ICTS

Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex structure. The moduli space of these curves (called pesudoholomorphic curves) is typically non-compact and not well-behaved. A nice compactification, due to Gromov, allows us to define certain invariants known as Gromov-Witten invariants.

The theory of Gromov-Witten invariants can be used to deform the usual cohomology ring structure of a symplectic manifold. This has connections, on the one hand, with enumerative geometry, and on the other hand, with deformation of associative algebras and topological quantum field theory. Gromov-Witten invariants have deep connections with physics through the ideas of Mirror Symmetry. Using String Theory, physicists have made amazing predictions about the Gromov-Witten invariants of the quintic threefold. From a mathematical point of view, a large number of predictions are still open.

The study of pseudoholomorphic mappings (with Lagrangian boundary conditions) of Riemann surfaces with boundaries (disks, cylinders, pairs of pants) have turned out to be immensely powerful as a tool and have yielded great results. The original theory of Floer has been extended to study Floer homology of a non-compact symplectic manifold. The invariants arising from Floer theory of cotangent bundle have deep and surprising connections to the algebraic structures present in the free loop space of the ambient manifold.

In this program, several international experts will deliver series of 3 to 4 lectures on their work. To be more easily accessible to the participants, the lecture series by experts are being planned so that they start with simple introduction and gently build up, leading to open questions in their area of expertise. The topics of these lectures are interconnected and will often use the theory of pseudoholomorphic curves as their foundation. There will be 4 lectures each day for a total duration of about 5 hours. Informal lectures / discussions sessions in the evening, complementing the actual lectures, are also being planned for the benefit of the participants.

A preparatory school at NISER in July 2017 is separately being planned to acquaint interested participants with this subject and help them get the most out of the discussion meeting at ICTS. This is the link for the preparatory summer school at NISER.