc $D$ and give sufficient conditions for a bounded analytic function on $D^n$ to be uniquely determined by its values on a finite set of points. In terms of the Pick problem on $D^n$, we give sufficient conditions for a Pick problem to have a unique solution.

\indent The problem of the densest packing of space by congruent regular tetrahedra has a long history, starting with Aristotle's assertion that regular tetrahedra fill space, and continuing through its appearance in Hilbert's 18th Problem. This talk describes its history and many recent results obtained on this problem, including contributions from physicists, chemists and materials scientists. The current record for packing density is held by my former graduate student Elizabeth Chen, jointly with Michael Engel and Sharon Glotzer.

If A is an abelian variety over a p-adic field K then A has good
reduction if and only if the p-adic Tate module of A is a crystalline
representation of the absolute Galois group of K. As there are examples of
curves over K with bad reduction whose Jacobian has good reduction, the
Galois action on the p-adic etale cohomology of the curve does not
determine its reduction. We will discuss these issues and point
to a p-adic criterion of good reduction for curves.