# School

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

### Graduate School (July 1 to 3, 2019)

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 cluster 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:

- July 1 — Christopher Lazda (Uva)
- July 2 — David Holmes (Leiden)
- July 3 — Steffen Sagave (Nijmegen)

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 | F. Cattafi | R.Buring |

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**

Francesco Cattafi (Utrecht)

*An overview on Lie pseudogroups and geometric structures*

**Abstract:** A Lie pseudogroup $\Gamma$ is a collection of locally defined diffeomorphisms arising as solutions of a PDE. This object gives rise to the notion of $\Gamma$-structure on a manifold $M$: it is a maximal atlas whose changes of coordinates take values in $\Gamma$. For instance, the set of local symplectomorphisms of the canonical symplectic structure of $\mathbb{R}^{2n}$ forms a Lie pseudogroup, and the associated $\Gamma$-structure is a symplectic structure on $M$.

More generally, starting from a Lie subgroup $G \subseteq GL(n,\mathbb{R})$, one can define a Lie pseudogroup $\Gamma_G$, and see that a $\Gamma_G$-structure coincides with the standard notion of an integrable $G$-structure on $M$. Nevertheless, a number of geometric structures can be described by Lie pseudogroups and not by Lie groups. This leads us to a natural question: what is the counterpart of a non-integrable $G$-structure in the Lie pseudogroup world (which we call “almost $\Gamma$-structure”)? And when does an almost $\Gamma$-structure come from a $\Gamma$-structure?

In this talk we are going to review these notions and provide an answer to the questions sketched above. In particular, we present a new characterisation of formal integrability in the setting of $\Gamma$-structures; this will be obtained by introducing the concept of principal Pfaffian bundle and studying its prolongations to higher orders. For doing that, we draw inspiration from similar results for PDEs on jet bundles and for $G$-structures, which we are going to recover. This is joint work with Marius Crainic.

R. Buring (RUG)

*Factorization problems in deformation quantization and Poisson bracket deformations*

**Abstract:** We recall basic theory and illustrate the work of two morphisms that consecutively relate three differential graded Lie algebras: of unoriented graphs, of multivector fields, and of polydifferential operators on affine manifolds. In the Kontsevich graph calculus, which is the starting point, the Schouten bracket [[ , ]] is represented by the edge •−•. Graphs on a larger number of edges and vertices encode more elaborated structures in many arguments, also allowing perturbative expansions of algebraic structure deformations using Feynman path integrals. The two morphisms under study build around the Schouten bracket as their domain and target structure, respectively.

**Course description and prerequisites**

**D-modules on Riemann surfaces **– Chris Lazda (UvA)

*Course description*:

Abstract: The theory of D-modules generalises that of vector bundles with a flat connection, and provides a deeper connection between the geometry and topology of complex manifolds. I will illustrate some basis aspects of the subject by concentrating on the case of Riemann surfaces, where a lot of the general theory greatly simplifies, and can often be worked out directly ‘by hand’.

*Pre-requisites*:

*Reading material*:

**Curves, jacobians and the double ramification cycle** — David Holmes (Leiden)

*Course description*:

I will begin with some basic background on algebraic geometry; this will be presented in a slightly unusual way, so there may also be something of interest there for those students already familiar with these things.

I will then move on to discussing (various classes of) algebraic curves. Essentially, we want to study degenerating families of Riemann surfaces, so we need to have nice definitions of families, and also a good understanding of the singularities that will appear.

Then jacobians: moduli spaces of line bundles on these degenerating families. We will give a heuristic idea, then a formal definition. We will then see how the case of smooth curves behaves in a fairly familiar and comfortable way. We will then study the case of singular curves, where some interesting phenomena occur.

Finally, we will discuss the double ramification cycle on the moduli space of curves. In this part I may need to assume a bit more background knowledge than has been needed so far, as I hope to describe some recent developments in the area.

*Prerequisites:*

Basic category theory (at the level of the mastermath `intensive course’) is really essential. Also some basic commutative algebra (in particular localisation and tensor products of modules over a commutative ring).

We will spend the first hour or so introducing basic concepts from algebraic geometry, from a categorical perspective (hopefully we will give working definitions of schemes and algebraic spaces, as well as basic properties of morphisms between them). This is aimed at an audience with no background in algebraic geometry, but will certainly be easier to follow if you have seen a bit, so reading any part of any introductory text in algebraic geometry will likely help your intuition.

*Background reading recommendations:*

Mastermath intensive course on categories and modules, in particular the 4th part (Yoneda and adjunction).

For localisation and tensor products, your choice of commutative algebra book (or Wikipedia, for that matter). Atiyah-Macdonald [Commutative algebra] is a classic, I also like Altman-Kleiman [A term of Commutative Algebra].

For a bit of intuition on algebraic geometry there are many options; Hartshorne is popular, Vakil’s `The rising sea’ is wonderful, and is probably closer in spirit to what we will do, but takes longer to get to interesting things. The Stacks Project is the bible, unless you are better at French, in which case EGA may be easier (but it lacks hyperlinks).

**The stable module category** — Steffen Sagave (Nijmegen)

*Course description*:

The aim of this lecture series is to introduce the stable module category, discuss some of its properties such as the triangulated structure, and explain how it relates to group cohomology and Tate cohomology. Most of this material will be rather elementary and should thus be accessible to a broad audience. Nonetheless, this will provide an opportunity to learn some important notions from homological algebra that are relevant for Algebraic Geometry, and Algebraic Number Theory, and Algebraic Topology. If time permits, I will towards the end also give an outlook on related deeper results such as the stratification theorem of Benson-Iyengar-Krause or the recent applications of the Tate construction in stable homotopy theory.

*Prerequisites:*

– Familiarity with groups, rings, and modules

– Knowledge of basic homological algebra such as projective resolutions, chain complexes, cochain complexes, Ext-groups