Georgia Institute of Technology College of Sciences

There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk I will discuss some recent works on the existence of global solutions.

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.