This is part of a minicourse in the framework of the seminar.
Integral calculus: Berezin integral. Superdeterminant (Berezinian) and its properties.
"Super" language in conventional differential geometry: forms and multivector fields as functions on auxiliary supermanifolds \hat M=ΠTM and \check M=ΠT*M. Differential operations such as the exterior differential, Lie derivative, Schouten bracket, and Nijenhuis bracket in this language. Integration of forms as a Berezin integral. Example: integration over fibers.
Some more advanced applications and examples. May include: volumes of supermanifolds, index theorem, L-infinity algebras, Lie algebroids...
Contents of the previous lectures:
Lecture 1 (February 16):
Origins of supergeometry in physics: uniform description of fermions and bosons; supersymmetry. Idea of algebra/geometry duality. Examples of "non-set-theoretic" geometric objects such as "double point" and coordinate superspace. Necessary notions from Z2-graded algebra. Two examples of supermanifolds.
Lecture 2 (March 9):
The examples of supermanifolds (the supersphere and the tangent bundle with reversed parity ΠTM, for an ordinary manifold M) carefully worked out. Differential calculus in the super case. Definition of a supermanifold in the language of ringed spaces. Discussion of "points" of a supermanifold.