1. Course Background & Purposes

The concepts of tensor product, alternating form, and polynomial function occur in a variety of areas in analysis and geometry including Lebesgue integration, differentials of higher order, analytic functions, Grassmann manifolds, differential forms, currents, linear partial differential equations, curvatures, and varifolds. The present course develops these concepts from the unifying viewpoint of Grassmann algebra and places an emphasis on universal properties as well as functorial constructions and their naturality. Selected applications are indicated throughout the course.

2. Course Outline & Descriptions

Course outline   The following topics will be covered: tensor products, graded algebras, the exterior algebra of a vector space, alternating forms and duality, interior multiplications, simple m-vectors, inner products, mass and comass, the symmetric algebra of a vectorspace, and symmetric forms and polynomial functions. The exposition of the present course only treats the case of real vector spaces. Students with knowledge of more general fields or modules will however realise that much of theory is applicable in these cases as well.

Details of the course   The main reference text will be the instructor’s weekly updated lecture notes written in LATEX extracted from [Men23]. They are based on and expand the relevant parts of Federer’s treatise [Fed69]. Videos recordings will be made available to the participants of the course.

Prerequisites We employ basic linear algebra (vector spaces, linear maps, Cartesian products, direct sums, and bases). No knowledge of multilinear algebra (or determinants) is required. Enrolling and auditing requires the approval of the course instructor.

References

3. Grading

Individual oral examination conducted in English (100%).

4. Credit

This course offers 3 credits.