# School

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

### Graduate School (August 22 to 24, 2022)

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 graduate 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 aspect of the event is that it will provide the opportunity for graduate students within the GQT cluster to get to know each other and have academic interactions.

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

- Magdalena Kędziorek (Nijmegen) – TBA
- Johan Commelin (Freiburg) – Hands on with Lean
- Álvaro del Pino Gómez (Utrecht) – Transversality and its applications

**Schedule (tentative)**

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

Mon (Kedziorek) | Tue (Commelin) | Wed (del Pino Gómez) | |
---|---|---|---|

9:00-9:45: PhD talk – Schwarz (Leiden) | 9:00-9:45: PhD talk – Becerra (RUG) | ||

10:00-12:30: morning session | 10:00-12:30: morning session | 10:00-12:30: morning session | |

12:30-14:00: lunch | 12:30-14:00: lunch | 12:30-14:00: lunch | |

14:00-17:30: afternoon session | 14:00-17:30: afternoon session | 14:00-17:30: afternoon session | |

18:30: dinner | 18:30: dinner | 18:30: dinner |

**Course description and prerequisites**

**Magdalena Kędziorek – A crash course on model categories**

*Course description*: Model categories are a highly successful framework for homotopy theory, allowing for the use of homotopical constructions outside of the traditional contexts of spaces. In this lecture series we will motivate the general concept, outline how model categories are used in practice and present simple examples to illustrate the commonly encountered properties that many model categories possess, such as properness, cofibrant generation and stability. The main focus of the lectures will be on constructions of model categories with desired properties, for example by mixing existing model categories, using left and right Bousfield localisations or transfers along adjunctions.

Most of this material will be rather elementary and will be accessible to a broad audience. The lecture series is planned to give the audience a user-friendly guide on how to work with model categories, allowing for connection to the related areas of infinity categories and triangulated categories.

*Pre-requisites*:

– Familiarity with groups, rings and modules

– Knowledge of basic homological algebra such as chain complexes, projective modules

– Familiarity with basic category theory such as categories, functors, adjunctions

*Reading material*: There are many books and articles on model categories. We will mostly use (some of the material presented in):

– W.G. Dwyer and J. Spalinski “Homotopy theories and model categories”. I recommend looking at section 2 for some recollection on basic category theory.

– S. Balchin “A handbook of model categories”

– M. Hovey “Model categories”

**J. Commelin (Freiburg) – Hands on with Lean**

The details of the course can be found here.

*Course description*: This course will give a hands on introduction to the Lean interactive theorem prover. After a general introduction, we will look at mathlib (the main library of mathematics for Lean) with an eye towards GQT-relevant subjects. We will alternate lectures and exercise sessions, and there will be ample time for hands-on practice.

*Prerequisites*: None

*Background reading recommendations*:

See https://leanprover-community.github.io/learn.html and https://leanprover-community.github.io/get_started.html#regular-install. (It is not necessary to install Lean on your laptop to take part in the exercises. But it might give a better experience, compared to the

online version.)

**Á. del Pino Gómez (Utrecht) – Transversality and its applications**

*Course description*: Transversality is one of the cornerstones of Differential Topology. For instance, it is the main ingredient behind:

-Whitney’s embedding theorem, stating that any n-dimensional manifold can be embedded into (2n+1)-dimensional euclidean space.

-The existence/genericity of Morse functions. I.e. “most” functions have non-degenerate critical points (which are, in particular, isolated).

-The Pontryagin-Thom construction, which relates cobordism classes of (framed) submanifolds to homotopy classes of maps into the spheres.

In fact, it plays a major role, generally, in immersion/embedding theory, Morse theory (both in its finite and infinite-dimensional incarnations), real (smooth) singularity theory, and many other fields.

The main idea behind transversality is that “most” maps tend to be nicely behaved, both by themselves and with respect to other maps/submanifolds. You can imagine for instance two curves in R^3: they may intersect, but it is possible to push them a bit in order to make them disjoint. The reason is that R^3 is large compared to the two curves, so they have plenty of space to avoid one another.

The goal of the minicourse will be to explain the Thom transversality theorem and its proof (but maybe not in painful detail). This result (which is the first “major” transversality statement) is powerful enough to address the examples given above. I hope to be able to spend most of the time discussing applications.

*Pre-requisites*: It is sufficient if you are familiar with the basic theory of smooth manifolds and you know what a fibre bundle is. Knowledge of differential forms, (co)homology and homotopy is probably helpful but not strictly necessary.

*Reading material*: For background on manifolds you can refer to Lee’s “Introduction to smooth manifolds”. For a quick introduction to the minicourse (maybe even overlapping with some of the ideas I will explain), I very much recommend Milnor’s “Topology from the differentiable viewpoint”. For further reading after the minicourse you may want to check out Guillemin and Golubitsky’s “Stable Mappings and Their Singularities”.

**PhD talk – R. Schwarz (Leiden) – The double-double ramification cycle**

Abstract. The double ramification cycle is a Chow class that has been studied extensively these last decades; informally said it measures the locus of curves with marked points p_1,…,p_n where there exists a function f from C to the projective line with div(f) equal to the sum of a_i p_i, where the vector A=(a_1,…,a_n) is some set ramification data. That is, we’d like to study the locus where the divisor sum a_i p_i is actually principal. However, formally defining the DR cycle was not always easy and there are other problems. For example, you would expect considering the DR locus with respect to two vectors A and B simultaneously (the double-double ramification cycle) to yield the same answer if you’d consider the DR with respect to A and A+B. This however need not be true. The recent introduction of the language of log geometry to the DR cycle allows us to make constructions that solve these problems. I would like to give intuition for the DR cycle and the problems described above, and maybe (time permitting) end with some comments about the use of log geometry for this and the recent work [D. Holmes, R.S., Logarithmic intersections of double ramification cycles].

**PhD talk – J. Becerra (Groningen) – Universal quantum knot invariants
**

Abstract. Quantum topology studies low-dimensional topology via invariants produced with the auxiliary data of an algebraic gadget. In this talk I will focus on two such invariants for knots: one is the “universal tangle invariant” Z_D, which uses a particular algebra D depending on some parameter epsilon; and the other is the “Kontsevich invariant”, which uses some formal power series. It is a conjecture by Bar-Natan and van der Veen that the second term in the expansion in epsilon of Z_D of a knot is tantamount to the second term in the so-called “loop expansion” of the Kontsevich invariant of the knot. I will outline recent work showing that the conjecture is true for the class of knots that bound a compact, connected, orientable surface of genus one.