The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.

Title: Regularity via minors and applications to conformal maps. Abstract: Let f:\mathbb{R}^n \to \mathbb{R}^n be a Sobolev map; Suppose that the k-minors of df are smooth. What can we say about the regularity of f? This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors.

The moduli space of curves, first appearing in the work of Riemann in the 19th century, plays an important role in geometry. After an introduction to the moduli space, I will discuss recent directions in the study of tautological classes on the moduli space following ideas and conjectures of Mumford, Faber-Zagier, and Pixton. Cohomological Field Theories (CohFTs) play an important role. The talk is about the search for a cohomology calculus for the moduli space of curves parallel to what is known for better understood geometries.

The rational solutions on an elliptic curve form a finitely generated abelian group, but the maximum number of generators needed is not known. Goldfeld conjectured that if one also fixes the j-invariant (i.e. the complex structure), then 50% of such curves should require 1 generator and 50% should have only the trivial solution. Smith has recently made substantial progress towards this conjecture in the special case of elliptic curves in Legendre form. I'll discuss recent work with Lemke Oliver, which bounds the average number of generators for general j-invariants.

It is a major challenge in Combinatorial Geometry to understand the intersection structure of the edges in a geometric or topological graph, in the Euclidean plane. One of the few "tight" results in this direction is the the Crossing Lemma (due to Ajtai, Chvatal, Newborn, and Szemeredi 1982, and independently Leighton 1983). It provides a relation between the number of edges in the graph and the number of crossings amongst these edges. This line of work led to several Ramsey-type questions of geometric nature. We will focus on two recent advances.