Supergeometry and Its Applications. I

Dr Theodore Voronov (University of Manchester)

Frank Adams Room 1,

This is a mini-course in the framework of the seminar, tentatively consisting of three lectures.

Lecture 1.

Idea of algebra/geometry duality. Examples of "non-set-theoretic" geometric objects ("double point"; quantum groups; coordinate superspace). Supermanifolds. Examples. Necessary notions from Z2-graded algebra. Superdeterminant (Berezinian). Berezin integral.

Lecture 2.

Language of supergeometry in the conventional world of differentiable manifolds. 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 Berezin integral. Integration over fibers.

Lecture 3.

Some more advanced applications and examples (depending on how the previous lectures go). May include: volumes of supermanifolds, index theorem, L-infinity algebras, Lie algebroids...

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