# Higher Structures in China II

### From CRCG-Wiki

Recently, higher categories have appeared in many aspects of mathematics. Higher Structures in China is a series of conferences aimed at bringing together international and Chinese mathematicians in various intersecting areas of research to facilitate the exchange ideas and further the development of applications of higher structures in their respective fields.This is the second conference of the series, and it will be held at the Mathematics School & Institute of Jilin University, Changchun, Jilin Province, from August 8-10, 2011. The first conference of Higher Structures in China was held in October 2010.

The program will include mini-courses and expository and research talks by participants. We hope to also provide ample opportunities for participants to interact informally to explore common interests and to generate ideas for future collaboration. (Click here for more pictures).

## Contents |

## Orgnaizers

- Yunhe Sheng (Jilin University):
*ysheng888(at)gmail.com, +86-150-4405-4878* - Chenchang Zhu (Universität Göttingen)
- Weiwei Pan (Universität Göttingen)
- Zhangju Liu (Peking University)

## **Schedule**

### Schedule for Monday 8 Aug

09:10 - 09:30 | REGISTRATION and PHOTO | |
---|---|---|

Chair | Yunhe Sheng | |

09:40 - 10:30 | John Baez | Higher gauge theory, division algebras and superstrings I [slides] |

10:50 - 11:40 | Nicholas Proudfoot | Categorification via geometry I |

11:40 - 14:30 | LUNCH BREAK | |

Chair | John Baez | |

14:30 - 15:20 | Li Guo | Operads and Manin Products I [slides] |

15:30 - 16:20 | Dorette Pronk | Equivariant cohomology with local coefficients for representable orbifolds I[slides] |

16:20 - 16:40 | COFFEE BREAK | |

16:40 - 17:30 | Chenchang Zhu | Higher extensions of Lie algebroids, integration of Courant algebroids and string Lie 2-algebras [slides] |

18:00 - ??? | DINNER |

### Schedule for Tuesday 9 Aug

Chair | Chenchang Zhu | |
---|---|---|

09:00- 09:50 | John Baez | Higher gauge theory, division algebras and superstrings II [slides] |

10:00 - 10:50 | Nicholas Proudfoot | Categorification via geometry II |

11:10 - 12:00 | Ke Wu | Categorification of Heisenberg algebra and its application[slides] |

12:00 - 14:30 | LUNCH BREAK | |

Chair | Nicholas Proudfoot | |

11:10 - 12:00 | Chenming Bai | Lie analogues of Loday algebras and successors of operads[slides] |

15:30 - 16:20 | Dorette Pronk | Equivariant cohomology with local coefficients for representable orbifolds II[slides] |

16:20 - 16:40 | COFFEE BREAK | |

16:40 - 17:30 | Shusen Ding | Norm Inequalities for Differential Forms and Related Operators [slides] |

18:00 - ??? | Banquet |

### Schedule for Wednesday 10 Aug

Chair | Weiwei Pan | |
---|---|---|

09:00- 09:50 | John Baez | Higher gauge theory, division algebras and superstrings III [slides] |

10:00 - 10:50 | Nicholas Proudfoot | Categorification via geometry III |

14:30 - 15:20 | Li Guo | Operads and Manin Products II [slides] |

12:00 - 14:30 | LUNCH BREAK | |

Chair | Dorette Pronk | |

14:30 - 15:20 | Melchior Grutzmann | Cohomology theories of H-twisted Lie and Courant algebroids [slides] |

15:30 - 16:20 | Rongmin Lu | Higher algebraic structures in parabolic geometry |

18:00 - ??? | Dinner |

## Program

The scientific program starts on Monday, August 8th, and ends on Wednesday, August 10th.

We will also be organizing a trip to Changbai Mountain for participants from Aug. 3 to Aug 6.

**Minicourse by John Baez:**

*Tttle:*Higher gauge theory, division algebras and superstrings

*Abstract:*http://math.ucr.edu/home/baez/susy/

**Minicourse by Nicholas Proudfoot:**

*Title:*Categorification via geometry http://pages.uoregon.edu/njp/Changchun.html

*Abstract:*

- Lecture 1: A geometric realization of the regular representation of the symmetric group

- We construct two commuting actions of the symmetric group on the cohomology of the flag variety, and show that these are isomorphic to the left and right actions on the regular representation.

- Lecture 2: Categorification of the regular representation of the symmetric group

- Next we promote this action to a pair of braid group actions on a certain category of D-modules on the flag variety (Bernstein-Gelfand-Gelfand category O), such that when we pass to the Grothendieck group, we recover the regular representation of Lecture 1.

- Lecture 3: Geometric category O and symplectic duality

- All of the beautiful structure on BGG category O, including the two commuting braid group actions, come from the geometry of the cotangent bundle of the flag variety. In this lecture we show that, given any sufficiently nice symplectic variety (of which the cotangent bundle of the flag variety is a special case), we can construct a category that shares many of the properties of BGG category O.

- All of the material in Lectures 1 and 2 are due to other people. All of the material in Lecture 3 is joint work with Braden, Licata, and Webster.

**Minicourse by Dorette Pronk**

*Title:*Equivariant cohomology with local coefficients for representable orbifolds.

*Abstract:*Orbifolds are spaces that can locally be described as the quotient of an open subset of Euclidean space by a smooth action of a finite group. To obtain a notion of morphism between orbifolds that is appropriate to study orbifold homotopy theory, orbifolds can be represented by proper foliation groupoids (i.e., Lie groupoids with a proper diagonal and discrete isotropy groups). Two such groupoids represent the same orbifold precisely when they are Morita equivalent. More specifically, the category of orbifolds can be viewed as the bicategory of fractions of the 2-category of Lie groupoids with respect to the essential/Morita equivalences. An orbifold is representable when it can be obtained as the quotient of a manifold by the action of a compact Lie group. The subcategory of representable orbifolds is equivalent to the bicategory of fractions of the 2-category of translation groupoids with equivariant maps with respect to equivariant Morita equivalences. This prepares the way to generalize invariants from equivariant Bredon cohomology to orbifolds. We construct an equivariant fundamental groupoid and describe Bredon cohomology with twisted/local coefficients for orbifolds. This is joint work with Laura Scull.

**Talk by Melchior Grutzmann**

*Title:*Cohomology theories of H-twisted Lie and Courant algebroids

*Abstract:*In the first part I introduce the structure of H-twisted Lie and Courant algebroids together with examples, some of which are motivated by supermanifolds. In brief we ask for the usual axioms of Lie (resp. Courant) algebroids, but twist the Jacobi identity by a 3-form H with values in the kernel of the anchor map. In addition we have a closedness condition for this H. Already the twist over a point (the algebra) gives new structures.

- In the second part I introduce three cohomology theories for subclasses of H-twisted Lie algebroids. The first occuring from the naive idea of cutting down cochains until the naive differential from the Cartan formula (that works for Lie algebroids) squares to 0. Also examples will be given. The second one is for splittable H-twisted Lie algebroids which are in 1:1-correspondence to symplectic dg-manifolds of maximal degree 3. This structure is also known under the name Lie algebra with representation up to homotopy in the literature. The third one finally works for regular anchor maps where we can promote its kernel to a degree 2 vector bundle and the underlying vector bundle of the algebroid to degree 1. Then it is possible to introduce another dg-structure that defines a third kind of cohomology.

**Talk by Rongmin Lu**

*Title:*Higher algebraic structures in parabolic geometry

*Abstract:*Let G be a semi-simple Lie group, with Lie algebra g, and P be a parabolic subgroup with Lie algebra p. A parabolic geometry on a manifold M is then given by a principal P-bundle over M equipped with a principal g-valued connection satisfying certain conditions. It is known that a parabolic geometry carries a Lie algebroid structure, but recent progress made by various researchers have uncovered more general algebroid structures. In this talk, we give a survey of these developments.

**Talk by Shusen Ding**

*Title:*Norm Inequalities for Differential Forms and Related Operators

*Abstract:*Differential forms are extensions of functions, and have become invaluable tools for many fields of sciences and engineering, including theoretical physics, general relativity, potential theory and electromagnetism. Differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. For instance, some kinds of differential forms are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals, and certain geometric invariance. The theory about operators applied to functions has been very well studied. However, the study about operators, such as Green’s operator G, the Laplace-Beltrami operator , the maximal operator M, and the homotopy operator T, applied to differential forms just began. As we know, the operator is widely used to define the different versions of the harmonic equations. Green’s operator G is often applied to study the solutions of various differential equations and to define Poisson's equation for differential forms. These operators are critical tools to establish existence and regularity for solutions to PDEs, and to control oscillatory behavior on a manifold. In many situations, the process to solve a PDE or a system of PDEs involves integration or integral estimate (for numerical solutions) and the operators are often used to represent solutions. Hence, we have to investigate the involved operators and their compositions. The main purpose of this presentation is to introduce some recent results about differential forms, including inequalities with different norms, such the p L -norms, BMO norms, L-norms and Orlicz norms. We will also discuss inequalities for the related operators and their compositions.

## Registration

The deadline for registration is **May 31st, 2011**.

If you wish to register for Higher Structures in China II, please contact Yunhe Sheng at ysheng888(at)gmail.com.

We invite proposals for mini-courses or talks from all participants. If you wish to give a talk at Higher Structures in China II, please send the title and the abstract to Yunhe Sheng.

International participants of Higher Structures in China II are also invited to give their talk at Capital Normal University in Beijing (before or following the conference). Participants who wish to give an additional talk at Capital Normal University should contact Yunhe Sheng.

All local expenses of speakers and a limited number of conference participants (these include: meals, lodging, the conference excursion) will be covered by Jilin University.

## Sponsors

Higher Structures in China II is sponsored by the Mathematics School & Institute of Jilin University, Changchun, China and the Courant Research Center, Georg-August-Universität Göttingen.

## Accommodation

All participants will be housed in June Hotel, situated at 811 Xiuzheng Road, near the intersection of Qianjin Street and Xiuzheng Road (see it on Google maps).

A room will be booked for you once you've registered.

## Getting to Changchun

The city of Changchun is the capital of Jilin Province.

- By air: The Changchun Longjia International Airport is 31 km from the downtown.
- By train: The main train station, the Changchun Railway Station, is located in the city center and has lines connecting Changchun with most major cities in China. If you are planning to travel by train to/from Changchun, you may wish to consult with Yunhe Sheng, as booking train tickets in China may be a bit complicated for the uninitiated.

## Participants

- John Baez (Centre for Quantum Technologies)
- Melchior Grutzmann (Sun Yat-Sen University)
- Nicholas Proudfoot (University of Oregon)
- Dorette Pronk (Dalhousie University)
- Li Guo (Rutgers University)
- Shusen Ding (Seattle University)
- Rongmin Lu
- Du Li (Universität Göttingen)
- Ke Wu (Capital Normal Uni.)
- Zhixi Wang (Capital Normal Uni.)
- Jie Yang (Capital Normal Uni.)
- Na Wang (Capital Normal Uni.)
- Lifang Wang (Capital Normal Uni.)
- Xiaoyu Jia (Capital Normal Uni.)
- Binsheng Lin (Capital Normal Uni.)
- Yifan Zhang (Capital Normal Uni.)
- Wenjing Chang (CAS)
- Chengming Bai (Chern Institute)
- Deshou Zhong (China Youth Uni. for Political Sciences)
- Zhujun Zheng (South China Uni. of Technology)
- Fang Huang (South China Uni. of Technology)
- Shaohan Chen (South China Uni. of Technology)
- Wei Chen (South China Uni. of Technology)
- Yanbin Yin (Henan University)
- Yanjun Chu (Henan University)
- Runxuan Zhang (Northeast Normal University)