Laboratory for AI in Algebra at Bar-Ilan University
The laboratory focuses on using AI tools to enhance research in algebra by challenging long-standing problems in group theory, such as the Burnside problem and Kaplansky’s zero divisor conjecture, using techniques from algorithmic optimisation and reinforcement learning.
The laboratory partners with Nebius, Stevens Institute of Technology, and Caltech.
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Yitzhak Ginsburg Master’s student | Borys Holikov Master’s student | Kyrylo Muliarchyk Postdoc | Professor, director of the laboratory |
Research directions
Faithfulness of the Burau representation of a braid group for n = 4
For n = 3, it is known to be faithful; for n ≥ 5, it is known to be unfaithful. Determining its faithfulness for n = 4 remains an open problem.
We are working on computing kernels of the 4-strand Burau representation modulo p. For p = 2,3, this was done in 1991, and for p = 5 in 2023. We are currently applying reinforcement learning to search for kernel elements modulo 7, with the hope that these results can be generalized to larger moduli. In the most favorable scenario, this would show that the Burau representation for n = 4 is unfaithful.
This is a joint work with Alexei Miasnikov, Professor at Stevens Institute of Technology.
Searching for counterexamples to the Andrews–Curtis conjecture
The Andrews–Curtis conjecture states that any balanced presentation (with the same number of generators and relators) of the trivial group can be reduced to a trivial presentation by a finite sequence of standard Andrews–Curtis transformations.
A closely related analogue for finite groups is known to hold, but the general Andrews–Curtis conjecture remains an open problem.
The Akbulut–Kirby family is often treated as a primary source of difficult test instances. These presentations are known to define the trivial group, yet they can be very hard to explicitly trivialize via Andrews–Curtis transformations.
Our approach is to study these presentations through their homomorphic images in finite groups, for example, various classical and exceptional groups of Lie type over finite fields. Assisted by AI tools, we construct such images and explore the corresponding finite Andrews–Curtis graphs to search for short trivialization sequences that might suggest how to trivialize the original presentation. We hope this will provide computational evidence indicating which instances are genuinely difficult and where further search efforts should be focused.
This is a joint work with Alexei Miasnikov, Professor at Stevens Institute of Technology.
Finding counterexamples to Kaplansky’s zero divisor conjecture
The long-standing Kaplansky zero divisor conjecture states that, given a field K and torsion-free group G, the group ring K[G] has no non-trivial zero divisors (i.e. if ab = 0, then a = 0 or b = 0).
Disproving the zero divisor conjecture would also disprove the related Kaplansky idempotent conjecture (i.e. if a² = a in K[G], then a = 0 or a = 1) and the Kaplansky unit conjecture (i.e. if a is a unit in K[G], then it is a scalar multiple of some element g in G).
In 2021, Gardam announced a counterexample to the unit conjecture for a specific known group P with K = F₂. However, the zero divisor conjecture is known to hold in this group P.
In this work, we implement and improve upon Gardam’s original method to find a counterexample to the zero divisor conjecture.
This is joint work with Alexei Miasnikov, Professor at Stevens Institute of Technology.
RL Algorithmic Mixer
Most interesting search problems can’t be solved by brute force. But specialized approaches quickly run into the No Free Lunch Theorem – improving an algorithm in one dimension tends to make it worse in another.
RL Algorithmic Mixer is a framework we develop to use multiple specialized algorithms while an ML orchestrator exchanges working sets between them based on each algorithm’s suitability profile, instead of seeking one algorithm that is optimal across the entire problem space. This approach outperforms currently available single-algorithm methods on smaller instances of the Burnside problem.
