Georgia Institute of Technology College of Sciences

An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the story is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on a quantitative version of the “how many?” question.

We will present an introduction to gauge theory and classical Chern-Simons theory, a 3-dimensional topological gauge field theory whose quantization yields new insights about knot invariants such as the Jones polynomial. Then we will give a sketch of quantum Chern-Simons theory and how Witten used it as a 3-dimensional method to obtain the Jones polynomial, as well as how it may be used to obtain other powerful knot and 3-manifold invariants. No physics background is necessary.

The linear programming discriminant(LPD) estimator is used in sparse linear discriminant analysis for high dimensional classification problems. In this talk we will give a sufficient condition for the variable selection property of the LPD estimator and our result provides optimal bound on the requirement of sample size $n$ and magnitude of components of Bayes direction.

We explain a method for the computation of normally hyperbolic invariant manifolds (NHIM) in discrete dynamical systems.The method is based in finding a parameterization for the manifold formulating a functional equation. We solve the invariance equation using a Newton-like method taking advantage of the dynamics and the geometry of the invariant manifold and its invariant bundles. The method allows us to compute a NHIM and its internal dynamics, which is a-priori unknown.We implement this method to continue the invariant manifold with respect to parameters, and to explore different mechanisms of breakdown. This is a joint work with Alex Haro.