About Melanie Matchett Wood's Work
My research tackles some of the oldest, yet still unsolved, mysteries of mathematics. In ancient Greece, Diophantus asked how many integral solutions equations have (for example, one integral solution to y^2=x^3+3x+5 is x=4, y=9). My research works towards an answer by exploiting a connection between the number of solutions to equations and the geometry of the shapes they define. In order to count integral solutions, I need to count solutions in finite number systems. This work helps elucidate the structures underlying the encryption algorithms that protect all of our data online. My research aims to answer questions Gauss asked 200 years ago about how numbers factor, not only into integers such as 6=2×3 but also into more complicated numbers such as 6=(√7-1)x(√7+1). I use a probabilistic approach to the seemingly deterministic question of how integers factor, with the aim of showing that mysterious microlevel algebraic structures aggregate into clear global patterns of factorization.