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.

The high-frequency Helmholtz equation in a heterogeneous medium is a difficult problem in numerical analysis. Direct solvers are either unwieldy to set up for the problem as a whole, or are hard to link up in a domain decomposition framework. On the other hand, preconditioners for iterative solvers tend to suffer either from a lack of scalability, or from a convergence rate highly dependent on the frequency.  

In this work, we present a hybrid approach, which results in a highly scalable algorithm with sublinear runtime in the number of unknowns normally needed to represent the solution in the volume.

This approach uses efficient direct solvers locally in large subdomains and properly couples them with transmission conditions in the form of incomplete Green’s integrals. The coupling allows us to reduce the problem to a boundary integral equation, which is solved iteratively. The BIE is preconditioned by introducing a polarization of the waves thus achieving a fast convergence rate independently of the frequency and the number of subdomains. This approach is especially attractive when the local Green’s functions are precomputed, and a fast algorithm is available for their application.

We will discuss two different aspects of infection with the human papillomavirus (HPV).

Part 1: The Biology of Viral Clearance.

Clearance of anogenital and oropharyngeal HPV infections is attributed primarily to a successful adaptive immune response. To date, little attention has been paid to the potential role of stochastic cell dynamics in the time it takes to clear an HPV infection. We combine mechanistic mathematical models at the cellular level with epidemiological data at the population level to disentangle the respective roles of immune capacity and cell dynamics in the clearing mechanism.

Part 2: Nonlinear Cost Curves and Optimal Vaccination Strategies.

The effectiveness of vaccinating males against HPV remains a controversial subject. Many existing studies conclude that increasing female coverage is more effective than diverting resources into male vaccination. Several recent studies on HPV immunization provide evidence for the fact that marginal vaccination costs increase with coverage. We develop a stochastic agent-based modeling framework to revisit the male vaccination debate in light of these new findings. Within this framework, we assess the impact of coverage-dependent marginal costs of vaccine distribution on optimal immunization strategies against HPV.