丁香园AV

Since the proof of Fermat鈥檚 Last Theorem by Wiles and Taylor-Wiles, mathematicians have investigated a number of generalizations, one of which involves the same equation, but allows solutions to lie in an extension of the rational field. As part of his NSERC USRA project in Summer 2021, Aisosa Efemwonkieke investigated Fermat鈥檚 Last Theorem over real quadratic fields. The methods used consisted of a combination of Galois representations and modularity over real quadratic fields together with quadratic reciprocity over number fields, and required learning algebraic number theory in a short time frame. Aisosa鈥檚 ability to absorb the theory and methods together with his insights led to a result in almost every branch he chose, including some cases which were unexpected, and others which required developing new methods involving Hilbert symbols and reciprocity. The project was the basis for Aisosa鈥檚 honours thesis and has resulted in a coauthored paper which proves Fermat鈥檚 Last Theorem for infinitely many exponents in the case of two real quadratic fields, which are notable in the literature for not being completely solved asymptotically. The paper is currently available on arXiv and has been submitted to a research journal.

In Summer 2021, Sohrab collaborated on a fast implementation of an algorithm to compute values of Riemann theta functions and their derivatives, with specified characteristic, and to arbitrary precision. He particularly contributed to the implementation of Siegel reduction, which preconditions the lattice basis for better convergence.

The implementation is now available as free/open source from  and is the fastest tool currently available for tasks such as reconstructing a smooth plane quartic curve from its period matrix. It forms an important tool for numerical experimentations supporting research in the explicit Schottky problem.

Parsa joined from the University of Waterloo for a second remote research term in summer 2021, studying the Fredman-Khachiyan algorithm for generating monotone Boolean functions. Parsa was intrigued by the possible application of methods due to Adam Wagner on using reinforcement learning to find interesting examples in extremal combinatorics. He quickly learned the necessary background and found out there were several potential approaches, which he implemented and tested. Parsa was able to overcome a number of challenges, among transferring them from Wagner's setting of classical graph theory to the setting of hypergraphs. On the very last day, one of the computational experiments turned up an unexpected example of a hypergraph on 8 vertices that is interesting combinatorially and in the context of the Fredman-Khachiyan algorithm.

Parsa carried responsibility for all aspects of the projects, from implementation to writing up the results. He came to Vancouver in June 2022 and present this work at the Canadian O.R. Society's annual meeting, and was well-received by an audience which included researchers with both optimization and machine learning backgrounds.