Description: 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 $SL_n$.

Description: 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.

Description: 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.

Description: Proves that over the base-$b$ max-min semiring that 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.
Journal Reference Journal of Integer Sequences 28, 25.2.7. 2025.

Description: Inspired by our previous paper Unexpected Biases between Congruence Classes for Parts in $k$-indivisible Partitions (see above), 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.
Journal Reference: Journal of Number Theory 261, 299-311. https://doi.org/10.1016/j.jnt.2024.02.003 2024.

Description: Studies the Erdos distinct angle problem and variants on finding the minimum number of distinct angles between nnn points in the plane, which were previously unstudied despite the ubiquitousness of the Erdos distinct distance problem.
Journal Reference: Discrete and Computational Geometry. https://doi.org/10.1007/s00454-022-00466-w . 2023.

Description: Defines and studies the Bergman Game, which decomposes positive integers into their base-$\varphi$ expansions.
Journal Reference: The Fibonacci Quarterly (Proceedings of the 20th Conference), 60.5, 18-38. https://www.fq.math.ca/Papers1/60-5/baily.pdf . 2022.

Description: 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.
Journal Reference: The PUMP Journal of Undergraduate Research, Volume 5, 122-147. 2022.

Description: 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. However, 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 $k < \frac{6(t^2-1)}{\pi^2}$.
Journal Reference: Journal of Number Theory. 248, 310-342. https://doi.org/10.1016/j.jnt.2023.01.006 2023.

Description: 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 $r < s$.

Description: 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 nnn is the number of tokens in the initial state.

Description: 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 $\beta_d$.

Description: 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.