Georgia Institute of Technology College of Sciences

In this series of (3-5) lectures, I will talk about different aspects of a class of contact 3-manifolds for which geometry, dynamics and topology interact subtly and beautifully. The talks are intended to include short surveys on "compatibility", "Anosovity" and "Conley-Zehnder indices". The goal is to use the theory of Contact Dynamics to show that conformally Anosov contact 3-manifolds (in particular, contact 3-manifolds with negative α-sectional curvature) are universally tight, irrducible and do not admit a Liouville cobordism to tight 3-sphere.

Vaccination is an effective method to protect against infectious diseases. An important consideration in any vaccine formulation is the inoculum dose, i.e., amount of antigen or live attenuated pathogen that is used. Higher levels generally lead to better stimulation of the immune response but might cause more severe side effects and allow for less population coverage in the presence of vaccine shortages. Determining the optimal amount of inoculum dose is an important component of rational vaccine design. A combination of mathematical models with experimental data can help determine the impact of the inoculum dose. We designed mathematical models and fit them to data from influenza A virus (IAV) infection of mice and human parainfluenza virus (HPIV) of cotton rats at different inoculum doses. We used the model to predict the level of immune protection and morbidity for different inoculum doses and to explore what an optimal inoculum dose might be. We show how a framework that combines mathematical models with experimental data can be used to study the impact of inoculum dose on important outcomes such as immune protection and morbidity. We find that the impact of inoculum dose on immune protection and morbidity depends on the pathogen and both protection and morbidity do not always increase with increasing inoculum dose. An intermediate inoculum dose can provide the best balance between immune protection and morbidity, though this depends on the specific weighting of protection and morbidity. Once vaccine design goals are specified with required levels of protection and acceptable levels of morbidity, our proposed framework which combines data and models can help in the rational design of vaccines and determination of the optimal amount of inoculum.

I will discuss knot concordances in 3-manifolds. In particular I will talk about knot concordances of knots in the free homotopy class of S^1 x {pt} in S^1 x S^2. It turns out, we can use some of these concordances to construct Akbulut-Ruberman type exotic 4-manifolds. As a consequence, at the end of the talk we will see absolutely exotic Stein pair of 4-manifolds. This is joint work with Selman Akbulut.

The importance of analyzing big data and in particular very large networks has shown that the traditional notion of a fast algorithm, one that runs in polynomial time, is often insufficient. This is where property testing comes in, whose goal is to very quickly distinguish between objects that satisfy a certain property from those that are ε-far from satisfying that property. It turns out to be closely related to major developments in combinatorics, number theory, discrete geometry, and theoretical computer science. Some of the most general results in this area give "constant query complexity" algorithms, which means the amount of information it looks at is independent of the input size. These results are proved using regularity lemmas or graph limits. Unfortunately, typically the proofs come with no explicit bound for the query complexity, or enormous bounds, of tower-type or worse, as a function of 1/ε, making them impractical. We show by entirely new methods that for permutations, such general results still hold with query complexity only polynomial in 1/ε. We also prove stronger results for graphs through the study of new metrics. These are joint works with Jacob Fox.