Partial differential equations (PDEs) can be used to model numerous
physical processes, from steady-state heat distribution to the
formation of black holes. When the solution changes over time,
special techniques and considerations must be taken for their accurate
solution. In this talk, I will briefly introduce the concepts,
discuss a few of the concerns, and show some numerical results from
simple model problems.

We describe several corollaries of the Littlewood-Richardson refinements, including a method for multiplying two Schur functions with different numbers of variables and expanding the result as a sum of key polynomials. We use interactions between Schur functions and quasisymmetric Schur functions to prove a conjecture of Bergeron and Reutenauer. We show that their conjectured basis is indeed a basis for the quotient ring of quasisymmetric functions by symmetric functions, which also provides a combinatorial proof of Garsia and Wallach's results about the freeness and dimension of QSym/Sym. This is joint work with Aaron Lauve.

Neuroscientists have become increasingly interested in exploring
dynamic
relationships among brain regions. Such a relationship, when directed from
one region toward another, is denoted by ``effective connectivity.'' An fMRI
experimental paradigm which is
well-suited for examination of effective connectivity is the slow
event-related design.
This design presents stimuli at sufficient temporal spacing for determining
within-trial
trajectories of BOLD activation. However, while several analytic methods for
determining
effective connectivity in fMRI studies have been devised, few are adapted to
the
characteristics of event-related designs, which include non-stationary BOLD
responses and nesting of responses within trials and subjects.
We propose a model tailored for exploring effective connectivity
of multiple brain regions in event-related fMRI designs - a semi-parametric
adaptation of vector autoregressive (VAR) models, termed "stimulus-locked
VAR"
(SloVAR). Connectivity coefficients vary as a function of time
relative to stimulus onset, are regularized via basis expansions, and vary
randomly across subjects. SloVAR obtains flexible, data-driven estimates of
effective
connectivity and hence is useful for building connectivity models when prior
information
on dynamic regional relationships is sparse. Indices derived from the
coefficient estimates can also be used to relate effective connectivity
estimates
to behavioral or clinical measures. We demonstrate the SloVAR model
on a sample of clinically depressed and normal controls, showing that
early but not late cortico-amygdala connectivity appears crucial to
emotional control and
early but not late cortico-cortico connectivity predicts depression severity
in the depressed group, relationships that would have been missed in a more
traditional VAR analysis.

Reto Muller investigated a new geometric flow which consists
of a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target closed manifold $N$, given by $\frac{\partial}{\partial t} g = - 2 Ric + 2 \alpha \nabla \phi \bigotimes \nabla \phi, \frac{\partial}{\partial t}\phi = \tau_{g}\phi $, where $\alpha$ is a positive coupling constant. This new flow shares many good properties with the Ricci flow.

We will define useful definitions of growth on an algebra. In particular, we will consider Gelfand-Kirillov (GK) dimension. After stating some nice properties of GK dimension of algebras, we will sketch a combinatorial proof of the Bergman Gap Theorem.

Chemotaxis is characterized by the directional cell movement following external chemical gradients. It plays a crucial role in a variety of biological processes including neuronal development, wound healing and cancer metastasis. Ultimately, the accuracy of gradient sensing is limited by the fluctuations of signaling components, e.g. the stochastic receptor occupancy on cell surface. We use concepts and techniques from interrelated disciplines (statistics, information theory, and statistical physics) to model the stochastic information processing in eukaryotic chemotaxis. Specifically, we address the following issues:

\begin{enumerate} \item What are the physical limits of the gradient estimation? \\ \item How much information can be reliably gained by a chemotaxing cell? \\ \item How to optimize the chemotactic performance? \\ \end{enumerate}

Through answering those questions, we expect to derive extra insights for general biological signaling systems.

A multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{R}^n$. Obvious examples of non-negative polynomials are squares and sums of squares. What is the relationship between non-negative polynomials and sums of squares? I will review the history of this question, beginning with Hilbert's groundbreaking paper and Hilbert's 17th problem. I will discuss why this question is still relevant today, for computational reasons, among others. I will then discuss my own research which looks at this problem from the point of view of convex geometry. I will show how to prove that there exist non-negative polynomials that are not sums of squares via ``naive" dimension counting. I will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares.

Knots are embeddings of circles in the three dimensional sphere. We will discuss an equivalence relation called knot concordance and the group of equivalence classes under connect sum

We will discuss the structure of the knot concordance group, finite order concordance classes and open problems in the area.