Graduate Research Seminar

November 13th, Elsa Magness: "An overview of elliptic curves with complex multiplication"

October 9th, AJ Vargas: "Symplectic geometry applied to C^\infty inscription problems"

September 25th, Lindsay Dever: "Ambient prime geodesic theorems"

June 5th, General Discussion: "Teaching"

During this discussion we will cover what it's like to teach in general, and during a pandemic.

May 1st, Lindsay Dever: "Equidistribution of holonomy in hyperbolic 3-manifolds"

The study of hyperbolic 3-manifolds draws deep connections between number theory, geometry, topology, and quantum mechanics. The closed geodesics of a hyperbolic 3-manifold are an important invariant and are parametrized by both their length and holonomy. Sarnak and Wakayama showed in 1999 that holonomy is equidistributed throughout the circle. In this talk, I’ll present new results including a refined count of length and holonomy which implies equidistribution in shrinking intervals. This will be illustrated with a numerical example.

April 10th, Daniel White: "Intro to Deep Learning: Neural Nets and Backpropagation"

Inspired by the behavior of neurons in our own brains, neural networks were proposed in the middle of the last century as a way to model the human decision-making process. This approach has been implemented successfully over the past few decades to tackle problems ranging from translation to image recognition. In recent years, Google’s DeepMind team made waves in the AI research community with their Alpha series, culminating in 2017’s AlphaZero. This program employs a trainable neural network that can achieve superhuman performance on any two-player game of perfect information, such as Chess or Go, in a matter of hours without any initial training data. DeepMind has been applying the fruit of their research to important scientific problems like disease detection and protein folding.

In this talk, we’ll develop the basic architecture of a neural network, which can be written in only tens of lines of code. The difficulty, as we will discuss, lies in the problem of training the network to complete a specific task. Depending on the problem domain and constraints such as the availability of training data and computation time, algorithms can sometimes be used to coax the network into a functional state. In particular, we will investigate the mathematics behind backpropagation, the so-called workhorse algorithm, which may be used when adequate training data is available.

Coding experience and advanced mathematics are not required for this talk; the main tools necessary are a few fundamental facts from multivariable calculus and linear algebra. Intuition and hand-waving are encouraged.

March 6th, AJ Vargas: "An Introduction to Matrix Groups"

It is well known that the ring of linear operators on Euclidean $n$-space is equivalent to the ring of $n\times n$ matrices over the real numbers. Hence, the group of invertible linear operators on Euclidean $n$-space can be endowed with the structure of a smooth manifold (more specifically, with the structure of a Lie group), so that one can use theorems of both algebra and smooth topology as tools for understanding these special and very interesting groups. In this talk, we survey some algebraic/topological properties enjoyed by these linear automorphism groups and various subgroups thereof.

February 14th, Elsa Magness: "Counting points on curves the Frobenius way"

January 31st, Lindsay Dever: "Chebyshev's Bias"

How many primes are congruent to 1 modulo 4 versus 3 modulo 4? Dirichlet proved in 1837 that there are infinitely many primes in each congruence class; in fact, the primes are equidistributed between these two classes. However, Chebyshev observed in 1853 that there are “more” primes congruent to 3 modulo 4. Even though the number of primes in each class is roughly the same, 3 modulo 4 is almost always “winning the race”. Assuming the Generalized Riemann Hypotheses and Grand Simplicity Hypothesis, Rubenstein and Sarnak showed in 1994 that this bias exists. In this talk, I’ll discuss the distinction between equidistribution and bias and give an overview of the results of Rubenstein and Sarnak.

December 12th, Daniel White: "Resonators: How L-functions Vibrate"

The Riemann zeta function can be expressed as a Dirichlet series in the half-plane of absolute convergence, and as a result, exhibits strong oscillatory behavior there. This behavior continues into the critical strip via the functional equation, an area of the complex plane where L-functions enjoy intense study. A decade ago, Soundararajan developed the resonator method, which exploits these oscillations to detect large values of the zeta function along the critical line. At its core, the idea is simple, and is essentially the same phenomenon that collapsed the Tacoma Narrows Bridge in 1940. We will discuss his approach and highlight concepts over technical details.

December 5th, AJ Vargas: "A Symplectic Perspective on the Inscribed Rectangle Problem"

In this talk we will analyze, from a somewhat modern point of view, a generalization of Toeplitz's old conjecture which proposes that to any reasonable closed curve in the plane, there corresponds a square each of whose vertices belong to the given curve. In particular, we will first give an overview of some classical and recent results surrounding the problem, then show how to reduce the problem to a question of whether certain Lagrangian submanifolds of 8 dimensional Euclidean space intersect, proposing finally some possibly new channels for the pursuit of a solution.

November 21st, Lindsay Dever: "Growth of Torsion in Homology Groups"

Every homology group decomposes into a free abelian group and a torsion subgroup. For hyperbolic 3-manifolds, computational results reveal that the torsion part is in fact very large. In addition, the torsion subgroup has deep and interesting connections to number theory. In this talk, I’ll discuss the connection between number theory and topology via Kleinian groups, and I’ll present results of Bergeron and Venkatesh, who proved that for certain arithmetic subgroups, the size of the torsion subgroup grows exponentially with the volume of the corresponding manifold.

November 7th, Isaac Sundberg: "Khovanov Homology"

Khovanov homology associates a bi-graded module to any link in the 3-sphere. In this talk, we describe Khovanov's construction of this module using a (1+1)-dimensional topological quantum field theory (TQFT). Along the way, we discuss the fundamentals of TQFTs and sufficient conditions for their well-definedness. This leads to a final discussion of different TQFTs used in the same setting to yield other homology theories (e.g. Lee Homology or Bar-Natan's equivalent definition of Khovanov homology using cobordisms).

October 24th, Lindsay Dever: "Weyl's Law: From Isometries to Eigenvalues"

Complex-valued, square-integrable functions on compact, hyperbolic 3-manifolds decompose uniquely into special functions called automorphic forms. These special functions are eigenfunctions of the Laplacian operator which are invariant under a discrete group of isometries. In general, it is a hard problem to determine the eigenvalues explicitly, but we do know how many eigenvalues to expect asymptotically. The asymptotic formula for the number of eigenvalues is known as Weyl's law. The proof of Weyl's law involves the Selberg trace formula, which connects these eigenvalues to geometric information about the manifold. In this expository talk, I will introduce hyperbolic 3-manifolds, define automorphic forms, and give a proof of Weyl's law for compact, hyperbolic 3-manifolds.

October 3rd, Daniel White: "What the Heck(e) is a Character?"

A homomorphism from a group to the unit circle is known as a character. Number theorists often consider subgroups of Q and their associated characters to extract arithmetic information about the integers. Familiar characters on these groups include the exponential function and Dirichlet characters. It turns out that all continuous* characters on Q can be built from these -- a direction consequence of the unique embedding of Q into C. It is natural to ask whether the characters associated to finite extensions of Q yield information about the corresponding ring of integers in the same way. The answer is a resounding "yes!", and even more, we obtain a whole new cast of (Hecke) characters as we examine such extensions.

*With respect to some fancy topology with archimedean and p-adic ingredients.

September 19th, AJ Vargas: "Configurations of the Plane"

A linear symplectic form on a Euclidean space is a non-degenerate, alternating tensor of rank 2 (equivalently an invertible, skew-symmetric matrix). It turns out that every even dimensional Euclidean space admits a linear symplectic form, and the reason is essentially because one can perform rotations by 90 degrees in a plane. In this talk, we examine a few key examples of linear symplectic forms and their features, all the while imagining typical elements of even dimensional Euclidean spaces as finite sequences (i.e. configurations) of points in a plane.

September 12th, Isaac Sundberg: "The Mazur Cork"

The Mazur Cork is a contractible 4-manifold whose history leads one through many delightful topics in low-dimensional topology: the Poincaré conjecture, handlebodies, and Kirby calculus (to name a few). In this talk, we discuss these topics and prove that the Mazur cork is a contractible 4-manifold that is smoothly distinct from the 4-ball.