Course outline The content of the present course splits into two parts as follows, where an emphasis indicates review topics which will not be examined. Firstly, regarding Grassmann algebra, we cover the exterior algebra of a vector space, alternating forms and duality, interior multiplications, simple m-vectors, and inner products. Secondly, regarding rectifiability, we study differentials and tangents (differentiation, Whitney’s extension theorem, approximate differentiation, Rademacher’s theorem, pointwise differentiation of higher order, factorisation of maps near generic points, submanifolds of Euclidean space, tangent vectors, and relative differentiation) and area for Lipschitzian maps (Jacobians, area of maps of Euclidean space, rectifiable sets, approximate tangent vectors and differentials, area of maps of rectifiable sets, Cartesian products, equality of measures on rectifiable sets, and rectifiable sets and manifolds of class 1).

Details of the course The main reference text will be the instructor’s weekly updated lecture notes written in LATEX, see [Men20]. They are based on and expand the relevant parts of Federer’s treatise [Fed69] and, regarding pointwise differentiation of higher order, the presentation of [Men19]. Grading is solely determined by individual oral examinations conducted in English.

Prerequisites We continue the course Geometric Measure Theory I which treated covering theorems, derivatives, Carathéodory's construction, graded algebras, the symmetric algebra of a vector space, symmetric forms, and polynomial functions.