ZMP Seminar (Summer term 2024):
Generalised Geometry and Homological Algebra
The notion of a generalized complex manifold was introduced by Hitchin as a common framework in which complex manifolds and symplectic manifolds can be described from a unified perspective.
Deformations of a projective variety X, considered as a generalised complex manifold, provide an extended deformation theory of X that agrees with the first order deformations of the category of coherent sheaves on X.
In this seminar we will first study fundamentals of generalised geometry and derived categories in order to be able to look at this coincidence.
We will then consider further notions of generalised geometry, like generalized Kähler or generalized Calabi-Yau structures, and the generalized Ricci flow, and look at applications to physics, in particular cosmic inflation.
We plan to talk about generalized complex structures and their deformations (2 talks) , derived categories and their deformations (2 talks) and then the generalised Ricci flow (2 talks). Some references are below.
Talks 1 & 2
- Marco Gualtieri, Generalized complex geometry, PhD thesis Oxford, 2003.
- Marco Gualtieri, Generalized complex geometry, Annals of Mathematics 174 (2011), 75–123.
- Vicente Cortés, Generalized Geometry, MSc lecture course summer semester 2023.
Talks 3 & 4
- D. Huybrechts: Fourier-Mukai Transforms in Algebraic Geometry, OUP (2006).
- J J. Block: Duality and equivalence of module categories in noncommutative geometry I, arXiv:0509284
- M. Gerstenhaber, S. Schack: Algebraic Cohomology and Deformation Theory, Springer (1988).
- Y. Toda: Deformations and Fourier-Mukai transforms, J.Differential Geometry (2009).
Talks 5 & 6
- Mario Garcia-Fernandez and Jeffrey Streets. Generalized Ricci Flow. University Lecture Series. AMS, 2020.
- Jeffrey Streets, Charles Strickland-Constable, Fridrich Valach: Ricci flow on Courant algebroids, arXiv:2402.11069
Further reading on connections between generalized geometry and string theory as well as T-duality:
- Anthony Ashmore, Charles Strickland-Constable, David Tennyson and Daniel Waldram,
Heterotic backgrounds via generalised geometry: moment maps and moduli, arXiv:1912.09981.
- Gil Cavalcanti and Marco Gualtieri, Generalized complex geometry and T-duality, A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings and Lecture Notes, American Mathematical Society (2010), p. 341--366.
- David Baraglia and Pedram Hekmati, Transitive Courant algebroids, String structures and T-duality, Adv. Theor. Math. Phys. 19 (3) (2015), 613--672.
- Vicente Cortés and Liana David, T-duality for transitive Courant
algebroids, Journal of symplectic geometry, Volume 21, Number 4, 775--856, 2023.
For questions about the seminar, or to get involved in any way, please contact Vicente Cortés or Julian Holstein.
The planned ZMP seminars and colloquia this semester are as follows: