Convex Bodies, Algebraic Geometry, and Symplectic Geometry

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This project concerns questions at the intersection of algebra, geometry, and combinatorics. The main areas to be explored are algebraic and symplectic geometry. Algebraic geometry is one of the oldest and most fundamental branches of mathematics. It has a wide range of applications from coding and data security to high energy physics and string theory. The principal objects of study are algebraic varieties, the sets of solutions of systems of algebraic equations in several variables; classical examples are ellipses, parabolas, and hyperbolas. Symplectic geometry is the modern framework of classical mechanics. It also plays an important role in modern quantum theories in physics. The main objects of study are symplectic manifolds, which are abstractions of the notion of phase space from physics. An important class of symplectic manifolds are algebraic varieties. Through this connection, the research project explores interactions between algebraic geometry and symplectic geometry, with the goal of advancing understanding in both areas.A central theme of this project is to associate convex bodies, known as the Newton-Okounkov bodies, to algebraic varieties, encoding information about the geometry of the variety. It gives a general framework to extend the scope of convex geometry methods from toric varieties to general varieties. The project will explore applications of this area in symplectic geometry and other directions. Some fundamental topics to be addressed in the project are: (1) constructing Hamiltonian torus actions on general projective varieties; (2) a general approach to the "principle of independence of polarization" in geometric quantization for a large class of projective varieties; (3) establishing connections with tropical geometry and computational algebra; and (4) investigating the notion of entropy from the point of view of the theory of Newton-Okunkov bodies. Among the applications, the project is expected to contribute to mathematical physics and quantum mechanics, via geometric quantization. The project will also investigate connections with tropical geometry, a branch of algebraic geometry connected to convex optimization.