Talks
I have given a few talks within the University of Nebraska-Lincoln Math Department, including at the Commutative Algebra Research Seminar (CARS), The Graduate Student Seminar (GSS) and our Quantum Interdisciplinary Research Group 3 (QIRG 3).
CARS, University of Nebraska - Lincoln, Spring 2026
Title: The Weak Lefschetz Property + Some Disappointment with Initial Ideals
Abstract: It is known that if R/in_< (I) has the Weak Lefschetz Property for some monomial order <, then R/I has the Weak Lefschetz Property as well. The converse is known not to be true. Murai posed the question; if we put the restriction that the monomial order < has to make in_<(I) a square-free monomial, can we make the converse true? The answer is no. We will go over a result and subsequent example in Hongmiao Yu's Paper "On the Weak and Strong Lefschetz Properties For R/in_<(I_t)" that demonstrates this.
QIRG3, University of Nebraska - Lincoln, Fall 2025
Title: Realizing Linear Operators Using Arrays of Mach-Zender Interferometers
Abstract: This talk goes over the results of Rech and Zeilinger's Paper "Experimental Realization of Any Discrete Unitary Operator" and opens a discussion on how the results translate from the true Quantum case to the case of the Mach-Zender Interferometer from Dr. Peter Dowben's earlier presentation for QIRG3.
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.