ACC Seminar: Progress Report on Summer Research and AI/Math Lab

ABSTRACT

Presentation 1:

In this half of the seminar, we discuss progress on my summer research in Model Theory with Professor Alexei Miasnikov and on groups with the AI lab.

The former concerns my work on definability and interpretations between algebraic structures and a number of associated geometric spaces, which come in many forms. There are two main results, each on different objects. First, there is an affine version of our previous result on regular bi-interpreting a field and its projective space. Second, it is shown that a free group of rank at least 2 is interpretable in its lattice of subgroups.

The latter concerns my programming and experimenting with the Knuth-Bendix completion algorithm for finitely presented groups. Here, we aim to use reinforcement learning to assist what is generally only a partial solution to the word problem. Our aim is to be able to target particular words for reduction when a general solution to the word problem is out of reach.

Presentation 2:

In this half of the seminar, we discuss progress on two projects in the AI/Math Lab.

The first project concerns the Kaplansky Zero Divisors conjecture, which asserts that all group rings of a torsion-free group over a field have no non-trivial zero divisors. We discuss various computational approaches to disproving this and the related Kaplansky units conjecture and discuss attempts at trying to disprove the Zero Divisors conjecture using Reinforcement Learning (RL) methods.

The second project attempts to extend the U-MATH benchmark for evaluating how LLMs perform at solving university-level math problems. We discuss the original U-MATH benchmark and the opportunities and challenges offered by integrating Gradarius' deterministic grading system into the pipeline, as well as the current progress on implementing this integration.

BIOGRAPHY

Brett Berger is a Stevens Institute of Technology PhD Mathematics student. The research discussed in this talk was completed under the direction of Professor Alexei Miasnikov.

Winston Lee is a Stevens Institute of Technology PhD Mathematics student. The research discussed in this talk was completed under the direction of Professor Alexei Miasnikov.

Attendance: This is a technical talk open to all.
A campus map is available at https://tour.stevens.edu.
Additional information is available at https://web.stevens.edu/algebraic/.