In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik’s conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these products.
@article{celik2024observationsgeometrycurvesblaschkeproducts,title={Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them},author={Celik, Mehmet and Duigin, Mathis and Guo, Jia and Luo, Dianlun and Spinelli, Kamryn and Zeytuncu, Yunus and Zhu, Zhuoyu},issn={2195-3724},url={http://dx.doi.org/10.1007/s40315-025-00579-2},doi={10.1007/s40315-025-00579-2},journal={Computational Methods and Function Theory},publisher={Springer Science and Business Media LLC},year={2025},note={Authored with undergraduates. Accepted to Computational Methods and Function Theory.},}
Approximation of L-functions associated to Hecke cusp eigenforms
We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function and point out the role of an equidistributional notion in ensuring the approximation is well-defined, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives, such as the locations and orders of zeros. Together with the Euler product expansion of the L-function, the family of approximations also encodes some of the key features of the L-function such as its functional equation. As an example, we apply this method to the L-function of the modular discriminant and demonstrate that the approximation successfully locates zeros of the L-function on the critical line. Finally, we derive via Mellin transforms a convolution-type formula which leads to precise error bounds in terms of the incomplete gamma function. This formula can be interpreted as an alternative definition for the approximation and sheds light on Matiyasevich’s procedure.
@article{HUANG202560,title={Approximation of L-functions associated to Hecke cusp eigenforms},journal={Journal of Number Theory},volume={272},pages={60-84},year={2025},issn={0022-314X},doi={https://doi.org/10.1016/j.jnt.2025.01.014},url={https://www.sciencedirect.com/science/article/pii/S0022314X25000381},author={Huang, An and Spinelli, Kamryn},keywords={Hecke eigenforms, L-function approximation, Mellin transforms},archiveprefix={arXiv},primaryclass={math.NT}}
2022
CR embeddability of quotients of the Rossi sphere via spectral theory
Henry
Bosch, Tyler
Gonzales, Kamryn
Spinelli, and
2 more authors
We look at the action of finite subgroups of SU(2) on S^3, viewed as a CR manifold, both with the standard CR structure as the unit sphere in \mathbbC^2 and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of SU(2) to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.
@article{bosch2021crembeddabilityquotientsrossi,title={CR embeddability of quotients of the Rossi sphere via spectral theory},author={Bosch, Henry and Gonzales, Tyler and Spinelli, Kamryn and Udell, Gabe and Zeytuncu, Yunus E.},year={2022},archiveprefix={arXiv},primaryclass={math.CV},journal={International Journal of Mathematics},volume={33},number={02},pages={2250014},doi={10.1142/S0129167X22500148},url={https://doi.org/10.1142/S0129167X22500148},note={Published as an undergraduate.}}
A Tauberian Approach to Weyl’s Law for the Kohn Laplacian on Spheres
Henry
Bosch, Tyler
Gonzales, Kamryn
Spinelli, and
2 more authors
We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral.
@article{bosch2020tauberianapproachweylslaw,title={A Tauberian Approach to Weyl's Law for the Kohn Laplacian on Spheres},author={Bosch, Henry and Gonzales, Tyler and Spinelli, Kamryn and Udell, Gabe and Zeytuncu, Yunus E.},archiveprefix={arXiv},primaryclass={math.CV},url={https://arxiv.org/abs/2010.04568},volume={65},doi={10.4153/S0008439521000163},number={1},journal={Canadian Mathematical Bulletin},year={2022},pages={134–154},note={Published as an undergraduate.}}
2021
Manifolds with bounded integral curvature and no positive eigenvalue lower bounds
Connor C.
Anderson, Xavier Ramos
Olivé, and Kamryn
Spinelli
We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on L^p norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian \lambda_1. This example shows that the assumption of smallness of the L^p norm of the curvature is a necessary condition to derive Lichnerowicz and Zhong-Yang type estimates under integral curvature conditions.
@article{anderson2021manifoldsboundedintegralcurvature,title={Manifolds with bounded integral curvature and no positive eigenvalue lower bounds},author={Anderson, Connor C. and Olivé, Xavier Ramos and Spinelli, Kamryn},year={2021},archiveprefix={arXiv},primaryclass={math.DG},url={https://arxiv.org/abs/2103.11970},journal={PUMP Journal of Undergraduate Research},number={4},pages={222-235},note={Published as an undergraduate.}}