'Et moi, ... si favait III mmment en revenir, One service mathematics has rendered the je n'y serais point aile:' human race. It has put CXlUImon sense back Iules Verne where it belongs. on the topmost shelf next to the dUlty canister lahelled 'discarded non- The series i. divergent; therefore we may be able to do something with it. Eric T. Bell O. Hesvi.ide Mathematics is a tool for thOUght. A highly necessary tool in a world where both feedback and non� linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com� puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d't!tre of this series.

Contenuti

I: Foundations.- I — Elements of graded algebra.- 1. Graded algebraic structures.- 2. Graded algebras and graded tensor calculus.- 3. Matrices.- II — Sheaves and cohomology.- 1. Presheaves and sheaves.- 2. Sheaf cohomology.- 3. de Rham, Dolbeault, and ?ech cohomologies.- 4. Graded Ringed spaces.- II Supermanifolds.- III — Categories of supermanifolds.- 1. Graded manifolds.- 2. Supersmooth functions.- 3. GH? functions.- 4. G-supermanifolds.- IV — Basic geometry of G-supermanifolds.- 1. Morphisms.- 2. Products.- 3. Super vector bundles.- 4. Graded exterior differential calculus.- 5. Projectable graded vector fields.- 6. DeWitt supermanifolds.- 7. Rothstein’s axiomatics.- V — Cohomology of supermanifolds.- 1. de Rham cohomology of graded manifolds.- 2. Cohomology of graded differential forms.- 3. Cohomology of DeWitt supermanifolds.- 4. Again on the structure of DeWitt supermanifolds.- VI — Geometry of super vector bundles.- 1. Connections.- 2. Super line bundles.- 3. Characteristic classes.- 4. Characteristic classes in terms of curvature forms.- VII — Lie supergroups and principal super fibre bundles.- 1. Lie supergroups.- 2. Lie supergroup actions.- 3. Principal superfibre bundles.- 4. Connections.- 5. Associated super fibre bundles.

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