Undergraduate Research
I enjoy working with UW-Eau Claire students on research projects in mathematics and mathematics education. If you're a UWEC student and any of the projects below interest you, feel free to send me an email or stop by my office to chat about them!
Student Validation of AI-Generated Proof
Proofs are the building blocks of mathematics. For a long time, humans were the main producers of proofs but computing developments in the second half of the 20th century led to computer-assisted proofs of famous mathematical results like the four color theorem.
When OpenAI released its large language model artificial intelligence chatbot ChatGPT in the fall of 2022, anyone could use a computer to generate many, many proofs of mathematical statements in a very short time. There's a catch though - AI models like ChatGPT run on text-prediction. When you ask ChatGPT "what is 2+2?", there isn't a calculator running in the background to compute 2+2. Instead, the model is using statistics to predict that the most likely text following "2+2" is 4, based on training data from across the internet.
What this means is that ChatGPT has no way to verify the mathematical validity of the proofs you generate with it. In fact, you can ask ChatGPT to prove a claim you know to be false and it will still generate an argument!
Our research team is interested in the ways that undergraduate mathematics students respond to AI-generated proofs. We're researching how knowing that a proof was generated by ChatGPT might impact the way a student validates a mathematical proof.
This project involves:
reading existing research papers in the discipline of mathematics education
using ChatGPT and other generative AI tools to generate proofs of mathematical claims
designing and conducting survey and interview research
analyzing data using quantitative (statistical) and qualitative (descriptive) methods
You might be interested in this project if you ...
have taken/are taking at least one course with a focus on proof writing (for example, Discrete Mathematics (Math 314) or Linear Algebra (Math 324))
are willing to complete the UWEC IRB training on conducting research with human subjects
are a math-secondary education major or want to learn more about mathematics education scholarship
Symmetry, Graphs, and the Difficulty of a Crossword
Math is all fun and games! Many of the games and puzzles that you know and play have underlying mathematical structures that can be explored. Those interested in recreational mathematics have used mathematics research techniques to study chess, Go, Hex, Sudoku, Rubik's cubes, and more!
Most crossword puzzles are an n x n grid of black and white squares that follows a few simple rules:
all words (strings of white squares) must be at least 3 letters long
rotating the unfilled grid by 180 degrees gives the same grid; in other words, puzzles have 180 rotational symmetry
every white square must be included in a down and across word
Crossword puzzles can be viewed with this mathematical lens; in fact, we can represent any crossword puzzle as a bipartite graph where vertices represent words and edges connect words that share a cell.
Our team is using graph-theoretic techniques to describe the difficulty of a crossword puzzle, as measured by the grid shape. We're especially interested in the question of whether puzzles with other symmetries (90 degree rotational, diagonal, vertical, horizontal) are more difficult than puzzles with 180 degree rotational symmetry.
This project involves:
reading existing mathematical research papers on the mathematical features of crossword puzzles
using techniques from graph theory and combinatorics to explore crossword puzzles
writing proofs or code that describe the mathematical structures of crossword puzzles
You might be interested in this project if you ...
have taken/are taking at least one course with a focus on proof writing (for example, Discrete Mathematics (Math 314) or Linear Algebra (Math 324))
like crossword puzzles or similar games
enjoy graph theory and combinatorics
have coding experience that you want to apply to puzzles and games
Homotopical Combinatorics
Let G be a finite group. We can study the subgroup lattice of G which is a partially ordered set under inclusion. It turns out that the information in these lattices can assist in the study of an important object in the field of homotopy theory called a G-N-infinity operad, first defined by Blumberg-Hill.
Recently, work of Rubin and Balchin-Barnes-Roitzheim has shown that the information of a G-N-infinity operad is really captured by a combinatorial object called a G-transfer system on the subgroup lattice. An overview of homotopical combinatorics can be found in this recent article in the Notices of the AMS.
This is a new area of study, so relatively little is known about transfer systems on subgroup lattices. Research on this topic could explore counting and classifying transfer systems on new groups.
This project involves:
reading existing mathematical research papers on transfer systems
exploring counting and characterization questions about pictures like the one seen here
writing proofs or code that describes allowable transfer systems on a lattice
You might be interested in this project if you ...
have taken proof-based mathematics courses
enjoyed Algebra (Math 425), combinatorics, or graph theory
are interesting in thinking about more abstract mathematical topics
UWEC undergrads have the opportunity to get paid for doing research with faculty members (during the school year or in the summer) and to travel to local or national conferences to present their results. To learn more about these opportunities, visit the Office of Research and Sponsored Programs.