Elements of Supergeometry

This course develops the theory of supergeometry, from its algebraic foundations to geometric and analytic structures. After discussing super linear algebra, superalgebras, and Lie superalgebras, we introduce the sheaf-theoretic approach to superspaces and supermanifolds. We cover differential geometry and analysis on supermanifolds, and the theory of Lie supergroups and homogeneous superspaces. We then turn to elements of algebraic supergeometry, covering projective superspaces, super Grassmannians, and algebraic supergroups. In the final weeks, we will explore selected specialized topics — such as the classification of simple Lie superalgebras, Q-manifolds and their connections to Poisson geometry, or the theory of supercurves and super Riemann surfaces — depending on class interest and time.

17th September ~ 17th December, 2026

Weekday	Time	Venue	Online	ID	Password
Thursday	10:40 - 12:15	Shuimo	Zoom 17	442 374 5045	BIMSA
Thursday	13:30 - 15:05	Shuimo	Zoom 17	442 374 5045	BIMSA

Basic graduate algebra and differential geometry and familiarity with Lie groups/Lie algebras and sheaves. Some algebraic geometry is helpful for the final third of the course.

∙ Super linear algebra
- Super vector spaces
- Tensor products, symmetric and exterior powers
- Supermatrices and the Berezinian
- Bilinear forms and duality

∙ Algebraic constructions
- Superalgebras and modules; supercommutative superalgebras
- Clifford algebras and super Brauer group
- Lie superalgebras

∙ Supermanifolds as ringed spaces
- Superspaces and supermanifolds (smooth, complex, algebraic)
- Global structure of supermanifolds

∙ Analysis on supermanifolds
- Inverse function theorem, immersions, submersions
- Integration theory: differential and integral forms, Berezin integral, de Rham complex, Stokes theorem

∙ Differential geometry of supermanifolds
- Tangent bundles; vector fields and ordinary differential equations
- Distributions and Frobenius theorem

∙ Lie supergroups and homogeneous spaces
- Definitions, functor of points approach
- Lie superalgebra of a supergroup
- Actions and homogeneous superspaces

∙ Algebraic supergeometry
- Functor of points and representability
- Projective superspaces
- Super Grassmannians and super flag varieties
- Algebraic supergroups

∙ Specialized topics (selection depends on progress and interest)
- Simple Lie superalgebras; basics of representation theory
- Q-manifolds and connections with Poisson geometry, Lie algebroids, BV formalism
- Complex structures and CS-manifolds
- Supercurves and super Riemann surfaces; super moduli spaces

Yes

English