Geometry and Representation Theory of Symplectic Resolutions

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Algebraic geometry is the study of solution sets of systems of polynomial equations called varieties. Symplectic varieties are varieties equipped with a notion of length so that around each point the variety looks like an even-dimensional vector space. Symplectic structures arise naturally in classical mechanics, the branch of physics that describes the motion of macroscopic objects. The main goals of this research project are to use tools from linear algebra and geometry to better understand and exploit natural symmetries enjoyed by symplectic varieties known as symplectic resolutions. The work is expected to strengthen connections of the field with geometric representation theory, combinatorics, and string theory. In more detail, this research project consists of four interrelated subprojects. The first subproject is to study symplectic duality between pairs of symplectic resolutions. This duality is closely related to mirror duality for 3-dimensional gauge theories. The second subproject is to prove a conjecture relating the intersection cohomology of a symplectic cone to the quantum cohomology of its symplectic resolution. The third subproject is to classify hypertoric varieties using zonotopal tilings, much in the same way that toric varieties are classified by fans. The fourth subproject is to study a new invariant of a matroid called its Kazhdan-Lusztig polynomial.