A real number x is called diophantine if its distance to rationals p/q is large relative to q -- more precisely, if for every d > 0 there is a positive C such that for every reduced rational p/q, we have |x - p/q| > Cq^{-2-d}, or equivalently |qx-p| > Cq^{-1-d}. Almost all reals have this property. Furthermore, almost every pair (x_1, x_2) has the property that for every d > 0 there is a C such that |q_1x_1+q_2x_2 -p| > C||q||^{-2(1+d)} for all p, q_1, q_2. In this talk, we discuss a noncommutative analog of this property for elements of SO(3). Namely, a pair (A,B) is called diophantine if there exists a constant D such that for every positive integer n and every reduced word W of length n in A, B, A^{-1}, B^{-1}, we have ||W - E|| > D^{-n}, where E is the identity matrix. It is conjectured that almost every such pair (in the sense of Haar measure) is diophantine. We will present a paper of Kaloshin and Rodnianski, in which the weaker bound D^{-n^2} is obtained.

Phyllotaxis, the arrangement of phylla (leaves, bracts, seeds) near the shoot apical meristems of plants has intrigued and mystified natural scientists for over two thousand years. It is surprising that only within the last two decades have quantitative explanations emerged that describe the wonderful architectures which are observed. I will give an overview of two types of explanation, teleological and mechanistic, one based on rules which posit that each new phyllo be placed according to some optimal packing principle and the other which uses plain old biophysics and biochemistry to build mechanistic models which lead to pattern forming pde's. One of the stunning new results is that, while the latter is richer, both approaches lead to completely consistent results. This may well have broader ramifications in that it suggests that nature may use instability driven patterns to achieve optimal outcomes.

The talk should be accessible to students and colleagues in other disciplines.

Some useful ``categories" in symplectic geometry, candidates for being the domains of quantization functors, are ones in which the morphisms X --> Y between symplectic manifolds are relations, rather than maps.  These are submanifolds of X x Y having nice geometric properties with respect to the product of the symplectic form on X and the negative of the symplectic form on Y.

An obstruction to getting actual categories is that the set-theoretic composition of relations does not preserve the class of manifolds, due to possible failures of transversality.

In this talk, I will describe several approaches to resolving the transversality problem, concentrating on the linear case.  Although the composition of linear relations is always linear, the composition operation itself fails to be continuous until it is modified to take nontransversality into account.

The talk will be based in part on work with David Li-Bland and Jonathan Lorand, available on the arXiv.

Recent advancements in computational fluid dynamics have enabled researchers to efficiently explore problems that involve moving elastic boundaries immersed in fluids for problems such as cardiac fluid dynamics, fish swimming, and the movement of bacteria. These advances have also made modeling the interaction between a fluid and an electromechanical model of an elastic organ feasible. The tubular hearts of some ascidians and vertebrate embryos offers a relatively simple model organ for such a study. Blood is driven through the heart by either peristaltic contractions or valveless suction pumping through localized periodic contractions. Models considering only the fluid-structure interaction aspects of these hearts are insufficient to resolve the actual pumping mechanism. The electromechanical model presented here will integrate feedback between the conduction of action potentials, the contraction of muscles, the movement of tissues, and the resulting fluid motion.