Geometry and Quantum Theory (GQT)

School

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

Graduate School (June 4 to 8, 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:

  • June 4 — Gil Cavalcanti (UU): Generalized complex geometry
  • June 5 — Hessel Posthuma (UvA): Index theory
  • June 6 —Walter van Suijlekom (Nijmegen): Non-commutative geometry

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 Talk by PhD Student Talk by PhD Student
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

June 5, 9:00 – 9:45 Nikolay Martynchuk (Universiteit Groningen)
Monodromy in integrable systems

June 6, 9:00 – 9:45 Davide Alboresi (Universiteit Utrecht)
Holomorphic curves in log-symplectic manifolds

 

Course description and prerequisites

Introduction to generalized complex geometry — Gil Cavalcanti (UU)
Course description: In this short lecture series we will cover the basics of generalized complex geometry and a couple of the first relations found to physics. Namely, we will introduce
1) the geometry of the double tangent bundle,
2) generalized complex and generalized metrics,
3) generalized Kahler structures.

We will study the basic features with which these structures endow manifolds such as decomposition of forms, exterior derivative and cohomology. We will then move one to prove the following results
1) equivalence between generalized Kahler structures and the bihermitian structures of Gates, Hull and Rocek,
2) generalized complex structures are preserved under T-duality.

Reading material:
[1] Marco Gualtieri. Generalized complex geometry. Ph.D. thesis 2003, ArXiv 0401221
[2] Marco Gualtieri. Generalized Kahler geometry. Generalized Kähler geometry. Comm. Math. Phys. 331 (2014), no. 1, 297–331.
[3] Gil Cavalcanti and Marco Gualtieri. Generalized complex geometry and T-duality. A celebration of the mathematical legacy of Raoul Bott, 341–365, CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, 2010.
[4] Gil Cavalcanti. Introduction to generalized complex geometry (http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf)

 

Index theory — Hessel Posthuma (UvA)
Course description: In this course I will give an introduction to index theory. The first part will center around the statement of the Atiyah-Singer theorem for  the index of an elliptic operator on a compact manifold, its generalizations and applications. In the second part I will give an overview of the different proofs of the theorem, using either K-theory, heat-kernels or ideas from physics (quantization), or noncommutative geometry.

 

Non-commutative geometry — Walter van Suijlekom (Nijmegen)
Course description: We will start the morning sessions by presenting a “light” version of noncommutative geometry by looking at finite noncommutative metric spaces (aka finite spectral triples) and their classification. We discuss bimodules and Morita equivalences in this context, and explain how to take a “product” of a finite spectral triple and a Morita equivalence bimodule to obtain a new finite spectral triple.
We will then move on to the general notion of KK-cycles and introduce the key structure given by the interior Kasparov product, of which the above product with a Morita bimodule is an example. We will avoid all functional analytical technicalities by working in the finite-dimensional case, even though towards the end we will briefly sketch how to extend the formalism to that case. This will be illustrated by concrete geometric examples coming from gauge theory and index theory.

Pre-requisites: Some Hilbert space theory, differential geometry.

Reading material: W.D. van Suijlekom. Noncommutative Geometry and Particle Physics. Springer, 2015 (a draft is available at www.waltervansuijlekom.nl).