# Topology and Geometry of Configuration Spaces

## Wintersemester 2013

- Blockseminar -

Despite its title a large part of the seminar will be concerned with algebra and representation theory. Recall that a homomorphism of a finite group G to the general linear group GL(V) of some finite dimensional complex vector space V is called a representation of G. In this seminar we want to first learn about the basic representation theory of the symmetric groups S_n.

In nature such representations often appear in sequences V_n, where V_n is a representation of S_n. The notion of representation stability recently introduced by Church and Farb makes precise how such a sequence of representations over different groups can stabilize. The main goal of the seminar is to learn the necessary language and study the proofs of some basic results concerning representation stability.

The last part of the seminar should then give a concrete application of the theory. The space of n distinct particles in a given space X, a subspace of the n-fold cartesian product of X, is called the nth configuration space of X and comes equipped with an obvious action of the symmetric group S_n. The cohomology groups of these spaces are therefore a prototypical example of a sequence of representations V_n. In the case, where X is a manifold of dimension >1 this sequence is representation stable by a recent result of Church.

Even though time may not allow a detailed treatment, we mention that the cohomology of the moduli spaces of curves with n marked points is another natural example of a sequence of representations.

## About the format

The block seminar will be held jointly by students from Freie Universität, Humboldt Universität,

and Polish students from Wroclaw and Warsaw Universities. It is organized as a cooperation of Gavril Farkas (HU), Holger Reich (FU)

and Tadeusz Januszkiewicz (IMPAN).

The block seminar will take place in Bedlewo, Poland on two weekends:

**Friday, December 6th, 2013 **till** Sunday, December 8th, 2013** and

**Thursday, January 16th, 2014 **till** Sunday, January 19th, 2014**.

For students from Berlin there will be at least three preparation meetings at 10:30 a.m. in the BMS Loft (Urania) on November 1st, November 15th, and November 29th, 2013, each preceding the BMS Fridays.

For the Polish students separate meetings will be scheduled.

Interested students should contact Gavril Farkas or Holger Reich via email.

## Prerequisites

The seminar will introduce students to a very active area of current research. For most of the seminar knowledge of the basic notions from algebra and topology is sufficient. Of course for the last talks some background in algebraic topology is required in order to talk about the cohomology of configuration spaces.

## Schedule

First trip: December 6th till December 8th, 2013

Fr | 20:00 | 1 | Representation theory of finite groups | (Myriam Mahaman) |

21:00 | 2 | Representation theory of the symmetric groups | (Constantin Scherr) | |

22:00 | BIRS-Video (Overview) | |||

Sa | 09:00 | 3 | Schur polynomials, the Littlewood-Richardson-rule, and its applications | (Stefano Filipazzi) |

10:30 | 4 | FI-modules | (Patrick Da Silva) | |

14:30 | 5 | FI#-modules | (Irene Schwarz) | |

16:30 | 6 | Stability degree, Part 1 | (Matthew Spong) | |

Su | 09:00 | 7 | Stability degree, Part 2 | (Daniel Lütgehetmann) |

10:30 | 8 | Stability degree and representation stability | (Peter Patzt) |

Second trip: January 16th till January 19th, 2014

Th | 20:00 | 9 | Review | (Sebastian Meinert) |

Fr | 09:00 | 10 | Noetherian property and consequences, Part 1 | (Dimitrios Patronas) |

10:30 | 11 | Noetherian property and consequences, Part 2 | (Mark Ullmann) | |

14:30 | 12 | Spectral sequences | (Karol Strzałkowski) | |

16:30 | 13 | Sheaves and derived functors, Part 1 | (Michał Marcinkowski) | |

18:00 | 14 | Sheaves and derived functors, Part 2 | (Łukasz Garncarek) | |

Sa | 09:00 | 15 | Cohomology of configuration spaces, Part 1 | (Fabian Lenhardt) |

10:30 | 16 | Cohomology of configuration spaces, Part 2 | (Emre Sertöz) | |

14:30 | 17 | Overview: the cohomology of mapping class groups | (Antonio Günzler) | |

16:30 | 18 | Ingredients: Birman exact sequence, Hochschild-Serre SS, Forgetful map | (Karl Christ) | |

18:00 | 19 | Proof of representation stability | (Swjatoslaw Gal) |

## Literature

### General literature:

Thomas Church, Jordan S. Ellenberg, Benson Farb: *FI-modules: a new approach to stability for S_n-representations*

Two workshop videos from *Cohomological methods in geometric group theory*:

Thomas Church: *Representation stability, FI-modules, and beyond*

Thomas Church: *Homological algebra of FI-modules*

### I. Representation theory of the symmetric groups

William Fulton, Joe Harris: *Representation Theory*, GTM 129 (general theory for finite groups in Part 1, symmetric group representations Chapter 4)

Gordon James, Martin Liebeck: *Representations and Characters of Groups*, Cambridge Mathematical Textbooks (easy to read introduction to the general theory)

Gordon James, Adalbert Kerber: *The representation theory of the symmetric group*, Encyclopedia of Mathematics and its Applications 16 (Sections 1.1 - 2.4 give a thourough introduction to the representation theory of the symmetric groups)

### II. FI-modules

Thomas Church, Jordan S. Ellenberg, Benson Farb: *FI-modules: a new approach to stability for S_n-representations*

Wolfgang Lück: *Transformation groups and algebraic K-theory*, LNM 1408 (Chapter II Section 9)

### III. Noetherian Property

Thomas Church, Jordan S. Ellenberg, Benson Farb: *FI-modules: a new approach to stability for S_n-representations* (Section 2.7)

Thomas Church, Jordan S. Ellenberg, Benson Farb, Rohit Nagpal: *FI-modules over Noetherian rings*

### IV. Configuration Spaces

John McCleary: *A User's Guide to Spectral Sequences*, Cambridge Studies in Advanced Mathematics 58

Burt Totaro: *Configuration spaces of algebraic varieties*, Topology 35 (1996), no.4, pp. 1057-1067

Thomas Church, Jordan S. Ellenberg, Benson Farb, Rohit Nagpal: *FI-modules over Noetherian rings* (Section 4)

Thomas Church: *Homological stability for configuration spaces of manifolds*, Invent. Math. 188 (2012), no. 2, pp. 465-504

### V. Moduli spaces

Rita Jimenez Rolland: *Representation stability for the cohomology of the moduli space M_g^n*

Ravi Vakil: *The moduli space of curves and its tautological ring*, Notices AMS 50 (2003), pp. 647-658

Carel Faber, Rahul Pandharipande: *Tautological and non-tautological cohomology of the moduli space of curves*, Handbook Of Moduli (Vol. 1), pp. 293-330

Carel Faber: *A conjectural description of the tautological ring of the moduli space of curves, in Moduli of curves and abelian varieties*, Aspects of Mathematics, Vieweg (1999), pp. 109-129

Richard Hain and Eduard Looijenga: *Mapping Class Groups and Moduli Spaces of Curves*, in Algebraic Geometry—Santa Cruz 1995, Proc. Symp. Pure Math. 62, Part 2 (1997), pp. 97-142

Joe Harris and Ian Morrison: *Moduli of curves*, Springer GTM 187

John Harer: *Stability of the homology of the mapping class groups of **orientable surfaces*, Ann. of Math. 121 (1985), pp. 215-249