J-Holomorphic Curves and Gromov-Witten Invariants (25 December 2017 to 04 January 2018)

source: International Centre for Theoretical Sciences       2018年1月4日
DATE & TIME: 25 December 2017 to 04 January 2018
VENUE: Madhava Lecture Hall, ICTS, Bangalore

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.

1:28:11 Classical background by Mohammed Abouzaid
1:26:43 Introduction to Contact Geometry by Dheeraj Kulkarni
3  53:40 Spectra and stable homotopy theory (Remote Talk) by Samik Basui
1:28:12 Introduction to h-principle by Mahuya Datta
1:29:44 Floer theory by Mohammed Abouzaid
1:34:41 Existence of Symplectic and Contact forms by Mahuya Datta
1:05:40 Legendrian and Transverse Knots by Dheeraj Kulkarni
1:35:25 Spectra by Mohammed Abouzaid
1:26:50 Introduction to legendrian contact homology using pseudo-holomoprhic... by Michael G Sullivan
10 1:06:00 Isocontact and isosymplectic immersions and embeddings by Mahuya Datta
11 1:04:49 A quick review of infinity algebras by Somnath Basu
12 1:30:24 Floer homotopy theory by Mohammed Abouzaid
13 1:24:28 Topological Strings and String Dualities by Rajesh Gopakumar
14 1:32:43 Knot contact homology and related topics by Michael G Sullivan
15 59:40 Introduction to Gromov-Witten Invariants by Ritwik Mukherjee
16 1:03:16 Rack theoretic invariants for Legendrian knots: First few steps by Dheeraj Kulkarni
17 1:24:21 Introduction to Gromov-Witten Invariants by Ritwik Mukherjee
18 1:28:45 Cellular legendrian contact homology by Michael G Sullivan
19 1:09:03 Moduli Space of Curves by Chitrabhanu Chaudhuri
20 1:37:09 Topological Strings and String Dualities (Lecture - 02) by Rajesh Gopakumar
21 1:29:41 Augmentations, generating families and micro local sheaves by Michael G Sullivan
22 1:19:32 Gromov–Witten Invariants and the Virasoro Conjecture (Remote Talk) by Ezra Getzler
23 1:23:28 Gromov–Witten Invariants and the Virasoro Conjecture - II (Remote Talk) by Ezra Getzler
24 1:31:44 Symplectic homology, algebraic operations on it and their applications by Janko Latschev
25 1:19:10 Lagrangian Floer theory by Sushmita Venugopalan
26 1:25:48 Fukaya category of a Hamiltonian fibration (Lecture – 01) by Yasha Savelyev
27 1:27:55 Fukaya category of a Hamiltonian fibration (Lecture – 02) by Yasha Savelyev
28 1:24:42 Gromov–Witten Invariants and the Virasoro Conjecture. III by Ezra Getzler
29 48:12 Lagrangian Floer theory (Lecture – 02) by Sushmita Venugopalan
30 1:33:41 Symplectic homology, algebraic operations on it and their applications by Janko Latschev
31 1:39:21 Transversality and super-rigidity in Gromov-Witten Theory by Chris Wendl
32 1:26:14 Transversality and super-rigidity in Gromov-Witten Theory (Lecture – 02) by Chris Wendl
33 1:32:03 Fukaya category of a Hamiltonian fibration by Yasha Savelyev
34 1:04:21 Symplectic homology, algebraic operations on (Lecture – 03) by Janko Latschev
35 1:31:15 Transversality and super-rigidity in Gromov-Witten Theory (Lecture - 03) by Chris Wendl
36 1:27:54 Symplectic homology, algebraic operations on (Lecture - 04) by Janko Latschev
37 1:36:10 Transversality and super-rigidity in Gromov-Witten Theory (Lecture - 04) by Chris Wendl

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