The joint poster of the summer school and the conference "Twists, generalised cohomology and applications" is available here.

There will be 3 lecture series (4 sessions per series):

André Henriques (Universiteit Utrecht, NL): "The string group via
conformal nets"

Notes are available here (for the first three lectures) and here (for the last)!

The string group String(n) is the 3-connected cover of the
orthogonal group O(n): it is an infinite dimensional topological group
whose homotopy groups vanish in degrees < 4 and that it is equipped
with a map to O(n) that is an isomorphism on homotopy groups above
degree 3. Given a real vector space with inner product V, a string
structure on V is a certain piece of extra data, call it S, with the
property that the automorphism group of the pair (V,S) is the string
group String(V).

The goal of this lecture series is to show that one can describe string
structures using defects between conformal nets (these are the
1-morphisms in a certain 3-category).

Conformal nets are functors from the category of 1-manifolds to that of
von Neumann algebras, and have their origins in a branch of
mathematical physics called algebraic quantum field theory. The
conformal nets that will be relevant for defining string structures are
the so-called "free fermion" conformal nets, and are among the simplest
ones to construct. There is a free fermion conformal net Fer(V)
associated to every real vector space V as above. The main result that
I will be presenting is that Fer(V)-Fer(R^n)-defects are string
structures.

Nitu Kitchloo (Johns Hopkins University, Baltimore, USA): "Geometry and
Topology of Kac-Moody groups"

Slides are available here!

Lecture 1 (The algebraic theory):

I will introduce various (equivalent) notions of loop groups. We will go
into the Lie theory of algebraic loop groups by introducing the affine
Lie algebra and the affine Weyl group. The example of the (untwisted)
affine lie algebra of rank two will be studied in some detail. This
corresponds to the group of loops on the compact Lie group SU(2).

Lectures 2,3 (The representation theory):

In these two lectures I will introduce an important class of
representations of the loop group known as positive-enery
representations. These decompose under a notion of level. We will
classify all positive energy representations of a given level, and
describe the Weyl-Kac character formula for irreducible representations.
I will illustrate these formulas with examples. Our final goal will be
to describe the geometric notion of fusion that induces a ring structure
on the Grothendieck group of positive energy representations of a fixed
level.

Lecture 4 (The homotopy theory):

In this lecture we will focus on the topology of the loop group and its
classifying space. We will describe several structure theorems that
pertain to the stable and unstable homotopy type of these spaces.

General remarks:

1) I will focus on loop groups since these are the groups most relevant
to field theory, but I will point out how most of this theory extends to
a larger class of groups known as Kac-Moody groups.

2) It will be helpful (though not necessary) if the audience is somewhat
familiar with the theory of compact Lie groups and their classifying
spaces.

Stephan Stolz (University of Notre Dame, Indiana, USA): "Field
theories and cohomology"

In these lectures, based on joint work with Peter Teichner, I want to present a conjectural picture of TMF(X) as homotopy classes of families of supersymmetric 2-dimensional Euclidean field theories parametrized by X. Here the notion of a supersymmetric Euclidean field theory is a refinement of the well-known definition of a topological field theory as a functor out of a bordism category. I will present evidence for the conjecture in form of an analogous description of de Rham cohomology and K-theory of X in terms of supersymmetric Euclidean field theories of dimension 1 and 2, respectively. In addition I will outline our proof that the partition function of supersymmetric 2-dimensional Euclidean field theory is a modular form with integral Fourier coefficients.

Tuesday

9:00-9:30 |
Registration |

9:30-10.30 | André Henriques |

11:00-12:00 |
Stephan Stolz |

14:00-15:00 |
Question session |

15:30-16:30 |
Nitu Kitchloo |

Wednesday

9:30-10.30 | André Henriques |

11:00-12:00 |
Stephan Stolz |

14:00-15:00 |
Question session |

15:30-16:30 |
Nitu Kitchloo |

Thursday

9:00-9:45 | André Henriques |

10:15-11:00 |
Stephan Stolz |

11:30-12:15 |
Nitu Kitchloo |

14:00-15:00 |
Question session |

15:00 - |
Excursion |

Friday

9:00-9:45 | André Henriques |

10:15-11:00 |
Stephan Stolz |

11:30-12:15 |
Nitu Kitchloo |

14:00-15:00 |
Question session |

15:00 - |
Teatime |

Registration is no longer available!

All lectures take place at the Mathematisches Institut - Westfälische Wilhelms-Universität in Münster (Germany). For a map click here. The nearest airport is Münster-Osnabrück (FMO). There is a bus (line S50) that takes about 40 min from the airport to the main train station of Münster. For train connections to Münster, check here; in the search form just write "ms" in the "Destination" field. This will determine the correct train station "Münster(Westf)Hbf". More information about transport to Münster can also be found here. For connections directly to our Mathematisches Institut please click here.

We have reserved a few double rooms (that can also be used as single rooms upon request) at Hotel Feldmann under the code "Summer school" until Sept. 7. Other standard options are: |

Hotel Am Schlosspark |

Hotel Bakenhof |

Hotel Jellentrup |

Hotel Busche am Dom |