Mahler's Conjectures date back to the 1930's while Bourgain's Conjectures to the 1980's. They both speculate that the cube and simplex are minimizers of certain functionals related to volumes of convex bodies in Euclidean space. While these problems are phrased purely in terms of (real) Euclidean space, in recent years it has been realized that they are deeply related to fundamental notions of several complex variables as well as complex geometry. I will attempt to describe some ideas from this beautiful story spanning over 80 years of mathematics.

Based on joint work with B. Berndtsson and V. Mastrantonis.

Bio: Rubinstein received his doctorate from M.I.T. in 2008. He then held positions at Johns Hopkins University and Stanford University before accepting a tenured position at the University of Maryland in 2012 where he has been since. His research has spanned problems originating from geometric analysis, algebraic geometry, microlocal analysis, convex analysis, and numerical analysis. Since 2012 he has been involved in initiatives for the development and expansion of Research Experiences for Undergraduates (REUs) and has edited a volume on ``Directions for Mathematics REUs" in 2016. In 2023 he received the University of Maryland Grand Challenges award for a project aimed at increasing the participation of underrepresented minorities and women in STEM.