2022 Ph.D Thesis Defenses

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 H. 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.

William Stagner

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

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