Mathematics Colloquia and Seminars

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Since Edward Waring stated his famous conjecture in his book "Meditationes Algebraicae" in 1770, Waring's problem has been of particular interest to mathematicians. As Charles Small put it in his survey "Indeed, it is one of those nasty gems, like Fermat's Last Theorem, which begins with a simply-stated assertion about natural numbers, and leads quickly into deep water.". The first half of our journey will be a quick introduction to Waring's problem together with our contribution. We present some new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of re-proving these results, we obtain another 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, one 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 present some sum-product formula results related to matrix rings over finite fields and explain the connection between these two interesting problems.