# Information

The seminar times/days are TBD. If you’re interested in speaking, please reach out to the organizer.

# Fall 2022 Talks

This semester is jointly organized by D. Zack Garza and Haiyang Wang.

TBD.

# Spring 2022 Talks

This semester was jointly organized by D. Zack Garza and Arvind Suresh.

## 01/25/2022

Speaker: Dino Lorenzini (25m)

## 02/01/2022

Speaker: Sarah Blackwell (20 or 50m)

Note: a second 20m slot is available if someone would like to speak.

## 02/08/2022

Speaker: Aleksander Shmakov (50m)

## 02/15/2022

Speaker: Peter Cassels (20m)

Speaker: Raemeon Cowan (20m)

## 01/22/2022

Speaker: Haiyang Wang (20m)

Speaker: D. Zack Garza (20m)

## 03/01/2022

Speaker: Jack Wagner (50m)

Speaker: Open!

Speaker: Open!

Speaker: Open!

Speaker: Open!

## 04/26/2022

Speaker: Peter Woolfitt (50m)

## 05/03/2022

Last day of classes, Friday schedule. Likely no talks

# Fall 2021 Talks

This semester was jointly organized by D. Zack Garza and Arvind Suresh.

## 9/7/2021

• Speaker: Peter Woolfitt:

• Title: A fun mystery topic!
• Speaker: Arvind Suresh:

• Title: On the realization problem

• Abstract: Given a rational representation ρ of the absolute Galois group G of a number field K, it is natural to wonder if there is an elliptic curve whose group of geometric points contains a subrepresentation isomorphic to ρ (we say the curve “realizes ρ”). For example, if ρ is the n−dimensional trivial rep, then E realizes ρ iff rank E(K) is at least n. In this talk, we give some positive results on the realization problem for higher−dimensional abelian varieties.

## 9/14/2021

• Speaker: Komal Agrawal
• Title: The Goldilocks Classification for ℕ

• Abstract: In this talk we look at the Goldilocks classification of the natural numbers arising from the sum-of-divisors function. We will then discuss other related concepts such as amicable numbers and k-sociable numbers. Also included is a cute proof of the divergence of the reciprocal sum of primes.

• Speaker: Erin Wood
• Title: Categorizing Teacher Moves For Supporting Student Reasoning

• Abstract: I’ll be discussing a framework developed by Ellis, Özgür, and Reiten to analyze teacher moves based on their potential for supporting student reasoning. I will give examples of the types of moves in each category, and talk about how this organizational structure may be useful in reflecting on and evaluating our own teaching moves.

## 9/21/2021

• Speaker: Andy Jenkins
• Title:
A partition-type classification of nilpotent orbits for classical simple Lie algebras

• Abstract:
Let $\mathfrak{g}$ be a simple Lie algebra of classical type and $G$ its associated connected reductive algebraic group. There is a natural action of $G$ on $\mathfrak{g}$ by conjugation, and if we restrict this action to the collection of nilpotent elements $\mathcal{N}$ of $\mathfrak{g}$, the orbits under this action are called nilpotent orbits. The study of the structure and geometry of nilpotent orbits has led to many important results in representation theory. In this talk, we will describe a classification of nilpotent orbits of $\mathfrak{g}$ in terms of partitions. As applications, we show how this parameterization describes a partial order on the collection of nilpotent orbits and give formulas for the dimensions of the nilpotent orbits.

• Speaker: Aleksander Shmakov
• Title: Some Trace Formulas for Modular Forms

• Abstract: Modular forms provide a bridge between the world of functional analysis and algebraic geometry. After recalling the definition of modular forms and Hecke operators, I will give two examples of trace formulas that relate Fourier coefficients of modular forms to point counts on elliptic curves over finite fields. Both trace formulas are a consequence of the Eichler-Shimura theorem but the second is somewhat more striking: it relates an infinite weighted sum of Fourier coefficients to a finite sum of point counts.

## 9/28/2021

• Title: On unit fraction decompositions

• Abstract: A unit fraction is a rational number of the form 1/n, where n is a positive integer. It can be proved that every positive rational number can be written as the sum of distinct unit fractions. In this talk, we will cover some known results about such decompositions, as well as discuss some open problems.

• Speaker: Haiyang Wang
• Title: Néron models of elliptic curves

• Abstract: Néron model was introduced by Andre Néron in 1961. It was used in defining Faltings’s height function, which played an important role in Faltings’s proof of the famous Mordell’s conjecture. In this talk, we will give a brief introduction to this topic. In particular, we will look at the Néron models of elliptic curves and some of their interesting properties.

## 10/5/2021

• Speaker: Freddy Saia
• Title: Isogeny volcanoes

• Abstract: In this talk, we will consider isogeny graphs of elliptic curves over finite fields. Certain subgraphs of these isogeny graphs have a “volcano” structure, which makes them amenable to explicit computations and is beneficial to applications in, for example, cryptography. I will not assume experience with elliptic curves or algebraic number theory; we will discuss the relevant facts on elliptic curves and orders in imaginary quadratic fields. Following this, we will investigate the structure of isogeny volcanoes, and time-permitting we will discuss an application.

## 10/12/2021

• Speaker: Dustin Kasser
• Title: Compact Sets Intersecting 3 Lines

• Abstract: I will be presenting a new result (not by me) on a sufficient condition for arranging compact sets so that they can all be intersected by only three lines.

• Speaker: Peter Cassels
• Title: The Hillman-Grassl Correspondence

• Abstract: Stanley’s hook length formula gives a nice generating function for enumerating reverse plane partitions for a given partition. In particular, this can be thought of as a generalization of the famed hook length formula originally proved by Frame, Robinson, and Thrall. Stanley’s result was reproven by Hillman and Grassl using purely combinatorial means. In this talk, after providing some quick background and context, I will demonstrate the algorithm discovered by Hillman and Grassl, and explain why how proves Stanley’s formula. If time permits, I will discuss some recent further generalizations of these results.

## 10/19/2021

• Speaker: Ye Tian
• Title: The Jones Polynomial

• Abstract: Mathematicians have been using polynomials to describe knots and links for the past century. Almost 60 years after the first such attempt by J. Alexander, Vaughan Jones discovered a new polynomial associated with knots and links in 1984. In this talk I will present the construction of Jones’ Polynomial as interpreted by Louis Kauffman and show that it is reasonably powerful in distinguishing different knots.

• Speaker: Zack Garza
• Title: Count All of the Things!

• Abstract: The Weil Conjectures predict that certain zeta functions attached to varieties defined over finite fields exhibit interesting properties reminiscent of L-functions, including an analog of meromorphic continuation, a functional equation, and sharp estimates for locations of zeros. We’ll discuss how these generating functions can be explicitly computed in a handful of examples, reducing to combinatorial point-counting of solutions to equations.

## 11/9/2021

• Speaker: Nagendar Reddy Ponagandla
• Title: 210 and an upper bound in Goldbach’s problem

• Abstract: It is clear that the number of distinct representations of a number ‘n’ as a sum of two primes is at most the number of primes in the interval [n/2, n-2]. Carl Pomerance conjectured and together with colleagues proved that 210 is the largest value of ‘n’ for which this upper bound is attained.

• Speaker: Dustin Kasser
• Title: An introduction to the Erdős–Szekeres conjecture

• Abstract: The Erdős–Szekeres conjecture (or as it is commonly known the “happy ending problem” as it resulted in a marriage of two mathemeticians) asks how many points must be required for every set of size at least K to contain a convex n-gon. A brief presentation of the n=3 and 4 cases will be covered, and then we will discuss the conjectured bound and the original proof that established an upper bound.

## 11/30/2021

• Speaker: Jack Wagner

• Title: My Favorite Functors

• Abstract: I will talk about my favorite functors. This talk will be for a general audience. You are encouraged to bring your favorite functors too.