Professor Kiumars Kaveh was awarded a 2016 Simon's Fellowship. The title of his project is "Convex bodies and algebraic geometry". This is a highly competitive program that provides funds to faculty for up to a semester long research leave from classroom teaching and administrative obligations. Kaveh's main areas of research are algebraic geometry and Lie theory. He is interested in combinatorial aspects of algebraic geometry and representation theory, and their connections with convex geometry. One of his current research interests is the theory of Newton-Okounkov bodies. This theory was initiated by seminal work of Andrei Okounkov (2006 Fields medalist).
Algebraic geometry is concerned with the study of algebraic varieties. An algebraic variety is the set of solutions of a number of polynomial equations in several variables (often considered over complex numbers). Toric varieties are a special class of varieties dened by binomial equations. In the past few decades, convex geometry and combinatorics of convex polytopes have played important parts in algebraic geometry, representation theory and symplectic geometry. This reaches its climax in the theory of toric varieties where there is a beautiful correspondence between the geometry of toric varieties and combinatorics of convex polytopes. The newly emerged theory of Newton-Okounkov bodies attempts to vastly generalize the correspondence between toric varieties and convex lattice polytopes. The Newton-Okounkov bodies are a far generalization of the (classical) notion of the Newton polytope of a (Laurent) polynomial in several variables. More>