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Grassmannian arrow pencils and Hasse-Witt invariants

Mirror symmetry is a duality arising in physics that relates distinct families of algebraic varieties. We analyze the geometry and arithmetic of highly symmetric Calabi-Yau hypersurface pencils in Grassmannian varieties and contrast their properties to classical mirror constructions associated to hypergeometric [Read More...]

Presenter: Ursula Whitcher, American Mathematical Society
Authors: Adriana Salerno, Ursula Whitcher, Chenglong Yu
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: September 30, 2023; 2:00 pm

Hook length biases and general linear partition inequalities

Connections between representation theory and the theory of integer partitions are well-known, and hook lengths of partitions sometimes play important roles in this context. Partly motivated by these connections, we consider the total number of hooks of fixed length in odd versus distinct partitions. We establish bias results, and show that there are more [Read More...]

Presenter: Amanda Folsom, Amherst College
Authors: Cristina Ballantine (College of the Holy Cross), Hannah Burson (University of Minnesota), William Craig (University of Cologne), Amanda Folsom (Amherst College), Boya Wen (University of Wisconsin).
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: September 30, 2023; 2:25 pm

Asymptotic Results on Subrings in Z^n of Corank at most k

The field of subgroup growth has become an active field in number theory. In this talk, we consider subring growth in the ring $\mathbb{Z}^n$. While the exact growth rate of the number of subgroups in $\mathbb{Z}^n$ is known, there is not even a conjecture about what the growth rate of the number of subrings in $\mathbb{Z}^n$ should be, though several [Read More...]

Presenter: Kelly Isham, Colgate University
Authors: Kelly Isham, Nathan Kaplan
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: September 30, 2023; 2:50 pm

Large Sums of Fourier Coefficients of Cusp Forms

It is a classical problem to consider the asymptotic behavior of the Fourier coefficients of modular forms and their hidden structures. One of the common approaches is to study the asymptotic of their summatory functions. In this talk, we establish an upper bound for the sum $\sum_{n\leq x} \lambda_f(n)$ of the Fourier coefficients of primitive cusp forms [Read More...]

Presenter: Naomi Tanabe, Bowdoin College
Authors: Claire Frechette, Mathilde Gerbelli-Gauthier, Alia Hamieh, Naomi Tanabe
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: September 30, 2023; 3:15 pm

Spectral Theory of Automorphic Forms Applied to String Scattering Amplitudes

In string theory, elementary particles are represented by vibrational modes of a string. Strings interact by various joining and splitting processes, and the probabilities that certain scattering processes occur are given by string scattering amplitudes. When computing a low-energy expansion of these string scattering amplitudes, coefficient functions arise [Read More...]

Presenter: Holley Friedlander, Dickinson College
Authors: Maryam Khaqan, Kim Klinger-Logan, Manish Pandey, Runqiu Xu
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: September 30, 2023; 3:40 pm

Wallpaper group structure of origami constructions

Origami is the art of folding paper into various patterns without cutting or tearing the paper. By viewing the paper as the complex plane, we record all intersection points to construct mathematical origami sets. Additionally, we include the various lines through the intersection points to create a repeating pattern that can be viewed as a wallpaper group. [Read More...]

Presenter: Sara Chari, St. Mary's College of Maryland
Authors: Sara Chari; Quinn Macauley
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 1, 2023; 9:45 am

Dynamic irreducibility over finite fields

We can ask whether a polynomial is irreducible over a particular field. If we consider the iteration of this polynomial (that is composing the polynomial with itself) we can explore if it remains irreducible over that base field. This phenomenon is called dynamical irreducibility or stability. We will explore what tools we have to discuss irreducibility over [Read More...]

Presenter: Bianca Thompson, Westminster University
Authors: Jamie Juul, Bella Tobin, Swati, Cigole Thomas, Tori Day, Rebecca DeLand, Malike Conteh, Michaela Fitzgerald, Sarah Szafranski, and Jasmine Camero
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 1, 2023; 10:10 am

Totally p-adic Numbers of Small Height

Let p be prime, and denote by Qp the field of p-adic numbers. We say that a number is totally p-adic if its minimal polynomial splits completely over Qp. It is certain that for any particular prime p and degree d, there is a smallest non trivial height of algebraic numbers that are totally p-adic and of degree d. When do we know what that smallest height is? [Read More...]

Presenter: Emerald Tatiana Stacy, Washington College
Authors: Emerald Tatiana Stacy
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 1, 2023; 10:35 am

Tantalizing Adventures with my Undergraduate Research Students on Numerous Numerical Sequences

This talk will focus on the research projects conducted with my undergraduate students on numerical sequences. My introduction to research in this area began in June 2018, when I stumbled on a YouTube video by a famous astrologer, named David Cochrane, and he made some number theoretic connections between the Fibonacci sequence and astrology. Cochrane is no [Read More...]

Presenter: aBa Mbirika, University of Wisconsin-Eau Claire
Authors: aBa Mbirika, Janee Schrader, and J"urgen Spilker
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 1, 2023; 11:00 am

An Exploration of Degeneracy in Abelian Varieties of Fermat Type

The term 'degenerate' is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this talk we will examine [Read More...]

Presenter: Heidi Goodson, Brooklyn College, City University of New York
Authors: Heidi Goodson
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 2, 2023; 8:30 am

How do points on plane curves generate fields? Let me count the ways.

In their program on diophantine stability, Mazur and Rubin suggest studying a curve $C$ over $\mathbb{Q}$ by understanding the field extensions of generated by a single point of $C$; in particular, they ask to what extent the set of such field extensions determines the curve . A natural question in arithmetic statistics along these lines concerns the size [Read More...]

Presenter: Renee Bell, Lehman College
Authors: Michael Allen, Robert J. Lemke Oliver, Allechar Serrano Lopez, Tian An Wong
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 2, 2023; 8:55 am

Good isogeny classes of elliptic curves

A rational elliptic curve $E$ is said to be good if $N_{E}^{6}<\max\{|c_{4}^{3}|,c_{6}^{2}\}$, where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. It is known that for each of the fifteen torsion subgroups $T$ occurring over $\mathbb{Q}$, there are infinitely many good elliptic [Read More...]

Presenter: Alexander Barrios, University of St. Thomas
Authors: Elise Alvarez-Salazar, Alexander Barrios, Calvin Henaku, Summer Soller
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 2, 2023; 9:20 am

Computing supersingular endomorphism rings using inseparable endomorphisms

We will discuss an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve E. We show that two calls to this algorithm compute a basis for a Bass suborder of the endomorphism ring of [Read More...]

Presenter: Jenny G. Fuselier, High Point University
Authors: Jenny Fuselier, Annamaria Iezzi, Mark Kozek, Travis Morrison, and Changningphaabi Namoijam
Symposium Year: 2023
Session: Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]
Presentation Time: October 2, 2023; 9:45 am

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