Derived Differential Geometry by Dominic Joyce & Applications of moduli spaces of J-holomorphic curves to symplectic geometry by Lino Amorim

2014.12.11 |

Date | Mon 03 Aug — Wed 12 Aug |

Time | 09:00 — 16:00 |

Location | Aud.D3 (1531-215), Department of Mathematics |

**Derived Differential Geometry **by

**Abstract:** Derived Differential Geometry is the study of derived smooth manifolds and orbifolds, where "derived" is in the sense of the Derived Algebraic Geometry of Jacob Lurie [12] and Toën-Vezzosi [14,15]. There are by now several different models for (higher) categories of derived manifolds and derived orbifolds: the "derived manifolds" of Spivak [13], the "d-manifolds" and "d-orbifolds" of Joyce [9,10], the "derived manifolds" of Borisov-Noel [1,2], and the "M-Kuranishi spaces" and "Kuranishi spaces" of Joyce [11]. For derived manifolds without boundary, these are all known to be roughly equivalent, at least at the level of homotopy categories.

Actually, a prototype version of derived orbifolds has been used for many years in the work of Fukaya, Oh, Ohta and Ono [4,5,6] as their "Kuranishi spaces", a geometric structure on moduli spaces of J-holomorphic curves in symplectic geometry, but it was not understood until recently that these are part of the world of derived geometry - see [11].

Derived manifolds are a (higher) category of geometric spaces which include ordinary smooth manifolds, but also many more singular objects - for instance, if *X*,*Y* are embedded submanifolds of a manifold Z, then the intersection *X* n *Y* has the structure of a derived manifold of dimension dim *X* + dim *Y* - dim *Z* (which can be negative), and any closed subset *S* of **R*** ^{n }*can be given the structure of a derived manifold of dimension

One reason derived manifolds and derived orbifolds are important is that many moduli spaces (families of isomorphism classes geometric objects) in differential geometry, and in complex algebraic geometry, can be given the structure of derived manifolds and derived orbifolds. For example, any moduli space of solutions of a nonlinear elliptic partial differential equation on a compact manifold is a derived manifold. Also, compact, oriented derived manifolds and derived orbifolds have "virtual classes", generalizing the fact that a compact, oriented n-dimensional manifold *X* has a fundamental class [*X*] in its top-dimensional homology group *H _{n}*(

This lecture course will define and discuss 2-categories of derived manifolds and derived orbifolds, and their applications to moduli spaces of solutions of nonlinear elliptic partial differential equations including *J*-holomorphic curves, and "counting" problems in differential geometry, and complex algebraic geometry. We explain two different ways to define these 2-categories, firstly using *C*^{∞}-rings, *C*^{∞}-schemes and* C*^{∞}-algebraic geometry [7,8], following Spivak [13], Joyce [9,10] and Borisov-Noel [1,2], or secondly using an "atlas of charts" (Kuranishi neighbourhoods) approach in [11], which is based on Fukaya et al. [4,5,6]. (See reading list below)

**Applications of moduli spaces of J-holomorphic curves to symplectic geometry **by

**Abstract:** The main goal of this course is to show how the moduli spaces of J-holomorphic curves can be used to associate to each Lagrangian submanifold an A-infinity algebra on its cohomology, following the work of Fukaya-Oh-Ohta-Ono.

We will start with a review of some basic concepts in symplectic geometry and an overview of the main questions in Lagrangian intersection theory. Then we introduce the moduli space of J-holomorphic curves and define Lagrangian Floer cohomology in Floer's original setting.

Next we analyze the difficulties encountered in trying to extend the definition of Floer cohomology to more general contexts. These lead us to consider A-infinity algebras. At this point we make a small digression into the homological algebra of A-infinity algebras, bimodules and categories.

We reach the heart of the course where we explain how by making the moduli spaces of J-holomorphic disks with boundary in a Lagrangian submanifold into Kuranishi spaces we can construct an A-infinity algebra on the cohomology of the Lagrangian. Additionally we define an A-infinity bimodule for each pair of Lagrangians.

Finally we organize all the Lagrangians for which Floer cohomology can be defined into an A-infinity category known as the Fukaya category of the ambient symplectic manifold. We end by discussing the Fukaya category's role in Kontsevich's Homological Mirror Symmetry Conjecture.

** **

**Social programme**

- Monday 3/8 at 6pm: Social networking incl. light supper. Free of charge.
- Tuesday 11/8 at 6pm: Special dinner at a restaurant in town. This costs DKK 375 (three course meal incl. a wine menu). Registration deadline Thursday 6 August 2015 - register at: auws.au.dk/default.aspx

** ****Included in the event**

- Morning and afternoon coffee/tea
- Lunches on weekdays
- Social dinner (first night)

In special cases, QGM is able to offer support for accommodation to junior researchers (PhD students and postdocs). Only applications received before 8 June 2015 will be considered.

** **

**Organizers**Jørgen Ellegaard Andersen (QGM, AU)

Niels Leth Gammelgaard (QGM, AU)

Organized in partnership with the Clay Mathematics Institute

**Participants**Alboresi, Davide (Utrecht University)

Amorim, Joana (Oxford Brookes University)

Bjerre, Mette (QGM, Aarhus University)

Bischoff, Francis (University of Toronto)

Branco, Lucas (University of Oxford)

Caputo, Raffaele (University of Hamburg)

Cotti, Giordano (SISSA)

Chan, Joseph (University of Melbourne)

Dvirnas, Albertas (Lund University)

Ens, Travis (University of Toronto)

Huang, Yang (QGM, Aarhus University)

Joonhyung, Kim (Hannam University)

Kanstrup, Tina (QGM/Aarhus University)

Kozhasov, Khazhgali (SISSA)

Marzioni, Simone, (QGM, Aarhus University)

Malusà, Alessandro (QGM, Aarhus University)

Matviichuk, Mykola (University of Toronto)

Rasmussen, Kenneth (QGM, Aarhus University)

Rembardo, Gabriele (Paris VI)

Ungheretti, Massimiliano (University of Copenhagen)

Vo, Huan (University of Toronto)

Wilkins, Nicholas (University of Oxford)

Zerouali, Ahmed (University of Toronto)

**Video**Video recordings of the lectures

**Reading list**

**Note:** A lot of these references are rather difficult to read for the uninitiated - the point of the lectures is to explain them. The most accessible are probably the survey papers [4], [8], [9], [14], plus [11], which needs only background in differential geometry.

[1] D. Borisov, *'Derived manifolds and Kuranishi models*', arXiv:1212.1153.

[2] D. Borisov and J. Noel, '*Simplicial approach to derived differential manifolds*', arXiv:1112.0033.

[3] E.J. Dubuc, '*C ^{∞}-schemes*', Amer. J. Math. 103 (1981), 683-690.

[4] K. Fukaya, '

[5] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, '

[6] K. Fukaya and K. Ono, '

[7] D. Joyce, '

[8] D. Joyce, '

[9] D. Joyce, '

[10] D. Joyce, '

[11] D. Joyce, '

[12] J. Lurie, '

[13] D.I. Spivak, '

[14] B. Toën, '

[15] B. Toën and G. Vezzosi, '