Geometry and Quantum Theory

Graduate School and Colloquium

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

Graduate School (3 to 5/Jul)

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:

3/Jul — Ana Ros Camacho: Topological field theories

4/Jul — Roland van der Veen: Knot invariants

5/Jul — Mingmin Shen: Derived categories in algebraic 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:

Course description and prerequisites

Topological field theories — Ros Camacho

Course description: We will present several algebraic and categorical notions concerning the functorial definition of topological field theories and then explore several examples in lower dimensions (like e.g. the two dimensional case and its relation with Frobenius algebras, Turaev-Viro and Reshetikhin-Turaev).

Pre-requisites: Familiarity with the basics of category theory should be enough.

Reading material:

- J. Kock, “Frobenius algebras and 2D topological quantum field theories”, http://mat.uab.es/~kock/TQFT.html

- M. Atiyah, "Topological quantum field theories”, Publ. Math. IHE ́S68 (1988) 175-186.

- R. Dijkgraaf and E. Witten, "Topological gauge theory and group cohomology”, Comm. Math. Phys. 129 (1990). http://math.ucr.edu/home/baez/qg-winter2005/group.pdf.

- J. Lurie, “On the classification of topological field theories”, Current Developments in Mathematics, Volume 2008 (2009), 129-280. https://projecteuclid.org/download/pdf_1/euclid.cdm/1254748657

Knot theory — Van der Veen

Course description: We present knot theory in such a way that its close relationship to Hopf algebras becomes apparent. More concretely, assembling a knot from elementary pieces may be interpreted as doing a computation in a special Hopf algebra. By actually constructing examples of such Hopf algebras using ideas from Poisson-Lie groups and statistical mechanics we obtain powerful tools to tackle topological questions.

Pre-requisites: multi-linear algebra.

Reading material:

S. Majid, A quantum groups primer, LMS 292.

Lecture notes, software and handouts.

Derived categories in algebraic geometry — Mingmin Shen

Course description: The derived category is a structure and language that originated from homological algebra. It has been widely used in mathematics. In this mini course, we start with a brief introduction to the derived category. Then we move on to its applications in algebraic geometry. Namely, we discuss the derived category of coherent sheaves. It is know that for varieties with ample canonical or anti-canonical sheaves, the derived category determines the variety itself. The case of trivial canonical sheaf is more intriguing and we will focus on the special case of K3 surfaces.

Pre-requisites: TBA

Reading material:

- Stacks project, Chapters "Homological algebra" and "Derived categories". http://stacks.math.columbia.edu

- D. Orlov, Derived categories of coherent sheaves and equivalence between them, http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf

Ph.D. talks

Organizers: Raf Bocklandt, Gil Cavalcanti and Maarten Solleveld.