Talks
I have given a few talks within the University of Nebraska-Lincoln Math Department: I have given three talks at our Commutative Algebra Research Seminar (CARS) and one talk at the Graduate Student Seminar (GSS)
CARS, University of Nebraska - Lincoln, Spring 2025
Title: Using Macaulay Duality to Determine the Strong Lefschetz Property for Artinian Gorenstein Algebras
Abstract: This is a talk in three parts. In the first, we will (re)introduce the Lefschetz Properties and make some observations about how they interact with Hilbert Functions. In the second section, we introduce the Macaulay Duality in order to categorize Artinian Gorenstein Algebras and make some observations about their nice Hilbert Functions. In the third and last section, we define a Hessian and leverage what we learned about Artinian Gorenstein Algebras and the Lefschetz Properties in the first two sections to make a general statement about whether or not a given Artinian Gorenstein algebra ought to have the Lefschetz Properties.
CARS, University of Nebraska - Lincoln, Summer 2024
Title: Algebras with the Strong Lefschetz Property as Representations of SL(2,C)
Abstract: We'll start with a review of the definitions of the Weak Lefschetz Property (WLP), Strong Lefschertz Property (SLP), and Strong Lefschetz property in the narrow sense (SLPn). We'll introduce representations of sl_2 in order to exposit a bit on interactions between tensor products and the Lefschetz properties.
Nota Bene: This talk was given over the summer to a small audience of my peers, all of which had seen a talk of mine on the Lefschetz Properties earlier. As such, the title I gave on the occassion of the talk was "The Lefschetz Properties Part 2: Electric Boogaloo". This is not exactly a descriptive title, so I have renamed it here.
CARS, University of Nebraska - Lincoln, Spring 2024
Title: The Jordan Type Theorem + An Anecdote
Abstract: In this talk I will introduce the Weak + Strong Lefschetz Properties and some other relevant properties in order to prove the Jordan Type Theorem. If I manage to do this in a timely fashion, I will show an example of the theorem making an accurate prediction and share an anecdote in which Kara and I wanted to see a different feature of the theorem and became extremely confused.
GSS, University of Nebraska - Lincoln, Fall 2023
Title: Deriving a Solution to the Linear Schrodinger Equation and Discussion on Soliton Solutions to the Nonlinear Schrodinger Equation
Abstract: Exactly what it says on the tin. We will use separation of variables to solve the linear Schrodinger equation for the hydrogen atom, first to establish the existence of a solution and then to form a more physically correct solution. We will then discuss the origins of the non-linear Schrodinger equation and why it's intriguing to mathematicians in the field of Partial Differential Equations, particularly the existence of travelling wave "soliton" solutions. This presentation is adapted from an 831 (partial differential equations, 1st year sequence) presentation, hence the focus on separation of variables, which was not taught that semester.
Research
I am interested in Commutative Algebra. I did some undergraduate research via the UCARE program at UNL, which I used as my honors thesis.