Convex Bodies, Fans and Algebraic Geometry

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In my research project, I plan to work on Newton polyhedra theory, the theory of Newton-Okounkov bodies, and tropical geometry. These theories connect convex geometry and geometry of fans (key objects from piecewise liner geometry) with algebraic geometry. These connections are very deep. They make sense even on the basic level of calculating geometrical volumes and counting the solutions of systems of algebraic equations***Newton polyhedral theory was started in 1975 with the celebrated Bernstein-Kushnirenko theorem, which gives a formula for the number of solutions of n polynomial equations in n variables using volumes of the Newton polyhedra associated with the equations. I found many proofs and extensions of this theorem, and nowadays it is often referred to as the BKK theorem where the last K stands for my name. ***I had always dreamed of extending the BKK theorem to arbitrary algebraic varieties, but it was not clear what kind of objects could replace the role of Newton polyhedra in such generality. It was only after a brilliant idea from A. Okounkov that appropriate objects have been defined and named Newton-Okounkov bodies. The theory of Newton-Okounkov bodies was systematically developed and generalized in my work with K. Kaveh (in particular we found the most general version of the BKK theorem) and independently by R. Lazarsfeld and M. Mustata. Since then there has been a burst of research activity in this area where many papers have appeared (and continue to appear). There have also been many conferences and workshops on this new subject.***Tropical geometry provides a wonderful relation between algebraic geometry and piecewise linear geometry. One of the original works in tropical geometry is the celebrated work of G. Mikhalkin, which explains how to solve algebraic problems by analyzing a planar diagram. A multidimensional version of this approach relates algebraic geometry with the geometry of fans. It could be considered as an extension of the BKK theorem (from complete intersections to general subvarieties in the torus).***I plan to address many concrete problems related to the Newton polyhedra theory, the theory of Newton-Okounkov bodies and tropical geometry. Some other subjects I plan to work on are my topological Galois theory which explains why many equations could not be solved by explicit formulas and my “theory of Fewnomials” whose concept is that “simple” and not cumbersome systems of equations should define sets with “simple” topology (this theory has proved to be a very powerful tool). This huge program involves research in very different areas and suits perfectly to attract young mathematicians. This research is of general interest for Pure Mathematics.