2025 Ph.D Thesis Defenses

Alex Nolte

Title: Some Geometric Aspects of Higher-Rank Teichmuller Theory
Thesis Advisor: Mike Wolf

We study qualitative aspects of the geometry of Hitchin representations of surface groups in PSL_n(R). The thematic core of this thesis is on Thurston-Klein geometric structures with distinguished foliations that are associated to Hitchin representations in SL₃(R) and PSL₄(R). This direction was initiated by Guichard-Wienhard’s work on PSL₄(R) Hitchin representations. We establish the following results on these foliated structures. First, we prove a rigidity theorem for the projective geometry of leaves of Guichard-Wienhard’s codimension-1 foliation associated to a PSL₄(R)-Hitchin representation.We then classify all foliations of domains of discontinuity in RP³ for PSL₄(R)-Hitchin representations that are geometrically similar to those studied by Guichard and Wienhard. Finally, we give a parallel framework to Guichard-Wienhard’s work in PSL₄(R) for geometric structures modeled on the space of full flags in R³ for SL₃(R)-Hitchin representations. We give a number of applications of our results and the surrounding circle of ideas. First, we resolve a question asked by Benz´ecri in 1960 on the point-set topology of the space C(RPⁿ) of projective equivalence classes of properly convex domains in RPⁿ for n ≥ 2. Next, we give explicit geometric constructions of flows constructed in the dynamical study of Hitchin representations. Finally, we construct asymmetric metrics on PSL_n(R)-Hitchin components for n > 3 and give a new formulation of Thurston’s asymmetric metric on Teichmu¨ller space. Finally, we study degree-n complex structures in the sense of Fock and Thomas. These are extensions of complex structures on surfaces whose deformation space T ⁿ(S) is conjectured to be canonically homeomorphic to the PSL_n(R) Hitchin component. Our main result in this direction is the construction of a canonical homeo-morphism from Fock-Thomas spaces T ⁿ(S) of higher complex structures to a bundle Bⁿ(S) of harmonic tensors over Teichmu¨ller space.

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Eliot Bongiovanni

Title: Extensions of Finitely Generated Veech Groups
Thesis Advisor: Christopher Leinginger

Given a closed surface S with finitely generated Veech group G and its π₁ (S)-extension Γ, there exists a hyperbolic space Eˆ on which Γ acts isometrically and cocompactly. The space Eˆ is obtained by collapsing some regions of the surface bundle over the convex hull of the limit set of G. Using the nice action of Γ on the hyperbolic space Eˆ, it is shown that Γ is hierarchically hyperbolic. These are generalizations of [Dow+23; Dow+24], which assume in addition that G is a lattice. Because finitely generated Veech groups are among the most basic examples of subgroups of mapping class groups which are expected to qualify as geometrically finite, this result is evidence for the development of a broader theory of geometric finiteness.

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Zhiyi Zhang

Title: Degeneration of Flat Metrics from k-Differentials
Thesis Advisor: Christopher Leinginger

This thesis explores the space of flat metrics sitting in the space of geodesic currents. The space of geodesic currents, considered as the completion of weighted closed curves, was first introduced by Bonahon to study hyperbolic 3–manifolds, and then used by him to provide an alternative description of Thurston’s compactification of Teichmu¨ller space. This space has since played a key role in the study of geometric structures on surfaces and their degenerations. We focus on the space of flat metrics from k–differentials on finite–type surfaces and its embedding into the space of geodesic currents. With this embedding, we describe a compactification of the space of flat metrics coming from k–differentials, for all k, where the boundary points are the mixed structures—currents that are a flat metric on a subsurface and a measured lamination on the complement. This generalizes the results of Duchin–Leininger–Rafi for the case when k = 2 and Ouyang–Tamburelli for the case when k = 3, 4.

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Ben Savoie

Title: A Graph-Theoretic Approach to Computing Ranks of Elliptic Curves over ℚ(i)
Thesis Advisor: Brandon Levin

Let K be a finite unramified extension of Q_p with p ≥ 5. In the first part of this thesis, we study the local geometry of the irreducible components in the reduced part of the Emerton–Gee stack for GL₂, which serves as a moduli space for two-dimensional mod p representations of Gal(K/K). We determine precisely which irreducible com- ponents are smooth, which are normal, and which have Gorenstein normalizations. We prove that the normalizations of these components admit smooth–local covers by Cohen-Macaulay and resolution-rational varieties, which are generally not Gorenstein. Finally, we determine the singular loci in the components, providing insights which up- date expectations about the conjectural categorical p–adic Langlands correspondence.

In the second part of this thesis, we introduce a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve E_b : y² = x³ + bx defined over Q(i), where b ∈ Z[i] and φ is a degree 2 isogeny of E_b. We begin by associating a weighted graph G_b to each curve E_b, whose vertices correspond to the odd Gaussian primes dividing b. The weights on the edges connecting these vertices are determined by the quartic residue symbols between these primes. We then establish a bijection between the elements of the φ-Selmer group of E_b and certain partitions of the graph G_b. This correspondence provides a linear-algebraic interpretation of the φ-Selmer group through the Laplacian matrix of G_b.

Using our algorithm, we explicitly construct several subfamilies of elliptic curves E_b over Q(i) with trivial Mordell–Weil rank. Furthermore, by combining our method with Tao’s Constellation Theorem for Gaussian primes, we prove the existence of infinitely many elliptic curves E_b over Q(i) with rank exactly 2. Additionally, we show that for each pair of rational twin primes (p, q), the curve E_pq considered over Q(i) has rank either 2 or 4, with the rank exactly 2 when p ≡ 5 mod 8. Lastly, we show that for each rational prime of the form p = a² + c⁴ (of which there are infinitely many), the elliptic curve E_−p over Q(i) has rank either 2 or 4, with rank exactly 2 if p ≡ 5 or 9 mod 16.

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Iris Emilsdottir

Title: Mind the Gap: Schwartzman Groups of Dynamically Defined Operators and Relater Problems
Thesis Advisor: David Damanik

The spectral properties of Schrödinger operators with ergodic potentials lie at the intersection of dynamical systems and spectral theory. These operators are fundamental in understanding quantum systems in disordered or aperiodic media.

This thesis focuses on gap labeling, a method for identifying admissible spectral gaps based solely on the underlying dynamics of the operator. We establish the sets of possible gap labels predicted by gap labeling theorems for various dynamical systems, including affine automorphisms of compact connected abelian groups, codings of rotations, and quasi-Sturmian subshifts.

For Schrödinger operators with potentials generated by the full shift, we analyze whether all labels allowed by the gap labeling theorem correspond to actual spectral gaps and identify potential obstructions to their realization. In contrast, for codings of rotations, we prove that every predicted gap is realized, extending known results for Sturmian subshifts.

This thesis contains joint work with David Damanik and Jake Fillman [26, 25].

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Jacob Kesten

Title: Division Algebras and Extended Frobenious Structures in Monoidal Categories
Thesis Advisor: Chelsea Walton

Due to the wide range of applications in logic, programming, and quantum physics, adapting algebraic objects to the monoidal setting has become an active area of current inquiry.

This thesis adds to this field of categorical algebra by exploring generalizations of division algebras and extended Frobenius algebras in monoidal categories. Division algebras were first introduced to the categorical setting in attempts to generalize structure results from classical algebra. Extended Frobenius algebras were introduced by Turaev and Turner in 2006 as a way to extend the correspondence between oriented 2-dimensional topological quantum field theories and commutative Frobenius algebras to the unoriented case. In this thesis, we explore the monoidal analogues of these objects. Concerning division algebras, we are especially interested in determining how analogues of the equivalent definitions of division algebras over a field relate in a variety of monoidal settings. We also find categorical and functorial constructions that interact well with division algebras and extended Frobenius algebras, and we use these constructions to produce examples.

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