Papers and Presentations


Links to Papers

The Weil bound for generalized Kloosterman sums of half-integral weight

Joint work with Nickolas Andersen and Gradin Anderson. Submitted.

Abstract: Let L be an even lattice of odd rank with discriminant group L′/L, and let α,β∈ L′/L. We prove the Weil bound for the Kloosterman sums Sα,β(m,n,c) of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen's identity for plus space Kloosterman sums with the theta multiplier system.

Available here.

Applying the Argument Principle to Count Zeros of Harmonic Functions with Poles

Joint work with Jennifer Brooks, Mary Jenkins, Alexander Lee, and Clay Whiffen. Submitted.

Abstract: Several recent papers investigate the way in which the number of zeros of a complex-valued harmonic function can depend on its coefficients by analyzing specific simple families. These families share two features: (1) the critical curve separating the sense-preserving and sense-reversing regions is a circle, and (2) the image of that circle is a well-understood parametric curve. In such cases, the harmonic analogue of the Argument Principle can be applied to count the zeros. In this paper, we illustrate the strengths and limitations of these techniques; we construct a new family of complex-valued harmonic functions with poles having some of these features. We obtain detailed zero-counting theorems for two subfamilies, and we illustrate how to obtain less detailed zero-counting theorems for the general family.

No permanent online link is currently available, but the PDF is available by request through email.

Zeros of a Family of Complex-Valued Harmonic Trinomials

Joint work with Jennifer Brooks, Michael Dorff, Alexandra Hudson, Erin Pitts, and Clay Whiffen. Bulletin of the Malaysian Mathematical Sciences Society, 2022.

Abstract: It is known that complex-valued harmonic polynomials of degree n can have more than n zeros. In Brilleslyper et. al., the authors consider a one-parameter family of complex-valued harmonic polynomials and determine, for different values of the real parameter, the number of zeros. We consider the same family but allow the parameter to be complex. As in the previous paper, our proof relies on the Argument Principle for Harmonic Functions and again requires us to find the winding number about the origin of a hypocycloid. The geometry is more complicated in this case, however. This additional complexity is reflected in the main theorem, which shows that the number of transitions in the number of zeros and the nature of these transitions depends on the argument of the complex parameter.

Available here.

Stieltjes interlacing of the zeros of jn

Joint work with William Frendreiss, Jennifer Gao, Austin Lei, Hui Xue, and Daozhou Zhu. Part of the 2021 Clemson University REU. Canadian Mathematical Bulletin, 2022.

Abstract: Let jn be the modular function obtained by applying the nth Hecke operator on the classical j-invariant. For n > m ≥ 2, we prove that between any two zeros of jm on the unit circle of the fundamental domain, there is a zero of jn.

Available here.

Interlacing Properties for Zeros of A Family of Modular Forms

Joint work with William Frendreiss, Jennifer Gao, Austin Lei, Hui Xue, and Daozhou Zhu. Part of the 2021 Clemson University REU. International Journal of Number Theory, 2023.

Abstract: Getz presented a family of level one modular forms fk for which all zeros lie on the unit circle in the fundamental domain. Expanding on work from Nozaki, Griffin et al., and Saha and Saradha, we show that the nonelliptic zeros of these fk satisfy two interlacing properties: standard interlacing, where the zeros of fk and fk+a alternate if and only if a ϵ {2,4,6,8,12} for sufficiently large k; and Stieltjes interlacing, where for l > k large enough, between any two zeros of fk, there is a zero of fl.

Available here.

A Stieltjes separation property of zeros of Eisenstein series

Joint work with William Frendreiss, Jennifer Gao, Austin Lei, Hui Xue, and Daozhou Zhu. Part of the 2021 Clemson University REU. Kyushu Journal of Mathematics, 2022.

Abstract: For k < l, let Ek(z) and El(z)be Eisenstein series of weights k and l respectively for SL2(z). We prove that between any two zeros of Ek(e) there is a zero of El(e) on the interval π/2 < θ < 2π/3.

Available here.

Cuspidal projections of products of Eisenstein series

Joint work with William Frendreiss, Jennifer Gao, Austin Lei, Hui Xue, and Daozhou Zhu. Part of the 2021 Clemson University REU. Journal of Number Theory, 2021.

Abstract: We show that the projection of a product of two or three Eisenstein series of level one onto the cuspidal subspace is not an eigenform unless the dimension of the cuspidal subspace is one.

Available here. Code referenced in the paper available at my github.

Non-convex geometry of numbers and continued fractions

Joint work with Nickolas Andersen, William Duke, and Zach Hacking. Functiones et Approximatio, 2023.

Abstract: In recent work, the first two authors constructed a generalized continued fraction called the p-continued fraction, characterized by the property that its convergents (a subsequence of the regular convergents) are best approximations with respect to the Lp norm, where p ≥ 1. We extend this construction to the region 0 < p < 1, where now the Lp quasinorm is non-convex. We prove that the approximation coefficients of the p-continued fraction are bounded above by 1/√5 + εp, where εp → 0 as p → 0. In light of Hurwitz's theorem, this upper bound is sharp, in the limit. We also measure the maximum number of consecutive regular convergents that are skipped by the p-continued fraction.

Available here for free or here for the journal version.


Presentations

  • Generalizing the Partition Number Formula to Mock Theta Functions, Part 1
    • Student Research Conference, Brigham Young University, February 2023 (session winner)
  • Zeros of a Family of Harmonic Functions with Poles, Part II
    • Student Research Conference, Brigham Young University, February 2023 (session winner)
  • Cuspidal projections of products of Eisenstein series
    • Young Mathematicians Conference, Ohio State University, August 2022
    • 34th Automorphic Forms Workshop, Brigham Young University (remote), March 2022
    • MATH REU Conference, Clemson University (remote), with William Frendreiss, July 2021
  • A Generalization of the Epstein Zeta Function to p-norms
    • Student Research Conference, Brigham Young University, March 2022
  • Zeros of a Family of Complex-valued Harmonic Polynomials, Part 2
    • Student Research Conference, Brigham Young University, March 2022
  • Non-Convex Geometry of Numbers and Continued Fractions
    • Student Research Conference, Brigham Young University, with Zach Hacking, February 2021 (session winner)
  • Zeros of Complex Harmonic Polynomials, Part 3
    • Student Research Conference, Brigham Young University, with Daniel South, February 2021
  • Shortcuts in Diophantine Approximation, Parts 1 and 2
    • Student Research Conference, Brigham Young University, with Zach Hacking, February 2020