- This event has passed.
CCMS Colloquium: Sinai Robins (University of São Paulo – Brazil)
Abstract: We introduce the geometry of numbers, beginning from the first principles, and proceeding to modern research topics in the field. Our point of view includes Fourier analysis on compact sets and in particular on convex bodies. We give some recent extensions of previous results by C. L. Siegel, and E. Bombieri, both of whom extended Minkowski’s first theorem for convex, centrally symmetric bodies.
A discrete version of these results allows us to give some new formulations for finite sums of discrete covariograms, for any finite set of integer points in Euclidean space. We’ll give visual examples, with pictures, of everything in dimension 2. On the other hand, a continuous application of these results allows us to shed additional light on the enumeration of lattice points in polytopes in R^d, and in particular some Ehrhart-type theory. This is joint work with Michel Faleiros.
Bio: Professor Sinai Robins completed his Ph.D. in Number Theory at UCLA. He has held tenured positions at Temple University and Nanyang Technological University and is currently a full professor at the University of São Paulo, Brasil. Professor Robins enjoys tackling problems in Discrete Geometry, Fourier Analysis, Number Theory, Combinatorial Optimization, and Data Science. Some of his recent research is reflected in his new 2024 book, called Fourier analysis on polytopes and the geometry of numbers: Part I, a gentle introduction. This book offers a unique, modern, Fourier-analytic perspective on the geometry of numbers. It explores the relationship between geometric properties of a polytope and its Fourier transform, provides new streamlined proofs of fundamental results, and invites undergraduates to explore these important ideas.
