These are all of my preprints and published articles. They are listed with the most recent first, and all of the research done during my undergraduate is at the end in its own section.
Click Abstract under any entry to expand it.
Papers
Universal braids for elliptic fibrations: from character varieties to Coxeter's factor groups
Abstract
Let $\pi : M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $\Delta \subseteq B$. We study the universal liftable braids for $\pi$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B, \Delta)$. When $B = S^2$, we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the $SL_2$-character variety for $(S^2, \Delta)$. When $B = D^2$ we classify when the subgroup of universal braids has finite index in the braid group $B_n = \operatorname{Mod}(D^2, \Delta)$, and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base $B$ associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.Stable cohomology of universal character varieties
Abstract
We study the universal $PGL_n$ character variety over $M_g$ whose fiber over a point $[C]$ is the space of $PGL_n$-local systems on the curve $C$. We use nonabelian Hodge theory and properties of Saito's mixed Hodge modules to show that the Leray-Serre spectral sequence for the projection to $M_g$ degenerates at $E_2$. As an application, we prove that the rational cohomology of these varieties stabilizes as $g$ goes to infinity and compute the stable limit. We also deduce similar results for the universal $G$-character variety over $M_{g,1}$ whose fiber over a punctured curve is the variety of $G$-local systems with fixed central monodromy around the puncture, for $G = GL_n$ or $SL_n$.How large is the braid monodromy of low-genus Lefschetz fibrations?
Abstract
Given a genus $g$ smooth Lefschetz fibration $\pi : M \to S^2$ with singular locus $\Delta \subseteq S^2$, we describe the subgroup $\operatorname{Br}(\pi)$ of the spherical braid group $\operatorname{Mod}(S^2,\Delta)$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\operatorname{Mod}(S^2,\Delta) : \operatorname{Br}(\pi)]$ is infinite. As an application of our methods, we prove that $[\operatorname{Mod}(S^2,\Delta) : \operatorname{Br}(\pi)] = \infty$ when $g = 1$, when $\pi$ is expressible as a self-fiber sum when $g \geq 2$, or when $\pi$ is a holomorphic genus $g = 2$ Lefschetz fibration whose vanishing cycles are nonseparating. In the genus $g = 1$ case, we relate the subgroup $\operatorname{Br}(\pi)$ to the action of $\operatorname{Mod}(S^2,\Delta)$ on the $\operatorname{SL}_2$-character variety for $S^2 \setminus \Delta$ and provide an alternate proof of the first application via recent work of Lam--Landesman--Litt.Torsion in the Braid Monodromy of Elliptic Fibrations
To appear in Algebraic and Geometric Topology (2026). arXiv pdf
Abstract
Given an elliptic fibration $\pi : M \to S^2$ with singular locus $\Delta$, one may associate to $\pi$ the subgroup $Br(\pi)$ of the spherical braid group $Mod(S^2,\Delta)$ consisting of those braids which lift to a fiber-preserving diffeomorphism of $M$. This paper analyzes the torsion elements in $Br(\pi)$, classifying the torsion in order $n = |\Delta|$ up to conjugacy in $Br(\pi)$ and showing there is no torsion of order $n-1$ and $n-2$. We do so by relating $Br(\pi)$ to the Hurwitz action on the $SL_2$ character variety for $(S^2,\Delta)$. These orders are particularly important because of Murasugi's theorem, which classifies the torsion in the spherical braid group.
Undergraduate research
Indecomposability Over the Max-Min Semiring
Journal of Integer Sequences 28, 25.2.7 (2025). pdfjournal
Abstract
Proves that over the base-$b$ max-min semiring almost all polynomials of a given degree are irreducible as the degree tends to $\infty$. Furthermore this paper proves that almost all power series over the max-min semiring are asymptotically irreducible.Parts in k-indivisible Partitions Always Display Biases between Residue Classes
Journal of Number Theory 261, 299-311 (2024). pdf journal
Abstract
Inspired by our previous paper "Unexpected Biases between Congruence Classes for Parts in $k$-indivisible Partitions", we examined the second-order term in the asymptotic for $D_k^\times(r,t;n)$ derived in that paper in more detail. In this paper, we confirm our earlier conjecture that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of $L$-functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing to conclude that there is always a bias towards one congruence class or another modulo $t$ among all parts in $k$-indivisible partitions of $n$ as $n$ becomes large.Unexpected Biases between Congruence Classes for Parts in $k$-indivisible Partitions
Journal of Number Theory 248, 310-342 (2023). pdf journal
Abstract
Using the circle method, we prove an asymptotic for $D_k^\times(r,t;n)$, which we define to be the number of parts among all $k$-indivisible partitions (those where no part is divisible by $k$) of $n$ which are congruent to $r$ mod $t$, when $k,t$ are coprime. We then observe that this implies the parts are asymptotically equidistributed; however, there is a bias in the second order term. Unlike previous biases of this type, the bias is highly unpredictable. We show that the bias reverts to the natural bias towards lower congruence classes for $k > \frac{6(t^2-1)}{\pi^2}$, and we explore the intricate properties of this bias for $k < \frac{6(t^2-1)}{\pi^2}$.Distinct Angle Problems and Variants
Discrete and Computational Geometry (2023). pdf journal
Abstract
Studies the Erdos distinct angle problem and variants on finding the minimum number of distinct angles between $n$ points in the plane, which were previously unstudied despite the ubiquitousness of the Erdos distinct distance problem.The Bergman Game
The Fibonacci Quarterly (Proceedings of the 20th Conference) 60.5, 18-38 (2022). pdfjournal
Abstract
Defines and studies the Bergman Game, which decomposes positive integers into their base-$\varphi$ expansions.Biases among Congruence Classes for Parts in $k$-regular partitions
Abstract
Using the circle method, we prove an asymptotic for $D_k(r,t;n)$, which we define to be the number of parts among all $k$-regular partitions (those whose parts have multiplicity at most $k - 1$) of $n$ which are congruent to $r$ mod $t$. We then observe that this implies the parts are asymptotically equidistributed; however, there is a bias in the second order term towards the lower congruence classes. We make this explicit and show that for $3 \leq k \leq 10, 2 \leq t \leq 10$ that for all $n \geq 1$ we have $D_k(r,t;n) > D_k(s,t;n)$ when $r < s$.Limiting Spectral Distributions of Families of Block Matrix Ensembles
The PUMP Journal of Undergraduate Research 5, 122-147 (2022). pdf
Abstract
Describes a new operation on matrices $\operatorname{swirl}(A, X)$ and its effect on the limiting spectral distribution of an ensemble, as well as providing a novel combinatorial proof that the ensemble of circulant Hankel matrices have the Rayleigh distribution as a limiting spectral measure.The Generalized Bergman Game
Abstract
Defines and studies the Generalized Bergman Game, which decomposes positive integers into their base-$\beta$ expansions, for $\beta$ a root of a non-increasing positive linear recurrence. Proves that the game terminates in $\Theta(n^2)$ time in the worst case, where $n$ is the number of tokens in the initial state.Large Sets are Sumsets
Unpublished draft, 2021. pdf
Abstract
Proves an explicit upper bound of $\left\lvert B\right\rvert - \alpha_d \log \left\lvert B\right\rvert$ on the size of irreducible subsets of a box $B$ in $\mathbb{N}^d$ for some computable constant $\alpha_d$. Furthermore this paper proves that this bound is asymptotically tight by providing a construction of an irreducible subset of any box $B$ in $\mathbb{N}^d$ of size $\left\lvert B\right\rvert - \beta_d \log \left\lvert B\right\rvert$ for some computable constant $\beta_d$.The Complexity of the Zeckendorf Graph Game
Unpublished draft, 2021. pdf
Abstract
Defines the Zeckendorf Graph Game, as well as the Generalized Zeckendorf Graph Game, and shows that the problem of deciding who has a winning strategy from a given initial state is PSPACE-Complete.