Georgia Institute of Technology College of Sciences

Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.

Knot Floer homology is an invariant for knots introduced by Ozsváth-Szabó and, independently, Rasmussen.  We will give a general introduction to the theory, sketching the definition and highlight its major properties and applications.

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in few cases by work of David--Philippon and DeMarco--Krieger--Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz--Rabinoff--Zureick-Brown).

Piecewise polynomials with certain global smoothness can be given by traditional finite element methods and also by neural networks with some power of ReLU as activation function. In this talk, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.

One of the most fundamental issues in fluid dynamics is whether or not an initially smooth fluid flow can evolve over time to arrive at a singularity -- a state for which the classical equations of fluid mechanics break down and the flow field no longer makes physical sense.  While proof remains an open question, numerical evidence strongly suggests that a singularity arises at the boundary of a flow like that found in a stirred cup of tea.  The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity.  A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model generalizes Burger's equation and captures key features in the mechanics of the blowup scenario. 

Drift analysis is the name for a collection of tools that allow to translate information about the one-step progress of a randomized process into information about first hitting times. Drift analysis is successfully used in the mathematical analysis of randomized search heuristics, most notably, evolutionary algorithms, but (for unclear reasons) much less in discrete mathematics or other areas of algorithms.

In this talk, I will give a brief introduction to drift analysis, show some classic and recent applications, and describe some open problems, both concerning drift methods and the mathematical runtime analysis of randomized search heuristics.

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. Specifically, neuronal networks at multiple scales utilize their structural complexities to achieve different computational goals. I will first introduce functional implications that can be inferred from a weighted and directed “single” network representation of the brain. Then, I will consider a more detailed and realistic network representation of the brain featuring multiple types of connection between a pair of brain regions, which enables us to uncover the hierarchical structure of the brain network using an unsupervised method.  Finally, if time permits, I will discuss computational implications of the hierarchical organization of the brain network, focusing on a specific type of visual computation- predictive coding.

Meeting Link: https://gatech.bluejeans.com/348270750