# School

The academic program consists of a 3-day graduate school and a 2-day conference.

### Graduate School (Nov 26 to 28, 2018)

The idea behind the GQT school is that participants will get a basic understanding of some central topics in GQT-research, including many important themes which may lie outside one’s own research specialization. Hence, we strongly encourage Ph.D. students to participate in all three days as a way to broaden their mathematical formation and increase awareness regarding the research carried out within the cluster.

Another important side to the event is that it will provide the opportunity for graduate students within the GQT to get to know and socialize with each other as well as have some academic interaction.

The school will consist of three days of lectures and exercise sessions; the topics and speakers are as follows:

- Nov 26 — Maarten Solleveld (Nijmegen)
- Nov 27 — Lennart Meier (UU)
- Nov 28 — Raf Bocklandt (Uva)

Besides the three minicourses, there will be one or two Ph.D. talks (given by Ph.D. students to Ph.D. students) on Tuesday and Wednesday morning before the beginning of the minicourse.

**Schedule**

During the school days the lectures and exercise classes will be arranged roughly as follows:

Mon | Tue | Wed | |
---|---|---|---|

9:00 – 9:45 | Arrival | Yongqi Feng (Nijmegen) | – |

10:00 – 11:00 | Lecture | Lecture | Lecture |

11:00 – 12:00 | Lecture | Lecture | Lecture |

12:00 – 12:30 | Discussion | Discussion | Discussion |

12:30 – 14:00 | Lunch | Lunch | Lunch |

14:00 – 15:00 | Lecture | Lecture | Lecture |

15:00 – 16:00 | Lecture | Lecture | Lecture |

16:00 – 17:00 | Exercises | Exercises | Exercises |

17:00 – 18:00 | Exercises | Exercises | Exercises |

18:00 – 18:30 | Solution of exercises | Solution of exercises | Solution of exercises |

18:30 | Dinner | Dinner | Dinner |

**Talks by PhD students**

November 27, 9:00 – 9:45: Yongqi Feng (Nijmegen), *On the formal degree and local Langlands **correspondence of supercuspidal unipotent representations*

**Course description and prerequisites**

**p-adic groups** — Maarten Solleveld (Nijmegen)

*Course description*:

P-adic groups are algebraic groups over a p-adic field. The archetypical example is GL(n,Q_p), where Q_p denotes the p-adic numbers. Such groups play an important role in algebraic geometry, representation theory and the Langlands program.

In this course we will discuss the structure of p-adic groups from several perspectives: number theory, topology and algebraic geometry. We will focus on reductive p-adic groups, whose algebraic structure can be described with maximal tori and Weyl groups. We will treat various decompositions of such groups,

some purely algebraic, some closer to p-adic geometry.

Many aspects of p-adic groups are completely different from algebraic groups over other fields (e.g. Lie groups). As an example, we will explain the action of GL2 (F) on a regular, infinite tree.

* Pre-requisites*:

Basic algebraic geometry and some knowledge of Lie groups and Lie algebras.

*Reading material*:

– Solleveld, “Lecture notes on the local Langlands correspondence”

(available at http://www.math.ru.nl/~solleveld/LLC.pdf)

– Serre, “Local fields” (chapters 1 and 2)

– Serre, “Trees” (chapter II.1)

– Springer, “Linear algebraic groups”

– Borel, “Linear algebraic groups”

**Stable homotopy theory** — Lennart Meier (UU)

*Course description*:

This lecture series will be an introduction to stable homotopy theory, i.e. to the homotopy theory of spectra. Spectra arose historically from different directions. For example, every homology or cohomology theory (like singular homology, K-theory or bordism) is represented by a spectrum. On the other hand, spectra can be studied by very similar methods as spaces and are even simpler in many respects.

The main aims of this lecture series will be to

* introduce spectra and their relationship to (co)homology theories,

* explain their role in the computation of bordism theories,

* introduce the stable homotopy category, providing a convenient framework to reason about spectra,

* connect them to duality theory, and

* talk about complex orientations and their relationship to formal group laws.

*Prerequisites:*

I will presume basic knowledge about the following topics:

* (Singular) (co)homology and the Eilenberg–Steenrod axioms

* homotopy groups

* differentiable manifolds

* vector bundles

References for the first three points include standard textbooks on algebraic topology like those of Bredon and Hatcher. For the basics of vector bundles see either the book by Milnor and Stasheff or https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html

*Background reading recommendations:*

A nice and modern texbook treatment of stable homotopy theory does sadly not exist. Older references are

* Adams: Stable homotopy and generalised homology (Parts II and III)

* Switzer: Algebraic Topology – Homology and homotopy

Online resources include:

* Malkiewicz: The stable homotopy category https://faculty.math.illinois.edu/~cmalkiew/stable.pdf

* Miller: Notes on cobordism http://www-math.mit.edu/~hrm/papers/cobordism.pdf

Course notes Lennart Meier: IntroStable

**Singularities and matrix factorizations** — Raf Bocklandt (UvA)

*Course description*: