Grassmann Algebra
2023/9/8-2023/12/22 9:10-12:10 on Fridays,
M210, Department of Mathematics, NTNU
Speaker:
Ulrich Menne 孟悟理 (National Taiwan Normal University)
Organizer:
Nan-Kuo Ho 何南國 (National Tsing Hua University)
Background & Purpose
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.
Outline
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. Auditing requires the approval of the course instructor.
References
[Fed69] Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. URL: https://doi.org/10.1007/978-3-642-62010-2.
[Men23] Ulrich Menne. Geometric measure theory, 2023. Lecture notes, National Taiwan Normal University.