Geometry and Topology Summer School

Symplectic topology is rooted in the study of dynamical systems arising from Hamiltonian mechanics. These systems are volume preserving, since they conserve the symplectic form, and they exhibit many rigidity properties, e.g. homological lower bounds on the number of periodic orbits. On the other hand, many naturally occurring dynamical systems in symplectic topology are not conservative, but instead conformally shrink and expand the symplectic form. These conformally symplectic systems include contactomorphisms and the flows of Liouville vector fields. The dynamical behavior of these systems is closely related to many open problems in symplectic topology. This mini-course will serve as an introduction to conformally symplectic dynamics and the many interesting open questions in the area.

Dirac discovered his eponymous equation almost one hundred years ago while investigating relativistic quantum mechanics. Since then (the Riemannian version of) the Dirac equation has played a central role in the development of geometry and topology. In this minicourse, after introducing the basic setup of spin geometry and discussing the basic properties of the Dirac equation, we will discuss its relation with differential geometry, algebraic topology and low-dimensional topology, with concrete applications in mind.

Heegaard Floer homology is an invariant for low-dimensional manifolds whose definition is based on constructions in Lagrangian Floer homology. Heegaard Floer homology can be used to study questions in knot theory, three-dimensional topology, and smooth four-dimensional topology. The purpose of this mini-course is to give an introduction to this invariant, with an eye towards both its applications and computational tools.

I will try to tell a few stories about what happens when one tries to redo topologiy to ask questions that ask more than whether something exists, but instead how much of some resource need to be spent to produce such an object. These are useful for answering quantitative questions sometime (I'll give examples), but more so, they open whole new areas for investigation with many unsolved questions and connections to other areas of mathematics.