Polynomial Method in Additive Combinatorics: Extension of CAP Sets Problem to More Than Three Points

The project with focus on the new polynomial method in (additive) combinatorics which has allowed long standing problems such as Kakeya problem over finite fields, CAP-set problem and Erdos distance problem to be resolved.

We will study the proof by Croot-Pach-Lev-Ellenberg-Gijswijt-Tao of the sub-exponential bound for cap-set problem concerning the maximal cardinality of sets which do not contain non-trivial three term arithmetic progressions in vector spaces over a finite field. We will also study the work of Lovett on lower bounds of a slice rank of a tensor with achieving a good lower bound on the latter resolving the CAP-set problem. Furthermore, we will extend on the work of Fish and Roy to include almost CAP-sets in more than three variables.

Georgio Hawi

The University of Sydney

Georgio Hawi is currently in his second year (in addition to some undergraduate mathematics units in 2018) of a Bachelor of Science/Master of Mathematical Sciences degree majoring in mathematics and physics at the University of Sydney. He has taken units from a variety of fields of mathematics including differential geometry, partial differential equations, Galois theory and complex analysis, which has allowed him to increase his versatility of both mathematical concepts and techniques. Throughout 2020–2021, he has undertaken four research projects with USyd in areas including eigenvalue perturbation theory, elliptic-curve cryptography, and dark matter (specifically axions and chameleons).

In the future, he plans to get his master’s degree and then a PhD, with which he intends to join the academic and research spheres. The AMSI Vacation Research Scholarship will allow him to further expand his horizons as well as consolidate his understanding and experiences regarding scientific research. Outside of mathematics, Georgio has many hobbies including chess, programming, gaming, and watching TV shows (his favourite of which is NCIS).

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