In neo-Riemannian musical theory, a particular collection of triadic chord transformations labeled by P, L, and R can be seen to form a group acting on the set of 24 major and minor triads. There are many other sorts of transformations between chords, and there are other sorts of chords besides major and minor triads.

In his GURM project, Christopher Heitmann sought other collections of chord transformations that would act on sets of chords as a group. He wrote code in both Python and GAP and developed a procedure to sift through a dataset of chord transformations and rule out certain combinations for violating necessary conditions for group structure. He was able to find a pair of chord transformations that acted as a group on a set of triads which included augmented and diminished triads.

Christopher presented his work on April 1, 2023 in a poster session at the Philadelphia Undergraduate Mathematics Conference (PUMC) and in a presentation on April 13, 2023 at the Eastern Pennsylvania and Delaware (EPaDel) Section Meeting of the Mathematical Association of America.

In "Gambling Under Unknown Probabilities as Proxy for Real World Decisions Under Uncertainty" by David Aldous and Thomas Bruss, the authors raise a question they call the allowance issue: in many betting scenarios, a gambler is trying to predict a certain probability as accurately as they can given limited information, but there is likely to be some error between their best predictive guess and the true probability. So, can these gamblers adjust their behavior to account for this discrepancy?

Gonghui Lin explored a particular type of betting game in which players must place a sequence of bets on the unknown probability of a coin coming up heads, gaining more information as the coin is repeatedly flipped and adjusting their prediction based on this information. A typical prediction strategy based on the Laplace Rule of Succession relies on the known sequence of past coin flips to determine the most likely probability that a coin will come up heads. In his GURM project, Gonghui developed a betting strategy by which a player can adjust the Rule of Succession prediction. He also developed Python code to simulate the betting game and show that his new strategy improves the player's score compared to the Rule of Succession strategy.

Gonghui presented his work on April 15, 2023 at the Eastern Pennsylvania and Delaware (EPaDel) Section Meeting of the Mathematical Association of America.

The Perfect Cuboid problem asks us to determine whether there exists a cube having integer valued lengths for its edges, face diagonals and space diagonal. As of today, it is still an unsolved problem.

In his GURM project, Dean Quach found another constraint on the possible prime factorizations of the product of the lengths of all edges, face diagonals and space diagonal. This new constraint gives more reason to believe that the perfect cuboid may not exist.

This work was presented on 1 April 2023, at the Phidadelphia Undergraduate Mathematics Conference (PUMC). You can see his presentation HERE.

In 2022, Julia Conigliari began her project by analyzing databases of real-world ranked preference election data. The project’s point of emphasis was the ways in which the same set of ballots can generate different electoral results depending on the choice of (even reasonable and widely-used) counting method. Julia’s work contributed to an article that has been submitted for publication. 

Liz’s project began as a collaboration including her peer – Julianna Sims. This work focused on variations of fixed point free nonexpansive maps on weakly compact and convex domains. The team succeeded during the GURM semester in constructing a new family of examples and clarifying the boundaries over what was possible with the technique. Liz presented this work at PUMC in April 2023, and at various Temple venues. This work has been submitted for publication.

Liz subsequently received two Frances Velay grants to expand on this work. Each Velay summer project led to new and interesting results and Liz continued to present summaries of her work at research symposia. 

Othmane Harraq’s project focused on a similar class of examples to the Abt-Fraioli project described above. A rigorous proof of one of the limitations of the technique was achieved during the GURM semester. 

Othmane’s tenacious work added further clarity to open questions in this area. 

In 2023, Nicholas Bader began his project...

In 24-25...