**Research Interests: Commutative Algebra and Number Theory**

**Research Interests: Commutative Algebra and Number Theory**

Dissertation Title: Decomposition of Modules and Tensor Products over Principal Subalgebras of Truncated Polynomial Rings

**Talks**

**Talks**

**UPCOMING:**

**Invited to give talks at the following conferences:**

Haynes-Granville-Browne Session of Presentations by Recent Doctoral Recipients, Joint Mathematics Meeting, January 2023

**GIVEN:**

**Decompositions of Modules over Principal Subalgebras of Truncated Polynomial Rings**

Abstract: ** **We investigate how modules decompose over principal subalgebras of certain truncated polynomial rings. In particular, we will investigate how a module decomposition may (or may not) change when we decompose over different principal subalgebras. Varying decompositions are related to the notion of rank varieties. Finally, we will examine how one might extend the notion of rank varieties to more general truncated polynomial rings.

**Homological Commutative Algebra and Related Talks, Georgia Southern University, June****2022****Commutative and Homological Algebra Market Presentations, January 2022 (Slides)**In November 2021, I had the pleasure of presenting a poster at the Blackwell-Tapia conference held at MSRI. The poster gives a brief introduction into my research area. You can access the pdf file by clicking the link above. First, we look at some examples of how modules decompose over principal subalgebras of certain truncated polynomial rings. In particular, we investigate how the decompositions for these modules can change (or not) due to our choice of our principal subalgebra. Then, we introduce the definition of a rank variety in terms of these particular modules. Finally, we investigate how the rank variety of a module relates to its decomposition by using its "representation matrix." Let me know if you'd like to hear more!

**Introduction to Continued Fractions**** (Lecture Notes)**

In April 2018, I gave a talk over Continued Fractions to a group of my fellow graduate students at Sam Houston State University. This was my project with Dr. Ken Smith for my Masters' degree. My notes are included in the link above. Continued fractions have many applications, including, but not limited to computing square roots, Pell's equation, representing irrational numbers and knot theory.

**Dominance and Periodicity for the Difference Equation U**_{n+1}** = max{α-U**_{n, }**β-U**_{n-2}**} (Poster)**

In August 2015, I participated in a summer undergraduate research program (SURP) at Texas Southern University (TSU). I worked with Drs. Willie Taylor and Roderick Holmes to prove that every real solution in the above difference equation is eventually periodic. We are still pushing to get it published. The link for the poster is provided above.

**Currently Reading:**

**Currently Reading:**

Steps in Commutative Algebra - R. Y. Sharp

Algebra I: Chapters 1-3 - N. Bourbaki

Computational Methods of Linear Algebra - V.N. Faddeeva

**Attended Conferences/Workshops**

**Attended Conferences/Workshops**

**UPCOMING:**

Homological Commutative Algebra and Related Topics - Summer 2022 - Savannah, GA

**ATTENDED:**

Joint Mathematics Meetings - Spring 2022 - Virtual

Southwest Local Algebra Meeting - Spring 2022 - Waco, TX

Blackwell-Tapia Conference - Fall 2021 - Berkeley, CA (MSRI)

Graduate Algebra Symposium - Fall 2021 - Virtual

National Association of Mathematicians MATHFest - Fall 2021 - Virtual

Free Resolutions and Representation Theory (ICERM) - Fall 2020 - Virtual

Math Summer Workshop for Achieving Greater Graduate Education Readiness (Math SWAGGER) - Summer 2020 - Virtual

Southwest Local Algebra Meeting (SLAM) - Spring 2020 - New Orleans, LA

Southwest Local Algebra Meeting (SLAM) - Spring 2019 - El Paso, TX

Gulf States Math Alliance Conference - Spring 2019 - Arlington, TX

Field of Dreams Conference - Fall 2018 - St. Louis, Missouri

Field of Dreams Conference - Fall 2017 - St. Louis, Missouri