Fall 2022/Spring 2023: I am co-organizing the Combinatorics student seminar with John Lentfer.

I am also co-organizing the Combinatorics research seminar with Andrés R. Vindas Meléndez, Christopher Ryba, and John Lentfer.

Please click on the paper title to show or hide the abstract.

• Generalized parking function polytopes (with John Lentfer and Andrés R. Vindas Meléndez). arXiv:2212.06885 [math.CO] (2022).

  Abstract: A classical parking function of length \(n\) is a list of positive integers \((a_1, a_2, \ldots, a_n)\) whose nondecreasing rearrangement \(b_1 \leq b_2 \leq \cdots \leq b_n\) satisfies \(b_i \leq i\). The convex hull of all parking functions of length \(n\) is an \(n\)-dimensional polytope in \(\mathbb{R}^n\), which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of \(\mathbf{x}\)-parking functions for \(\mathbf{x}=(a,b,\dots,b)\), which refer to as \(\mathbf{x}\)-parking function polytopes. We explore connections between these \(\mathbf{x}\)-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of \(\mathbf{x}\)-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain the first closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

Please click on the paper title to show or hide the abstract.

• Fourier Coefficients of Level 1 Hecke Eigenforms (with Rachana Madhukara), Acta Arith. 200, No. 4, 371--388 (2021). arXiv:2007.08683[math.CO].

  Abstract: Lehmer's 1947 conjecture on whether \(\tau(n)\) vanishes is still unresolved. In this context, it is natural to consider variants of Lehmer's conjecture. We determine many integers that cannot be values of \(\tau(n)\). For example, among the odd numbers \(\alpha\) such that \(|\alpha| \leq 99\), we determine that \( \tau(n) \not\in \{-9, \pm 15, \pm 21, -25, -27, -33, \pm 35, \pm 45, \pm 49, -55, \pm 63, \pm 77, -81, \pm 91 \}. \) Moreover, under GRH, we have that \(\tau(n) \neq -|\alpha|\) and that \(\tau(n) \notin \{9,25,27,39,75,81\}.\) We also consider the level 1 Hecke eigenforms in dimension 1 spaces of cusp forms. For example, for \(\Delta E_4 = \sum_{n = 1}^{\infty} \tau_{16}(n)q^n\), we show that \( \tau_{16}(n) \not\in \{\pm \ell: 1\leq \ell \leq 99, \ell \text{ is odd}, \ell \neq 33,55,59,67,73,83,89,91\} \cup \{-33,-55,-59,-67,-89,-91\}. \) Furthermore, we implement congruences given by Swinnerton-Dyer to rule out additional large primes which divide numerators of specific Bernoulli numbers. To obtain these results, we make use of the theory of Lucas sequences, methods for solving high degree Thue equations, Barros' algorithm for solving hyperelliptic equations, and the theory of continued fractions.

(Summer 2022) I am teaching Math 54.

(Spring 2022) I am a GSI for Math 54.