Equivariant and combinatorial techniques in algebraic and symplectic geometry

点击翻译按钮获取中文摘要

Many problems in both applied and pure Mathematics involve the solution of a system of equations. Algebraic Geometry is precisely the study of the space of solutions to systems of algebraic equations, and is therefore a core area of Mathematics. For instance, the proof of Fermat's Last Theorem gives properties of an algebraic- geometric object (an elliptic curve) associated to a non-trivial solution of Fermat's equation. Also, the theory of Mirror Symmetry in theoretical physics uses the language of algebraic geometry in a fundamental way. Algebraic geometry also has applications in quantum computing, mathematical biology, cryptography, and image and signal processing. Symplectic geometry, on the other hand, is the mathematical formulation of classical physics. The theory of symmetries and conservation laws within symplectic geometry has connections with representation theory, which can be thought of as the mathematical framework of quantum physics, and other areas of modern physics such as fluid mechanics. Combinatorial and convex geometry includes the study of polytopes, which are generalizations of the figures in plane geometry such as triangles, trapezoids, and parallelograms. The convex geometry of polytopes is important in many research areas, such as optimization theory. These three core areas of mathematics are related in many ways. The applicant proposes to study in detail several instances of these rich connections: Newton-Okounkov bodies, toric varieties and their stack analogues, and Goresky-Kottwitz-MacPherson theory. There are many aspects of the applicant's research program which form an effective training program for future scientists. The long-term benefits of this research program are two-fold: first, the results of this research will bring to light many new combinatorial techniques for analyzing the equivariant geometry of important spaces which arise in many real-world applications (e.g. Mirror Symmetry, cryptography, fluid mechanics, optimization theory), and secondly, the training aspects of this proposal will produce highly qualified personnel at the undergraduate, graduate, and postgraduate level, who possess competitive research and technical skills in important areas of geometry. **********