By dynamical coupling of data with known models, we determine underlying parameters and unmeasured state variables for a variety of systems, including Lorenz, Colpitts, and Hodgkin-Huxley neurons. The dynamic coupling is mediated by use of a cost function, which is minimized by optimization software (SNOPT, IPOPT) to achieve the desired synchronization. Considering measurement and model noise, we discuss the same problem where we introduced an 'action' in order to estimate states and parameters. This action is used along with Markov Chain Monte Carlo methods in order to sample from a probability distribution.

Skew Young tableaux are simple combinatorial objects
arising in the theory of symmetric functions and the representation
theory
of the symmetric group. The number $f^\sigma$ of (standard) skew Young
tableaux of skew shape $\sigma$ has a simple determinantal formula due
to
Aitken. We will discuss some situations for which there exist other
formulas or generating functions for $f^\sigma$. For instance, for certain
skew shapes $\sigma$ the number $f^\sigma$ can be described in terms of
Euler numbers (the number of alternating permutations of $1,2,\dots,n$)
using an analytic technique introduced by Elkies and further developed
by
Baryshnikov and Romik. Certain other sequences $\sigma_n$ of skew shapes
have simple generating functions for the numbers $f^{\sigma_n}$, based
on
a well-known connection between determinants and generating functions.

\indent The Jones polynomial can be understood in terms of the representation theory of the quantum group associated to $sl2$. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.

A dual equivalence for an arbitrary collection of
combinatorial objects endowed with a descent set is a relation for which
equivalence classes group together terms according to the Schur
expansion of the corresponding generating function. After outlining the
definition of dual equivalence, we'll present three main applications:
the Schur expansion of Macdonald polynomials, Schur positivity of
k-Schur functions (joint with S. Billey), and a combinatorial rule for
the Littlewood-Richardson coefficients of the Grassmannian in the
special case of a Schubert polynomial times a Schur function (joint with
N. Bergeron and F. Sottile).

\indent I will discuss the problem of efficiently constructing
curves of genus 1 and 2 over finite fields with a
prescribed number N of points. In both cases, there are algorithms that, at least heuristically and in practice, run in time polynomial in log N. They are of complex analytic nature, using CM-techniques. Time permitting, I will also explain why these techniques are provably insufficient to efficiently construct genus-2 Jacobians with a presecribed number of points.

Let $A/K$ be an abelian variety over a global field $K$.
For each place $v$ of $K$, one associates an integer $c(v)$ called the Tamagawa
number of the place, using the reduction of the abelian variety at $v$.
Let $c$ denote the product of the $c(v)'s$. Let $t$ denote the order of the
torsion subgroup of Mordell-Weil group $A(K)$. The ratio $c/t$ is a factor in
the leading term of the L-function of $A/K$ at $s=1$ predicted by the
conjecture of Birch and Swinnerton-Dyer.
We investigate in this talk possible cancellations in the ratio $c/t$.
For elliptic curves over $Q$. the smallest ratio $c/t$ is $1/5$, obtained only
by the modular curve $X_1(11)$.

\indent Ion channels are irresistible objects for biological study because they are ‘nanovalves of life’ controlling most biological functions, much as transistors control computers. Direct simulation of channel behavior in atomic detail is difficult if not impossible. Gaps in scales of time, volume, and concentration between atoms and biological systems are each ~1012. All the gaps must be dealt with at once, because biology occurs on all the scales at once.
Simple models are surprisingly successful in dealing with ion binding in three very different (and important) channels: the sodium channel that produces the signals of nerve and muscle and two cardiac calcium channels that control contraction. Amazingly, one model with the same three parameters accounts quantitatively for qualitatively different binding in a wide range conditions for two very different calcium and sodium channels. Binding free energy is an output of the calculation, produced by crowding charged spheres into a very small space. The model does not involve any traditional chemical ‘quantum’ binding energies at all.
How can such a simple model give selectivity when crystallographic wisdom and chemical intuition says that selectivity depends on the precise structural relation of ions and side chains? The answer is that structure is a computed consequence of forces in these correlated crowded systems. Binding sites are self-organized and at their free energy minimum. Different structures form in different conditions. Binding is a consequence of the ‘induced fit’ of side chains to ions and ions to side chains.
Equilibrium is death to biology. A variational approach is obviously needed to replace our equilibrium analysis and is well under way, applying the energy variational methods of Chun Liu, used to deal with highly correlated systems like liquid crystals.

In 1913, Major Percy MacMahon showed that the major index
and inversion number statistics are equidistributed over permutations.
A bijective proof of this fact was first given in 1968 by Dominique
Foata who constructed a recursive bijection on permutations such that
the major index of the source is the inversion number of image. In
2004, Jim Haglund made a major breakthrough in the theory of Macdonald
polynomials by conjecturing a formula for Macdonald polynomials,
proved shortly thereafter with Haiman and Loehr, that involved
relatively simple generalizations of the major index and the inversion
number. In this talk, we will show how a filtration of Foata's
bijection can be used to give a simple proof of Macdonald positivity
for certain cases and outline how this approach might be generalized.
Time permitting, we will outline how similar techniques might be
useful in giving a bijective proof of the so-called q,t symmetry of
Macdonald polynomials. This talk will be accessible to first year
graduate students and contains several open problems.