# School

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

### Graduate School (August 28 – 30, 2023)

The aim of the GQT school is to provide participants with a basic understanding of some central topics in GQT research, including important themes which may lie outside one’s own research specialization. Hence, we strongly encourage graduate students to participate all three days as a way to broaden their mathematical formation and increase awareness of the research carried out within the cluster.

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

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

- Gil Cavalcanti (Utrecht) – Towards differential log-geometry
- Ilke Canakci (VU) – Introduction to cluster algebras
- Peter Hochs (Nijmegen) – Dirac operators

**Schedule (tentative)**

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

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

9:00-9:25: Boaz Moerman
9:30-9:55: Maxime Wybouw |
9:00-9:25: Ronen Brilleslijper
9:30-9:55: Kevin van Helden |
||

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**

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Gil Cavalcanti (Utrecht): 28 August

*Title: *Towards differential log-geometry

*Abstract: *Log-geometry comes from algebraic geometry. Roughly it can appear as a way to study singular spaces, which become “sort-of” non singular in the log-context or as a way to study Fano varieties which then become more Calabi—Yau-like. In this school I will introduce a differential version of log-structures, that decouples the log-structure from any underlying complex/algebraic geometry. I will introduce the study complex and symplectic structures on log-manifolds and explain how log structures can help to understand T-duality for torus actions with fixed points. This course is based on a series of papers with Gualtieri, Klaasse and Witte.

===========

Ilke Canakci (VU): 29 August

*Title: *Introduction to cluster algebras

*Description: *These lectures concern the rapidly evolving theory of *cluster algebras*. Since their introduction by Fomin and Zelevinsky in 2001, cluster algebras have been studied worldwide, leading to remarkable connections in disciplines as diverse as combinatorics, algebraic, hyperbolic and symplectic geometry, representation theory, dynamical systems, and string theory.

The aim of the school is two-fold: firstly, we will introduce cluster algebras through the language of quivers providing ample examples along the way and secondly, we will discuss their classifications of finite type and finite mutation type. In particular, we will see connections to Dynkin diagrams, and to the combinatorics of triangulations of marked surfaces.

*Prerequisites:* None

*Reading material:* Here is a (non-exhausting) list of introductory material on cluster algebras.

- Marsh, Lecture notes on cluster algebras, https://ems.press/books/zlam/201?na
- Williams, Cluster algebras: an introduction, https://arxiv.org/abs/1212.6263
- Schiffler, Cluster algebras from surfaces, https://www.springerprofessional.de/en/homological-methods-representation-theory-and-cluster-algebras/15703952 or https://schiffler.math.uconn.edu/wp-content/uploads/sites/914/2019/03/LNCIMPA.pdf
- Fomin, Williams, and Zelevinsky, https://people.math.harvard.edu/~williams/book.html

===========

Peter Hochs (Nijmegen): 30 August

*Title:* Dirac operators

*Description:* Dirac operators are first-order linear differential operators whose squares are Laplace-type operators. They were introduced by Dirac in 1928 to describe the electron quantum mechanically. Apart from their applications in physics, Dirac operators have turned out to be relevant to geometry, topology and representation theory.

This course starts with the definition of Dirac operators and their basic properties. We then look at a special kind of Dirac operator: the Spin Dirac operator. The final goal is to understand the statement “a Spin manifold with nonzero A-hat genus does not admit Riemannian metrics of positive scalar curvature”, and to get some idea of how Dirac operators can be used to prove it.

*Prerequisites:* (Riemannian) manifolds, vector bundles, Hilbert spaces

*Reading material: *Lecture notes for this course: Dirac operators GQT Peter Hochs

More background information (including the proofs that we skip) can be found in these books:

* Berline, Getzler, Vergne: Heat kernels and Dirac operators

* Friedrich: Dirac operators in Riemannian geometry

* Gilkey: Invariance theory, the heat equation, and the Atiyah- Singer index theorem

* Lawson and Michelsohn: Spin geometry

===========

**PhD talks**

Boaz Moerman (UU): Adelic approximation of generalized integral points

Maxime Wybouw (RU): Double Centralizer Duality in Algebra and Topology

Ronen Brilleslijper (vU): A Hamiltonian formalism for elliptic PDEs

Kevin van Helden (RUG): Intrinsic torsion, Principal bundles and Spencer cohomology

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