# School

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

### Graduate School (January 18 to 20, 2021)

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 aspect of the event is that it will provide the opportunity for graduate students within the GQT cluster to get to know and (virtually) 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:

Note: unlike previous editions, each speaker gives a 45 minute talk every morning (18-20 January) and exercise/discussion sessions are in the afternoon (Zoom and gather.town)

- — Magnus Bakke Botnan (VU)
- — Gijs Heuts (Utrecht)
- — Victoria Hoskins (Nijmegen)

**Schedule**

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

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

9:30 – 10:15 | Heuts | Hoskins | Botnan |

10:30 – 11:15 | Botnan | Botnan | Heuts |

11:30 – 12:15 | Hoskins | Heuts | Hoskins |

14:00 – | Exercises | Exercises | Exercises |

**Course description and prerequisites**

**Representations of Quivers and Data analysis **– Magnus Bakke Botnan (VU)

*Course description*: Quiver representations permeate mathematics, and they have recently found applications in topological data analysis. In this short course I will introduce the basic language of quivers and their representations, give an intuitive explanation of the finite type-tame-wild trichotomy, as well as explaining how this relates to data science through persistent homology. Some time will be spent exploring data sets using the software RIVET.

*Exercises*: Part 1 Session 1

*Pre-requisites*: A good understanding of linear algebra over finite fields is necessary. Rudimentary knowledge of (simplicial) homology will be helpful.

*Reading material*: Most of the material can be found in my lecture notes: https://www.few.vu.nl/~botnan/lecture_notes.pdf . Further material on quivers and their representations can be found in any introductory book on said topic, e.g. Barot, Michael. “Representations of quivers.” Introduction to the Representation Theory of Algebras. Springer, Cham, 2015. 15-31.

**Rational homotopy theory** — Gijs Heuts (Utrecht)

*Course description*: A central problem in homotopy theory is to compute the homotopy groups of spheres, i.e., to describe all homotopy classes of maps between spheres. These groups are still unknown in general, but their rationalizations were completely calculated by Serre in the 50s. This is the starting point of rational homotopy theory: the study of properties of topological spaces that can be ‘detected’ by rational invariants, such as rational cohomology and rational homotopy groups. Work of Quillen and Sullivan proves that the study of rational homotopy theory can be completely translated into algebra. Quillen showed that one can ‘model’ rational homotopy theory via (differential graded) Lie algebras, whereas Sullivan provided an alternative in terms of (differential graded) commutative algebras, very much inspired by the de Rham complex of differential forms on a manifold. We will explore the basics of their results, the duality between Lie algebras and commutative algebras, connections to geometry, and perhaps an outlook on modern generalizations of the work of Quillen and Sullivan.

*Exercises:*

Part 1 Exercises1

Part 2 Exercises2

Part 3 Exercises3

*Prerequisites:*

*Background reading recommendations:*

Useful lecture notes and book:

Berglund, “Rational homotopy theory”, https://staff.math.su.se/alexb/rathom2.pdf

* *

**Parallels between moduli of quiver representations and vector bundles** — Victoria Hoskins (Nijmegen)

*Course description*: Moduli spaces provide geometric solutions to classification problems

and in this course we will study two moduli problems: moduli of quiver

representations and moduli of vector bundles over a smooth projective algebraic

curve. A central theme will be to explore the similarities between the geometry

of these moduli spaces. After describing the basic properties of these moduli

problems and outlining the constructions of their moduli spaces in algebraic

and symplectic geometry, we introduce hyperkähler analogues over the complex

numbers: moduli spaces of representations of a doubled quiver satisfying

certain relations imposed by a moment map and moduli spaces of Higgs bundles.

In the final lecture, we will survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of

these complex hyperkähler moduli spaces due to work of Crawley-Boevey and Van

den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting,

and work of Schiffmann in the bundle setting.

*Slides and exercises*:

Part 1 GQT Lecture 1

Part 2 GQT Lecture 2

Part 3 GQT Lecture 3

*Prerequisites: *The talks will be based on my lecture notes https://www.emis.de/journals/SIGMA/

2018/127/sigma18-127.pdf and the references therein. We will use some basic

tools and language from algebraic geometry, so some familiarity with the basics

of algebraic geometry or Riemann surfaces would be helpful; for example, some

knowledge of one of the books below would be useful.

This course should also be accessible to students with a background in

differential geometry.

*Background reading recommendations: *

P. Griffiths and J. Harris “Principles of Algebraic Geometry”

R. Hartshorne “Algebraic Geometry”

R. Miranda “Algebraic Curves and Riemann surfaces”