It is a celebrated result of Scott that every closed curve on a hyperbolic surface lifts to a simple closed curve on some finite cover. In light of this work, we pose the following question, What information about two covers X and Y of S can be derived by understanding how curves on S lift simply to X and Y? In this talk, we will motivate this question and explore its answer for regular finite covers of a closed hyperbolic surface. This work is joint with Tarik Aougab, Max Lahn, and Yang (Sunny) Xiao.

As it turns out, Grassmannians (and various related friends) are important for machine learning. We will see that Schubert varieties play a key role in a method for distinguishing subspaces of data.

The third of four talks explicating the early work of Maryam Mirzakhani. This is some of the mathematical background for the discussion of JT gravity later in the semester.

For a group acting on the circle by orientation-preserving homeomorphisms, there is an associated Euler class. I'll explain what information it contains as a (bounded) cohomology class. In the case of a surface group pi_1(S), the Milnor-Wood inequality states that the pairing |eu([S])| of the Euler class with the fundamental class is no more than the negative part of Euler characteristic of S. I'll explain this inequality from the bounded cohomology point of view.

Topological string theory was introduced by Bershadsky-Cecotti-Ooguri-Vafa in 1993. For most of the 2000's I tried to understand how the "partition function" they introduced as a formal series could be viewed as a geometric object beyond its series expansion. I still do not really understand this. Luckily, though, thinking about it forced me and my collaborators to learn various things about twistor spaces, which turned out to be a key clue to understanding what Kontsevich and Soibelman were saying about BPS states (generalized Donaldson-Thomas invariants), and to later applications of that theory to hyperkahler geometry / WKB analysis. Recently these ideas seem to be feeding back into the original questions about the topological string, though the complete picture is not yet clear, at least to me. I will try to describe some of this story. It involves joint work with many people including Boris Pioline, Murat Gunaydin, Andrew Waldron, Johannes Walcher, Davide Gaiotto, Greg Moore.

In this talk, we will look at the method of Second Quantization from an analytical viewpoint. While C^*-algebras and their applications to QFT and statistical mechanics have been studied intensely by the likes of V. Jones, Bratteli, Robinson, et al., the literature on concrete computations is fairly scarce. Many papers outline very few standard estimates they rely on for their analysis, but it remains unclear how to deal with more complex situations. I will try to show a constructive perspective on how to work with C^*-algebras and present how the formulation compares with problems in Calculus of Variations and also dynamical problems.