Colloquium: Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas

Dr. Yesim Demiroglu Karabulut from Harvey Mudd College will be giving a video conference colloquium lecture on "Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas" on Thursday Jan 30th, 3:30-5:00pm in CLE B007.

Abstract: In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of proving these results, we obtain an original result that provides an analogue of S{\'a}rk{\"o}zy's theorem in the finite field setting (showing that any subset $E$ of a finite field $\Bbb F_q$ for which $|E| > \frac{qk}{\sqrt{q - 1}}$ must contain at least two distinct elements whose difference is a $k^{\text{\tiny th}}$ power). Once we have our results for finite fields, we can apply some classical mathematics to extend our Waring's problem results to the context of general (not  necessarily commutative) finite rings. In the second half of our talk, we talk about some sum-product formulas related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.