Research

Constructing the Comparison Spectrum for Algebraic and Topological K-Theory of Banach Algebras

Ph.D. thesis research under the supervision of Dr. Markus Pflaum. Developed a rigorous construction of the relative K-theory of algebraic and topological K-theory of Banach algebras as a spectrum. Detailed the delooping map, providing clear formulation for negative relative K-theory groups and the spectrum as a connective spectrum.

Topological Data Analysis Applied to Materials Science Glass Data

REU/G research project under the supervision of Dr. Markus Pflaum. Studied the mathematical and computational foundations of topological data analysis (TDA) and its applications, including the DELTA group’s work on cyclohexane energy landscapes. Led a sub-project involving two undergraduate researchers in collaboration with the Coe College Physics Department, conducting a proof-of-concept TDA analysis of silica glass molecular conformation data. This work established a foundation for future investigations of the Kauzmann paradox.

Fault-Free Tileability of Rectangles, Cylinders, Tori, and M¨obius Strips with Dominoes

REU research project under the supervision of Dr. Jon White. Proved the fault-free tileability of all cylinders, Tori, and M¨obius Strips with dominoes. Collaborated on related problems in infinite plane tiling, hexagonal tiling, and maximum tile packing.

Segal's Γ-Spaces and the Sphere Spectrum

Independent research under the supervision of Dr. Agn`es Beaudry. Studied foundational topics in homotopy theory, algebraic topology, and category theory, focused on Proposition 3.5 of Categories and Cohomology Theories by Dr. Graeme Segal and the Barratt-Priddy-Quillen Theorem. This work informed later research within my Ph.D. thesis.

Formal Group Laws

Summer research assistantship under the supervision of Dr. Agn`es Beaudry. Studied foundational topics in chromatic homotopy theory, with a focus on p-adics, formal group laws, and Morava stabilizer groups.

Conductivity of Tellurium Vanadate Glasses

REU research project under the supervision of Dr. Mario Affiagato. Vitrified tellurium vanadate metallic glasses from raw compounds and performed conductivity measurements to investigate their electrical properties. This work informed later TDA research.

In this project, I designed and led a undergraduate students in developing interactive presentations and worksheets to promote mathematical thinking among high school and undergraduate audiences. I taught on mathematics pedagogy, active learning strategies, and the use of tactile learning materials.

One of my students had prior research in packing problems and number theory. They wrote two 50-minuite presentations. The first being a highly technical reserach talk, the second targeting outreach to a more general audience. They used a laser-cutter to create circle-packing tactiles for the audience. The talk centered on audience members playing with the tactiles and exploring solutions to accessable problems. This facilitated a deeper understanding and appreaciation, allowing the student to share their novel work with a general audiance in a meaningful way

Another of my students chose to study game theory. They reserached introductory lessons and activities, then created four interactive games to demonstrate core concepts in game theory. They developed props for each game and worked to streamline the blend between content, game instruction, and the playing of games by audiance memebers within their presentation.

Topological Data Analysis (TDA) is a very powerful and newer field of math that is sure to make big waves in the future. The only thing holding TDA back is applying it on more data sets.  Its a solution looking for problems.

This project is ideal for someone who has background in coding, or someone who want to complete interdisciplinary research with data from another field. So far, I have seen success in using TDA on chemistry, physics materials science, and cyber security web-traffic data.

Sona math research is currently not well explored in literature, but it appears to have connections to algebra, geometry, and topology.

Interested in Sona math research? Start with my worksheets under Materials -> Math Club: Sona Part 1 (2-3 hours) and Sona Part 2 (10+ hours of fun)

I have many open problems about Sona ready for you, or you can propose your own questions to solve!

I expect one summer or two semesters of research would be enough for you to publish new results.

Tiling is an amazing field of math research connecting to discrete math, combinatorics, algebra, topology, geometry, and more. Tiling itself has many subfields!

Fault free tiling with dominoes is my specialty, and I have interesting open problems ready for you.

I am also prepared to help with any other tiling interests of yours: tiling of the infinite plane, hexagonal tiling, maximal tile packing, etc.

Don't know where to start? Take a look at "Polyominoes: A Guide to Puzzles and Problems in Tiling" by George E. Martin.

In this project students would create interactive presentations or worksheets covering fun mathematical topics. Our goal is to reach a middleschool, highschool, or undergraduate audience. We will learn about various pedagogy topics, with a focus on the use of manipulatives/tactiles (physical demonstrating objects) and incorporating what we learn into our talks. I ran this project previously, in 2024 at CU Boulder, with amazing student outcomes!

Students may choose their topics from their own previous math research, a list of topics I provide, or they may propose their own topic.

Juggling and flow arts are inherently mathematical. The pattens, sequences, circular motions, links, and knots involved all follow a strict mathematical basis in how they function and thus connect deeply to the fields of group theory, topology, and geometry. For this project students will research current theory from a selection of juggling, poi and club swinging, tech and Russian fans, rope dart, and hoops. Some introductory ideas to look into include siteswap, prechac, and VTG notation.

• For a computer science minded student, we can work on a project which animates siteswap, prechac, or VTG patterns. See Siteswap Animator, SiteswapSim, and Library of Juggling (siteswap) or Passist (prechac) for inspiration.

• For a student interested in combining math and dance, this project could end in a recital of some kind.

• For a student interested in combing math and music, the math of juggling can equally be seen as math describing drumming or playing other instruments. We could focus on that connection and study the math of rhythms from a musical point of view.

• For a student interested in the physical act of juggling, this project could include development of new patterns, passing patterns, or publishing of a juggling video.

• For a student interested in digging deeper into the math, we could continue research further by looking into the more advanced connections with number theory, graph theory, linear algebra, combinatorics, probability theory, topology, and other algebra sub fields.

Crafting is inherently mathematical. Many disciplines of crafting involve rules, patterns, symetry, scale, knots, and geometric structures, connecting crafting to geometry, algebra, and topology! Students can research the math involved in crafts from a selection of knitting, macrame, weaving, origami, darting, smocking, string art, tessellations, or kinetic crafts like flexagons and the Hoberman sphere. Alternatively, students may take a mathematical concept they like and encode it into a craft, see previous math quilting projects from University of Kentucky and CU Boulder.

This project could result in a coresponding art project, or could remain strictly in mathematical theory.