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.

Online lectures and online classes are changing the landscape of math education. At New York University, we are creating a hybrid Calculus 1 course which will combine interactive online content with an in-class component involving lectures and problem sessions. We relate the structure of this course to that of other non-traditional calculus classes, and we discuss some potential advantages of this class over the standard lecture format.

There is a joke that when one counts something in linear algebra, the only possible results are 0, 1, and infinity.  What about in number theory?  Well, if we count the primes, we get infinity.  To get a finite number, we could try to count primes less than some fixed N.  It was proven in 1955 that something interesting happens sometime before N = 10^(10^(10^1000)).  Huh?!  What if we count gaps between primes?  Just in 2013, it was proven that something interesting happens for gaps less than 70,000,000.  What if we try to write down a really big prime number?  Right now, nobody can write down a prime bigger than 10^(10^8).  In this talk, we'll introduce some of these big numbers in number theory, we'll carefully say what they mean, and we'll describe progress on making the first two big numbers smaller and on making the third big number bigger.