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Kamryn Spinelli

Publications and source records attributed to Kamryn Spinelli.

7 recordsLinked to original sources
arXiv2026

Restriction theorems: from orbits and Chevalley to periods and Galois

Using a new approach based on Galois theory, we study subvarieties of complex representations of reductive groups which satisfy restriction properties on their invariant rings and function fields, along the lines of the Chevalley restriction theorem. For a certain well-behaved class of representations, we explicitly parametrize candidates for these restriction properties and explain a technique to understand their deformations in complex families. We also give algebraic and geometric characterizations of the Chevalley restriction property which clarify how this perspective connects back to previous orbit-theoretic approaches. Finally, we utilize these restriction properties to prove explicit formulas for period integrals of some Calabi-Yau families. The key insight is that the restriction property on function fields can be leveraged to locally interpolate between the algebraic and analytic settings. Using this technique, we lift hypergeometric period formulas from subfamilies to obtain novel explicit formulas for periods of Calabi-Yau double covers of projective spaces and elliptic curves in $\mathbb{P}^2$, expressed in terms of invariant functions on their parameter spaces.

math.AG↗
arXiv2025

Inferring Exocytosis Profiles from Cell Shapes Using a Dual-Configuration Model of Walled Cell Tip Growth

Tip growth in filamentous cells, such as root hairs, moss protonemata, and fungal hyphae, depends on coordinated cell wall extension driven by turgor pressure, wall mechanics, and exocytosis. We introduce a dual-configuration model that incorporates both turgid and unturgid states to describe cell wall growth as the combined effect of elastic deformation and irreversible extension. This framework infers exocytosis profiles directly from cell morphology and elastic stretches, formulated as an initial value problem based on the self-similarity condition. Applying the model to Medicago truncatula root hairs, moss Physcomitrium patens protonemata, and hyphoid-like shapes, we find that exocytosis peaks at the tip in tapered cells but shifts to an annular region away from the apex in flatter-tip cells beyond a threshold. The model generalizes previous fluid models and provides a mechanistic link between exocytosis distribution and cell shape, explaining observed variations in tip-growing cells across species.

q-bio.CB↗
arXiv2024

Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them

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.

math.CV↗
arXiv2024

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.

math.NT↗
arXiv2021

Manifolds with bounded integral curvature and no positive eigenvalue lower bounds

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 $λ_1$. 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.

math.DG↗
arXiv2021

CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of $\operatorname{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in $\mathbb{C}^2$ 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 $\operatorname{SU}(2)$ 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.

math.CV↗