Graduate Seminar on Advanced Geometry (S4D3)
Hauptseminar Geometrie (S2D1)
University of Bonn, Winter semester 2019/20
Instructors: Christian Blohmann [3], Lory Kadiyan
Time/venue: Wednesday 14:15-16:00, Max Planck Institute for Mathematics, seminar room
From the perspective of physics, the notion of factorization algebra can be seen as a framework to formulate classical and (perturbative) quantum field theory, which allows to extend the deformation quantizaton from classical mechanics to field theories. From the perspective of mathematics, a factorization algebra is a cosheaf on a manifold with values in vector spaces (or differential complexes) with an additional factorization property, which defines a homology theory and, therefore, produces topological invariants. This is a math seminar, so we will focus on the mathematical side and use the physics background as motivation and source of examples.
A background in basic differential geometry (sheaves on manifolds, de Rham cohomology) and homological algebra is assumed. Some basic knowledge of homotopical algebra will be helpful. A background in Physics is not necessary, but may be helpful for motivation (quantum mechanics, classical field theory, quantum field theory).
The seminar consists of short talks (ca. 30-45 minutes) by the participants on well-defined parts of the seminar material (e.g. a presentation of an important result with proof or an introduction of a new concept) which are framed and connected by short introductory talks by the seminar organizers.
If you are interested in participating you can send an email to blohmann@mpim-bonn.mpg.de [4] and/or sign up at the organizational meeting on October 1. In addition, you will have to register officially via Basis. If you have a preference for one or several of the talks, please let me know by email.
In order to obtain credit you will have to participate actively. This means that you will have to a) come to all seminar meetings and b) give one of the short talks listed below. The seminar is officially registered for both, Bachelor studies (S2D1) and Master studies (S4D3).
Remarks: The dates of the session are subject to changes and not strict since some topics take more time than an entire meeting and some less. Due to the dies academicus there will be no seminar on Dec 4, 2019.
Talks marked with * are more difficult.
1: The definition in explicit terms, basic properties, morphisms (Secs. 3.1.1 and 3.1.4 in [1]), Sebastian Meyer
2a: Proof that associative algebras are locally constant factorization algebras on $\mathbb{R}$ (Secs. 3.1.1 and 3.2 in [1]), Stefano Ronchi
2b: Bimodules and factorization algebras (Sec. 3.3.1 in [1]), Federica Bertolotti
3: Quantum mechanics as factorization algebra over $\mathbb{R}$ (Sec. 3.3.2 in [1]), Milan Kroemer
4a: The Lie algebra homology complex (App. A.4 in [1]), Yannick Burchart
4b: The enveloping algebra as factorization algebra (Prop. 3.4.1 in [1])
5*: Proof of Lem. 3.5.15 in [1], Lennart Ronge
6a: Twisted factorization envelope (Sec. 3.6.3 in [1])
6b: The Kac-Moody factorization algebra (Examples on p. 78 of [1]), Jan Nöller
7a: The divergence complex of a measure (Sec. 4.1.1 in [1]), Matthias Hippold
7b*: Derived tensor products, Jakob Kraasch
8: Proof of Lem. 4.2.4 in [1] computing the cohomology ring of classical observables on the line, Sid Maibach
9*: Proof of Prop. 4.3.3 in [1] recovering the Weyl algebra as factorization homology, Leonard Hofmann
10a: Proof of Prop. 4.5.2 in [1] computing the algebra of space observables
10b*: Relation of Chern-Simons theory with knot theory (Sec. 4.5.4 in [1]), Annika Tarnowsky
11*: Proof of Thm. 6.4.2 in [1] relating Hochschild homology to factorization homology on a circle
12a: Proof of Prop. 8.1.1 in [1] (enveloping algebras)
12b: Proof of Prop. 8.1.2 in [1] (Kac-Moody algebras)
Main reference
[1] Kevin Costello, Owen Gwilliam, Factorization Algebras in Quantum Field Theory, Vol. 1, Cambridge UP, 2017
Additional literature
[2] Owen Gwilliam, Factorization algebras and free field theories, PhD thesis, Northwestern University 2012
[3] Kevin Costello, Owen Gwilliam, Factorization Algebras in Quantum Field Theory, Vol. 2 (28 April 2016), manuscript, available online at https://people.math.umass.edu/~gwilliam/vol2may8.pdf [5], last accessed on 9/30/2019
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/41
[2] http://www.mpim-bonn.mpg.de/node/4234
[3] http://people.mpim-bonn.mpg.de/blohmann/
[4] mailto:blohmann@mpim-bonn.mpg.de
[5] https://people.math.umass.edu/~gwilliam/vol2may8.pdf