The period and index of a curve C are two quantities which describe the failure of C to have rational points. The mismatch between the two is of interest for its impact on the Shafarevich-Tate group of the Jacobian of C. The period-index problem is to determine what values of period and index are possible for a given genus g. We will give a complete answer when g=1, and an almost complete answer when g ≥ 2.

Absolute concentration robustness is a property that allows signaling networks to sustain a consistent output in the face of protein concentration variability from cell to cell.  This property is structural and can be determined from the topology of the network alone.  In this talk, I discuss this concept first for deterministic systems, and then set out to describe their stochastic behavior.  In the long term, the corresponding stochastic processes undergo an extinction event that eliminates the robustness. However, these systems have a transiently robust behavior that may be sufficient to carry out the necessary signal transduction in cells.  This work has been recently funded by NSF and graduate students are invited to inquire about working with me on this topic.

I will discuss three different contexts in which commutative
rings of functions and modules over them are replaced by their
non-commutative versions. One is the ring of differential operators, where
the modules correspond to systems of differential equations. The second
setting is geometric quantization which provides a baby version of the
Hilbert space in quantum mechanics. The third setting is deformation
quantization in symplectic geometry. I will explain a relation between these
three versions, although the reasons behind the relations are not quite well
understood.