Title: Introduction to Gauss sums

Abstract: Gauss sums are central objects in number theory. In this talk, I will introduce Gauss sums and show some of their astonishing properties.

Title: Tree Fitting Problem

Abstract: Given points and their measured distances, the tree fitting problem is to produce a tree graph which spans the points and "fits" the given distances. While such a seemingly simple problem was first introduced in the 1960s, the problem turns out to be hard so that there is still a lot of room for improvement. In this talk, we will explore the problem in various settings and introduce some results.

Title: Sullivan's dictionary on geometric prime number theorems

Abstract: The prime number theorem states that the number of primes of size at most T grows like T/log⁡T, proved by Hadamard and de la Vallee Poussin in 1896. For Gaussian primes, that is, prime ideals in Z[i], not only does the number of Gaussian primes of norm at most T grow like T/log⁡T but also the angular components of Gaussian primes are equidistributed in all directions, as proved by Hecke in 1920. Geometric analogs of these profound facts have been of great interest over the years. We will discuss effective versions of these theorems for hyperbolic 3-manifolds and for rational maps. Both Kleinian groups and rational maps define dynamical systems on the Riemann sphere and they are expected to behave analogously in view of Sullivan’s dictionary. We will explain how our theorems fit in this dictionary.

Title: Stability in geometric inequalities

Abstract: I will review some results on the stability of some geometric inequalities. The notion of stability considers the following question: Suppose that a function almost attains equality in some inequality for which minimizers are known. Can we prove that such a function is close to one of the minimizers? Over the past decade, the method of mass transportation made significant progress on this issue. In this talk, I will focus on the Brunn-Minkowski inequality and the isoperimetric inequality and give a quick overview of the proof strategy.

Title: Nuclear power- one hell of a way to boil water

Abstract: In this talk, I will give a glimpse into the world of nuclear reactor physics- starting from the fundamentals of the fission process, ending the physics of an entire core, We will touch on topics like safety in reactors, homogenization of the relevant equations, reactor dynamics and finally we'll talk about what happened in Chernobyl, time permitting.

Title: Flows, growth rates and veering triangulations

Abstract: The work of Thurston, Fried, McMullen, Mosher, Fenley and others weaves together a rich picture of fibrations and flows in 3-manifolds, linking growth rates of orbits, dilatations of pseudo-Anosov maps, and Thurston's norm on homology. I'll describe some of this picture and talk about some joint work with Michael Landry and Sam Taylor which develops a combinatorial model using the "veering triangulations" of Agol-Gueritaud, which permits some computations and refines our understanding a bit.

Title: Schur-Weyl duality and its generalizations

Abstract: Schur-Weyl duality was originally defined by Schur in 1901, which builds a correspondence between the representations of the symmetric group and the polynomial representations of the general linear group. In this talk, I will first walk through the classical Schur-Weyl duality, which states that the symmetric group and the general linear group generate each other’s centralizers. Then I will discuss some possible generalizations of the classical Schur-Weyl duality, for example, the duality between the orthogonal group and the Brauer algebra.

Title: 3 implies Chaos

Abstract: It is well known that chaos arises from 3-body problems (both in the planetary sense as in Reuben's talk and in the romantic sense). In this talk, we will discuss the (not) more general statement that "3 implies chaos". Topics to be surveyed are the classical example of the logistic function, connections to the Mandelbrot set and the classical results of Li--Yorke and Sharkovsky, giving a streamlined proof of the latter due to Burns-Hasselblatt.

Title: Algebraic Structure of Mapping Class Groups via Dynamics.

Abstract: Random walk on the mapping class group produces mapping classes with large translation lengths on Teichmüller space. In 2007, Maher and Tiozzo showed that they are not normal generators of the mapping class group. In contrast, Lanier and Margalit proved in 2018 that pseudo-Anosov mapping classes with small translation lengths are normal generators of the mapping class group. In this talk, we introduce related work and show that reducible mapping classes with small translation lengths also normally generate the mapping class group using simple combinatorics. This is joint work with Hyungryul Baik and Chenxi Wu. We also discuss an analogous question for asymptotic translation length on curve graphs and related work if time permits.