Generalised geometry - NMAG597

• Title: Generalised geometry
• Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2025
• Semester: winter
• E-Credits: 4
• Hours per week, examination: winter s.:2/1, C+Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: taught
• Language: English
• Teaching methods: full-time
• Guarantor: Mgr. Fridrich Valach, Ph.D.
• Teacher(s): Mgr. Fridrich Valach, Ph.D.
• Class: M Mgr. MSTR > Volitelné
• Classification: Mathematics > Geometry, Mathematics General

Annotation

English

We will develop the theory of Courant algebroids, known as generalised geometry. This will lead to a natural unification of various geometric setups (i.e. symplectic and complex structures can be both regarded as generalised complex structures), and at the same time provide a solid framework for understanding dualities in string theory and dynamics of supergravity. We will discuss both the original vector-bundle interpretation as well as the more conceptual graded-geometric approach, whose knowledge proves to be an immensely powerful tool in many contexts in geometry and mathematical physics.

Last update: Šmíd Dalibor, Mgr., Ph.D. (19.05.2025)

Czech

Rozvineme teorii Courantových algebroidů, běžně známou také jako zobecněná geometrie. Na jedné straně to povede k přirozenému sjednocení různých geometrických konceptů (například symplektické i komplexní struktury lze považovat za zobecněné komplexní struktury), a zároveň to poskytne solidní rámec pro pochopení dualit v teorii strun a dynamiky supergravitace. Představíme jak původní interpretaci v řeči vektorových bandlů, tak i koncepčnější přístup gradované geometrie, jehož znalost se ukazuje jako nesmírně účinný nástroj v mnoha kontextech geometrie a matematické fyziky.

Last update: Šmíd Dalibor, Mgr., Ph.D. (19.05.2025)

Course completion requirements

prezentace/ústní skouška

Last update: Valach Fridrich, Mgr., Ph.D. (23.05.2025)

Literature

M. Gualtieri, Generalized Complex Geometry, Oxford University DPhil thesis (2004), https://arxiv.org/abs/math/0401221

https://fridrichvalach.com/materials/gg.pdf (draft of Lecture notes from Special Topics Lectures at the Imperial College London, 2023)

Last update: Valach Fridrich, Mgr., Ph.D. (23.05.2025)

Teaching methods

blackboard lectures

Last update: Valach Fridrich, Mgr., Ph.D. (23.05.2025)

Syllabus

Preliminary schedule:

• Prerequisites in differential geometry (Cartan calculus, vector bundles, ...) and linear algebra

• Courant algebroids: definition, examples, classification results, Dirac structures, generalised complex structures

• Graded-geometric interpretation: supermanifolds, NQ symplectic manifolds, connection with Poisson structures, Lie algebroids, ...

• Elements of generalised Riemannian geometry

• Supergravity and dualities in string theory

Last update: Valach Fridrich, Mgr., Ph.D. (23.05.2025)