## 2023 Ph.D Thesis Defenses

**Tam Cheetham-West**

**Title:** Finite Quotients of Hyperbolic 3-Manifold Groups

**Thesis Advisor: **Alan Reid

This thesis provides further evidence of the seemingly very close relationship be-tween the geometry of a finite-volume hyperbolic 3-manifold and the profinite completion of its fundamental group.

**Ethen Gwaltney**

**Title:** Stahl-Totik regularity and exotic spectra of Dirac operators

**Thesis Advisor: **Milivoje Lukic

This thesis motivates and presents three novel results in the spectral theory of one-dimensional Dirac operators, each of which concerns various forms of exotic or distinguished spectral characteristics. First, we consider the possibility of embedded eigenvalues in the absolutely continuous spectrum of a Dirac operator with operator data of Wigner-von Neumann type. Second, we demonstrate the genericity of Cantor spectrum when the operator data is chosen to be limit-periodic. Third, we provide for the Dirac operator setting an analogue of Stahl-Totik regularity, which, among other things, provides a lower bound on the thickness of the spectrum in terms of the operator data when the data is taken to be uniformly locally square integrable.

**Connor Sell**

**Title:** Cusps and commensurability classes of hyperbolic 4-manifolds

**Thesis Advisor: **Alan Reid

It is well-known that the cusp cross-sections of finite-volume, cusped hyperbolic n-manifolds are flat, compact (n − 1)-manifolds. In 2002, Long and Reid proved that each of the finitely many homeomorphism classes of flat, compact (n − 1)-manifolds occur as the cusp cross-section of some arithmetic hyperbolic n-orbifold; the orbifold was upgraded to a manifold by McReynolds

in 2004. There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This thesis provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type in any dimension. Further, we extend this result to find commensurability classes of hyperbolic 5-manifolds that avoid some compact, flat 4-manifolds as cusp cross-sections, and classes of non-arithmetic manifolds in both dimensions that avoid some cusp types

**Asgeir Valfells**

**Title:** Local Criteria in Polyhedral Minimizing Problems

**Thesis Advisor: **Bob Hardt

This thesis will discuss two polyhedral minimizing problems and the necessary local criteria we find any such minimizers must have. We will also briefly discuss an extension of a third minimizing problem to higher dimension. The first result we present classifies the three-dimensional piecewise linear cones in R4 that are mass minimizing w.r.t. Lipschitz maps in the sense of Almgren’s M (0, δ) sets as in Taylor’s classification of two-dimensional soap film singularities. There are three that arise naturally by taking products of R with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no p.l. minimizers outside these five. The second result we present is an assortment of criteria for edge-length minimizing polyhedrons. The aim is to get closer to answering a 1957 conjecture by Zdzislaw Melzak, that the unit volume polyhedron with least edge length was a triangular right prism, with edge length 22/3311/6 ≈ 11.896. We present a variety of variational arguments to restrict the class of minimizing candidates.

**Chunyi Wang**

**Title:** Direct and Inverse Spectral Theory for the Hamiltonian System with Measure Coefficients

**Thesis Advisor: **David Damanik

This thesis discusses the direct and inverse spectral theory of Hamiltonian systems with measure coefficients, which can cover more singular cases. In the first part, we define self-adjoint relations associated with the systems and develop Weyl-Titchmarsh theory for these relations. Then, we develop subordinacy theory for the relations and discuss several cases when the absolutely continuous spectrum appears. Finally, we develop inverse uniqueness results for Hamiltonian systems with measure coefficients by applying de Branges’ subspace ordering theorem. Overall, this thesis contributes to the study of Hamiltonian systems with measure coefficients, expands the self-adjoint operator theory to a more general class of physical models, and investigates common spectral properties among different model

**Harshit Yadav**

**Title:** Functorial constructions of Frobenius algebras in the Drinfeld center

**Thesis Advisor: **Chelsea Walton

Frobenius algebras in vector spaces are classical algebraic structures. However, because of their discovered connections to various fields, including computer science and

topological quantum field theories, there is a growing interest in exploring their generalizations within the framework of monoidal categories. Inspired by these connections, this thesis delves into the problem of functorially constructing ‘nice’ Frobenius algebra objects in such categories. We introduce unimodular module categories and employ them to provide a functorial construction of Frobenius algebras in the Drinfeld center of a finite tensor category. We also classify unimodular module categories over the category of representations of a finite dimensional Hopf algebra

**Kenneth Zheng**

**Title:** Brauer groups of a family of nonnegative Kodaira dimension elliptic surfaces

**Thesis Advisor: **Anthony Varilly-Alvarado

We explore the Brauer groups of the elliptic surfaces given by y2 = x3 + t6m + 1 over Q for m = 2, 3. When m = 2, the resulting surface is K3, and when m = 3, the surface is honestly elliptic with Kodaira dimension 1. We compute the algebraic Brauer groups of these surfaces by studying the action of Gal(Q/Q) on their Neron-Severi groups. Following the work of Gvirtz, Loughran, and Nakahara [GLN22], we find bounds for the exponents of transcendental Brauer groups of these surfaces. The transcendental Brauer group is closely related to the transcendental lattice. The argument begins with an explicit description of the basis of the respective transcendental lattices and reinterpreting elements of these lattices as elements in rings of integers. From this, we bound the transcendental Brauer group. These bounds apply more generally to the surfaces given by y2 = x3 + A1t6m + A2 for Ai ∈ Z and m = 2, 3

## 2022 Ph.D Thesis Defenses

**Austen James**

**Title:** A Bayesian Approach to Computing Brauer Groups of Cubic Surfaces

**Thesis Advisor: ** Tony Varilly-Alvarado

We present an algorithm for computing Brauer groups of cubic surfaces. The algorithm takes as input an equation f (x, y, z, w) = 0 for a cubic surface X over Q and a confidence threshold 0.5 < ρ < 1, and outputs the Brauer group of X, Br X/ Br Q and a confidence level ψ > ρ for the result. The algorithm runs by sampling lifts of Frobenius at many primes of good reduction and relies on Chebotarev’s density theorem and Bayesian inference to produce, with confidence ψ > ρ, a subgroup of W (E6). This subgroup represents the action of Galois on the geometric Picard group of X, from which we compute Br X/ Br Q. We give a description of this algorithm and a proof that it terminates, as well as an implementation in Magma. We also examine the speed of such an approach relative to existing methods and explore how the Bayesian technique of this algorithm can be applied to answer questions concerning the Galois and Brauer groups of other classes of surfaces.

**Yikai Chen**

**Title:** Mathematical Results for Michell Trusses

**Thesis Advisor: **Robert Hardt

Given an equilibrated vector force system $\mathbf{F}$ of finite mass and bounded support, we investigate the possibility and properties of a cost minimizing structure of given materials that balances $\mathbf{F}$. Our work generalizes and reinterprets results of Michell’s paper in 1904 and Gangbo’s recent work where the given equilibrated force system occurs on a finite set of points and the balancing structure consists of finitely many stressed bars joining these points. Such a bar corresponds to an interval $[a,b] \subset \mathbb{R}^n$ having a multiplicity $\lambda \in \mathbb{R}$ where $|\lambda|$ indicates the stress density on the bar and $\sgn(\lambda)$ indicates whether it is being compressed or extended. While there exists a finite bar system to balance any given equilibrated finite force system, Michell already observed that a finite cost-minimizing one may not exist. In this thesis, we introduce two new mathematical representations of Michell trusses based on one-dimensional finite mass varifolds and flat $\R^n$-chains. Here one may use a one-dimensional signed varifold to model the balancing structure so that the internal force of the positive (or compressed) part coincides with its first variation while the internal force of the negative (or extended) part coincides with its negative first variation. For the chain model, we use the subspace of structural flat $\mathbb{R}^n$ chains in which the coefficient vectors are a.e. co-linear with the orientation vectors. The net force then becomes simply the $\mathbb{R}^n$ chain boundary and so cost-minimization becomes precisely the mass-minimizing Plateau problem for structural chains. For either model, a known compactness theorem leads to existence of optimal cost-minimizers as well as time-continuous cost-decreasing flows.

**Giorgio Young**

**Title:** Some results on the spectral theory of one-dimensional operators and associated problems

**Thesis Advisor: **Milivoje Lukić

This thesis discusses results in the area of spectral theory of Schrödinger operators, and their discrete analogs, Jacobi matrices, as well as some closely associated problems. The first result we present relates to the quantum dynamics generated by a particular class of almost periodic Schrödinger operators. We show that the dynamics generated by Schrödinger operators whose potentials are approximated exponentially quickly by a periodic sequence exhibit a strong form of ballistic transport. The second result exploits the connection between the KdV hierarchy and one-dimensional Schrödinger operators to prove a uniqueness result for the KdV hierarchy with reflectionless initial data via inverse spectral theoretic techniques. The third and fourth results concern orthogonal and Chebyshev rational functions with poles on the extended real line. In the process of extending some of the existing theory for polynomials and exploring some of the new phenomena that arise, we present a proof of a conjecture of Barry Simon’s. This thesis contains joint work with Benjamin Eichinger and Milivoje Lukić.

**Nicholas Rouse**

**Title:** On a conjecture of Chinburg-Reid-Stover

**Thesis Advisor: **Alan Reid

We study a conjecture of Chinburg-Reid-Stover about ramification sets of quaternion algebras associated to hyperbolic 3-orbifolds obtained by (d,0) Dehn surgery on hyperbolic knot complements in S^3. For a sporadic example and an infinite family, we prove that the set of rational primes p such that there is some d such that the quaternion algebra associated to the (d,0) surgery is ramified at some prime ideal above p is infinite. This behavior is governed by the Alexander polynomial of the knot, and we investigate its connection to reducible representations on the canonical component of the character variety and the failure of a certain function field quaternion algebra to extend to an Azumaya algebra over the canonical component. We further provide a more general framework for finding such examples that one may use to recover the infinite family.

**Stephen Wolff**

**Title:** The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces

**Thesis Advisor: **Anthony Várilly-Alvarado

In this thesis we study the inverse Galois problem for del Pezzo surfaces of degree one and for algebraic K3 surfaces. We begin with an overview of how the question of the existence of k-points on a nice k-variety leads, via Brauer groups, to the inverse Galois problem. We then discuss an algorithm to compute all finite subgroups of the general linear group GL(n,Z) up to conjugacy. The first cohomologies of these subgroups are a superset of the target groups of the inverse Galois problem for any family of nice k-varieties whose geometric Picard group is free and of finite rank. We apply these results to algebraic K3 surfaces defined over the rational numbers, providing explicit equations for a surface solving the only nontrivial instance of the inverse Galois problem in geometric Picard rank two. Next we study representatives from three families of del Pezzo surfaces of degree one, searching for 5-torsion in the Brauer group. For two of the three surfaces, we show that the Brauer group is trivial when the surface defined over the rational numbers, but becomes isomorphic to Z/5Z or (Z/5Z)^2 when the base field is raised to a suitable number field. For the third surface, we show that its splitting field has degree 2400 as an extension of the rational numbers, a degree consistent with 5-torsion in the Brauer group.

**Gilliam Stagner**

**Title:** Filling links and minimal surfaces in 3-manifolds. **William Stagner**

**Thesis Advisor: **Alan Reid

This thesis studies this existence of filling links 3-manifolds. A link L in a 3-manifold M is filling in M if, for any spine G of M disjoint from L, \pi_1(G) injects into \pi_1(M - L ). Conceptually, a filling link cuts through all of the the topology 3-manifold. These links were first studied by Freedman-Krushkal in the concrete case of the 3-torus M = T^3, but they leave open the question of whether a filling link actually exists in T^3. We answer this question affirmatively by proving in fact that every closed, orientable 3-manifold M with fundamental group of rank 3 contains a filling link.

**Leonardo S. Digiosia**

**Title:** Cylindrical contact homology of links of simple singularities

**Thesis Advisor: **Joanna Nelson

In this talk we consider the links of simple singularities, which are contactomoprhic to S^3/G for finite subgroups G of SU(2,C). We explain how to compute the cylindrical contact homology of S^3/G by means of perturbing the canonical contact form by a Morse function that is invariant under the corresponding rotation subgroup. We prove that the ranks are given in terms of the number of conjugacy classes of G, demonstrating a form of the McKay correspondence. We also explain how our computation realizes the Seifert fiber structure of these links.

**Shawn Williams**

**Title:** Extensions of the Fox-Milnor Condition

**Thesis Advisor: **Shelly Harvey

The search for slice knots is an important task in low dimensional topology. In the 1960s, Fox and Milnor proved a theorem stating that the Alexander polynomial of a slice knot satisfies a special factorization. A decade later, Kawauchi extended this theorem for the multivariable Alexander polynomial of slice links. This factorization, known as the Fox-Milnor condition, has been used and generalized many times as an obstruction to a link being slice. In this defense, we will see two more extensions of this condition, first to the multivariable Alexander polynomial of 1-solvable links, and then for the first order Alexander polynomial of ribbon knots.