Georgia Institute of Technology College of Sciences

In this talk, we obtain new computational insights into two classical areas of statistics: generalization and sampling. In the first part, we study generalization: the performance of a learning algorithm on unseen data. We define a notion of generalization for non-converging training with local descent approaches via the stability of loss statistics. This notion yields generalization bounds in a similar manner to classical algorithmic stability. Then, we show that more information from the training dynamics provides clues to generalization performance.   

In the second part, we discuss a new method for constructing transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

We'll begin with a primer on hyperbolic and stable polynomials, which have been popular in recent years due to their many surprising appearances in combinatorics and algebra. We will cover a sketch of the famous Branden Borcea characterization of univariate stability preservers in the first part of the talk. We will then discuss more our recent work on multivariate hyperbolic polynomials which are invariant under permutations of their variables and connections to this Branden Borcea characterization.

Zoom Link: https://gatech.zoom.us/j/99596774152