New Frontiers in Symplectic Topology

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Symplectic geometry originated in the study of classical mechanics as a general setting for studying conservative dynamics. Surprisingly, in the last few decades, mathematicians have found symplectic structures are relevant in many areas very far from mechanics, including gauge theory, algebraic geometry, representation theory and string theory. The proposed research uses symplectic geometry as a bridge between some of these disparate areas of mathematics.The proposal has three strands, which are quite distinct in nature, but are tied together by ideas from symplectic geometry.In the first strand of the research, we examine a very curious conjecture at the interface between Hamiltonian dynamics and birational geometry. In birational geometry, there is an important class of singular spaces called "compound Du Val (cDV) singularities" which arise in Mori's famous minimal model program for classifying 3-dimensional algebraic varieties. If you look very close to these cDV singularities, you find a natural class of dynamical systems (Reeb flows on the link) and it seems in examples that the dynamics of the Reeb flow tells you about whether you can resolve the singularity by introducing only 1-dimensional curves. We aim to prove a strong version of this conjecture, first in a simple case (compound A_n) and then in general.In the second strand of the research, we study a class of 4-dimensional spaces from algebraic geometry (algebraic surfaces of general type). Surfaces of general type have very complicated topology, and provide a wonderful testing ground for our understanding of 4-dimensional space. There has been a lot of progress recently in understanding how such spaces can degenerate, and we want to use this to answer some long-open topological questions about these 4-dimensional spaces.In the third strand of the research, our goal is to give a construction of topological invariants of low-dimensional manifolds using algebraic geometry. Our approach is informed by the homological mirror symmetry conjecture which relates symplectic geometry to algebraic geometry.