Contact Geometry of Links of Singularities and Affine Varieties

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The aim of this project is to start a program establishing a connection between two different areas of geometry. The first area is called symplectic geometry and the second area is called algebraic geometry. Symplectic geometry involves the study of classical physical systems, for instance pendulums, moving particles, other devices with possibly many moving parts. It gives a nice way of describing the position and velocity of these systems in a unified way. Algebraic geometry is the study of solutions of certain fundamental equations called polynomial equations. These equations are used in many areas of mathematics, and even outside of mathematics, such as physics and computer science. The solutions to these equations have a shape, which in some places is a nice smooth object, but also can have regions which are not smooth called singular points. In this project, we explore a deep relationship between these singular points and certain families of classical physical systems. Many properties of singularities have corresponding purely dynamical interpretations in these physical systems which will be explored further as very little is currently known. The study of singularities is especially important in algebraic geometry at the moment due to recent advances in an program called the minimal model program. Part of this project also involves teaching graduate students how to use various fundamental tools from symplectic geometry. Such tools are featured in this program. The primary aim of this project is to understand the relationship between symplectic/contact geometry and affine algebraic geometry. This relationship will be studied using pseudo-holomorphic methods that were originally introduced by Gromov and have now developed into important tools such as Gromov-Witten invariants, Floer homology, and Symplectic Field Theory. Mumford gave us a criterion to tell us when a complex surface is smooth at a particular point or not from its link. The PI proved a higher dimensional version of Mumford's result where the link now has a natural contact structure. Part of the aim of this proposal is to start a program relating the algebraic properties of higher dimensional singularities with their links as contact manifolds in the same way that Mumford's paper did so for surface singularities. Much of this work is also directly related to recent advances in the minimal model program. As a starting point we hope to generalize at least parts of a result by Neumann giving us the topology of a resolution from the link. In a related project, we hope to understand the McKay correspondence for quotient singularities with crepant resolutions using Floer theoretic ideas. This might lead us to extend this correspondence to quotient singularities that do not have crepant resolutions, and even to more general singularities. The PI will give many examples of contact manifolds not contactomorphic to links of singularities. The PI will also look at affine varieties from a symplectic perspective by trying to understand how properties such as log Kodaira dimension and rational connectedness relate to the symplectic structure. This project also involves teaching graduate students how to apply pseudo-holomorphic methods in various situations.