# Seminars & Events for Graduate Student Seminar

##### Spectral theory of the Laplacian and number theory

In this talk, I will discuss the interplay between the geometry of Riemannian manifolds and properties of eigenfunctions and eigenvalues of the Laplacian. Of particular interest is the relationship between the chaotic behavior of the manifold, in terms of ergodic geodesic flow, and chaotic behaviour of eigenfunctions. Connections will be made between eigenfunctions of the Laplacian and number theory via the theory of modular forms.

##### Min-max theory and least area minimal hypersurfaces

Minimal hypersurfaces are one important way to understand the structure of the ambient manifold and min-max theory is a powerful method for constructing them. I will present the basic ideas of min-max theory, a theorem of Calabi-Cao as an application and a possible extension of the latter.

##### Fibered links in S^3

In this talk I will define what it means for a link to be fibered and discuss the construction of fiber surfaces for fibered links through plumbing and twisting. I will give a very brief introduction to sutured manifolds and then present one method of detecting fibered links via sutured manifold decomposition.

##### Ramanujan graph

We discuss the geometry and spectral properties of expander graphs. We'll define Ramanujan graphs and show why they are optimal "expander graphs". Next, we'll explain the construction of explicit Ramanujan graphs by Lubotzky, Phillips, and Sarnak, and relate geometric and spectral properties of LPS Ramanujan graphs to well-known facts in number theory.

##### Embeddings into Euclidean space and related obstructions

We will discuss several results about bi-Lipschitz embeddings of metric spaces into Hilbert space, motivated by related classic theorems in Banach spaces. Emphasis will be given on techniques for proving non-embeddability and Ramsey-type theorems in this context.

##### Composition laws

We will first look at Gauß's classical composition law for binary integral quadratic forms and some of its applications. Then, we describe a few special cases of Bhargava's higher composition laws.

##### Geometric identities on moduli spaces and their applications

We will discuss the Bridgeman-Kahn and Mirzakhani-McShane identities on moduli spaces of bordered Riemann surfaces. As applications, we will look at Bridgeman’s proof of the classical Abel identity and Mirzakhani’s work on Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.

##### Applications of the Chebotarev density theorem

The Chebotarev density theorem is a basic and useful theorem in number theory. I will introduction some applications of CDT to modular forms based on Deligne’s theorem on l-adic representation attached to modular forms, and make connections with conjectures of Maeda and Lang-Trotter.

##### Polynomials, Representations, and Stability

I will introduce recent work of Church, Farb, and others on homological/representation stability, primarily through the elementary and explicit example of configurations of points in the plane. I will explain how one can use homological stability to say interesting things about polynomials over finite fields and I will derive analogies between number theory and topology, e.g., that the Riemann zeta function is related to an iterated loop space of projective space!

##### Yang-Mills flow and entropy

We introduce the Yang-Mills functional and flow, and discuss its history, motivation and past results. We then discuss a notion of entropy and introduce a Yang-Mills soliton, sketching some joint work with Jeff Streets.

##### Calculus of variations, Sobolev spaces, and geometric measure theory

We will discuss how the theory of Sobolev spaces and geometric measure theory are frameworks that arise more or less in the same way in the study of certain variational problems. I will provide some concrete examples, of which the main ones will be minimal surfaces and some elliptic PDEs.

##### An invitation to conformal geometry – from Gaussian curvature to Q-curvature

In the study of surface geometry, the uniformization theorem and the Gauss-Bonnet formula are of central importance. The former provides standard geometric models while the latter connects geometric and topological properties of a surface. In this talk, I will discuss higher-dimensional generalizations of these two results in a conformal-geometry context. In particular, I will talk about the Yamabe problem, the four-dimensional Gauss-Bonnet-Chern formula, and Q-curvature as important examples.

##### Hyperbolic volumes

The volumes of hyperbolic 3-manifolds form a peculiar subset of the reals. I will talk about the structure of this set and how each volume is secretly the real part of a complex number.

##### The geometric theory of regularity

Understanding the regularity of solutions to geometric variational problems (like minimal submanifolds, harmonic maps, and Yang-Mills connections) is of fundamental importance to geometry and neighboring fields. In this talk, we'll introduce the major tools used to analyze the singularities of such objects, and discuss what's known and what remains unanswered in this vast and lovely area of geometric analysis.

##### On Salem and Pisot numbers

Salem and Pisot numbers are two remarkable class of algebraic integers. We'll talk about their special number-theoretic properties and their connection to other parts of mathematics such as graph theory, hyperbolic geometry and dynamical systems. At the end, we'll talk about relevant conjectures which have resisted proof for many years.

##### y^2 = x^3 + Ax + B

Solving the titular equation in integers x and y is a problem that goes back to Diophantus himself in his famous book on, yes, Diophantine equations. Bhargava and collaborators have recently made progress on the expectation that, when sampling over random A and B, one should expect no ***rational*** solutions half the time, and infinitely many (but minimally so, in a sense) rational solutions the other half of the time. So what of integral solutions? Siegel showed long ago that there are always finitely many integer solutions to such an equation (when the discriminant is nonzero). In this talk I will show that there are at most boundedly many (< 66) on average.

##### The Shafarevich conjecture

The Shafarevich conjecture (now a theorem of Arakelov and Parshin) is a statement concerning families of smooth algebraic curves. It says that for any curve C, there are only finitely many admissible families of smooth curves of given genus over C, and that when such families exist, the base curve C has to be hyperbolic. In this talk, I will introduce the relevant notions and explain the proof of the hyperbolicity of the base curve. If time permits, I will also mention some higher-dimensional generalizations of the conjecture.

##### Things to do with Things

We introduce a Thing, and do some things with Things. Things were the original brainchild of Vladimir Drinfeld in the context of function field arithmetic, but have since reached new levels of abstraction, spurring entirely new things to do with Things in number theory, arithmetic algebraic geometry, and beyond. Time permitting, we will discuss both some theory and application of Things, and hopefully leave the audience with some new things to think about Things.

##### An introduction to semialgebraic geometry

A semialgebraic set is a subset S of R^n defined by a finite sequence of polynomial equations P=0 and inequalities Q > 0, or any finite union of such sets. In this talk, I will introduce some basic properties of semialgebraic sets, including that they are closed under the projection operation and triangularizable. If time permits, I will show an explicit bound for the number of connected components of a real algebraic set as a function of the degree of the equations and the dimension of the ambient space.

##### Recent sphere-packing successes

We will discuss how to arrange non-overlapping, equal-sized spheres so as to cover the greatest possible fraction of Euclidean space of a fixed dimension. We hope especially to outline the spectacular breakthrough of Viazovska in 8 dimensions and the solution one week later in 24 dimensions by herself with Cohn, Kumar, Miller, and Radchenko.