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source: Centre International de Rencontres Mathématiques 2015年8月7日

Jean-Morlet Chair - Research Talks - Lalonde/Teleman

Semester on 'Moduli Spaces in Symplectic Topology and Gauge Theory'

May - October 20154

General themes

Many areas of modern geometry lead naturally to moduli spaces classifying certain geometric objects and a minimality problem on these moduli spaces. For instance, Perelman's proof of the Poincaré conjecture uses the Ricci flow, hence a parabolic evolution equation on the moduli space of metrics. The same setting is also present in higher dimensions, where Aubin, Yau, Tian and Donaldson established fundamental existence theorems. In Contact-Symplectic topology, the focus of the program, the same concepts are also behind the recent spectacular proof by Taubes of the full Weinstein conjecture on the existence of closed orbits of the Reeb flow on contact manifolds, which uses implicitly Seiberg-Witten theory hence moduli spaces of monopoles and the Embedded Contact Homology developed by Hutchings. The ubiquitous Floer theory is present almost everywhere in symplectic topology, but has also found extraordinary applications in low-dimensional differential topology. The theory of $J$-holomorphic curves, which is the core of the Gromov-Witten theory, has been used in almost complex geometry at all levels, but also by Welschinger, Kharlamov, Itenberg and Salomon to derive new real enumerative invariants.

The proposed theme semester at CIRM will be a hub dedicated to the study of these questions that lie at the heart of the current developments in symplectic and differential topology, as plenary talks at the ICM's 1998, 2002, 2006 and 2010 show.

http://lalondeteleman.weebly.com

Nick Sheridan: Counting curves using the Fukaya category 1:11:23

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In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a ‘mirror manifold’. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his ‘homological mirror symmetry’ conjecture and stated that it would ‘unveil the mystery of mirror symmetry’. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is ‘less mysterious’ than the previous proof. This is based on joint work with Sheel Ganatra and Tim Perutz.

Recording during the thematic meeting: "Jean-Morlet Chair: Moduli spaces in symplectic topology and in Gauge theory" the June 4, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

Sheel Ganatra: The Floer theory of a cotangent bundle, the string topology of the base and... 1:06:22

Kai Cieliebak: On a question by Michele Audin 1:03:08

Tim Perutz: From categories to curve-counts in mirror symmetry 1:07:36

Emmy Murphy: Existence of Liouville structures on cobordisms 1:05:11

Jake Solomon: The degenerate special Lagrangian equation 1:06:16

Mohammed Abouzaid: Nearby Lagrangians are simply homotopic 58:56

John Pardon: Virtual fundamental cycles and contact homology 1:01:41

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