Federer, Herbert: Geometric Measure Theory.For our purposes, the Introduction and §§ 1, 3, 4 are more than enough. This books presents lots of topics from a standard algebraic topology course, up to spectral sequences and characteristic classes, in the language of differential forms. Bott, Raoul Tu, Loring: Differential Forms in Algebraic Topology.Springer 2001.Ĭoncise presentation of all the basic results you may skip the chapters on manifolds. Important topics: exterior derivative, integration, Stokes’ theorem, de Rham cohomology. Until January 17, you should get familiar with differential forms. Paolini, E.: A short proof of the minimality of Simons cone. Relation between currents and Caccioppoli sets in Chapter 4. Giaquinta, Mariano Modica, Giuseppe Souček, Jiří: Cartesian Currents in the Calculus of Variations I.Maggi, Francesco: Sets of finite perimeter and geometric variational problems.Giusti, Enrico: Minimal surfaces and functions of bounded variation.Publish or Perish 1970, 1979, 1999.Ĭontains derivations of the first and second variation formula. Spivak, Michael: A Comprehensive Introduction to Differential Geometry, Volume Four.This and the following are more broad, complete introductory texts Gentle standard introduction, including basics of submanifold geometry. Nice exposition with lots of pictures, originated as an accessible companion to Federer’s Geometric Measure Theory. Morgan, Frank: Geometric Measure Theory.Historical overview and several proofs in the plane. Blåsjö, Viktor: The isoperimetric problem.Popular science article on recent progress on multi-bubble conjectures. Klarreich, Erica: ‘Monumental’ Math Proof Solves Triple Bubble Problem and More.Lots of 2d solutions of the minimal surface equation in 3d Euclidean space. Sheet 1 (for discussion on 21 25 October) Literature Introduction Optional excercise will appear from time to time in lecture. Physicists, advanced bachelor students, and PhD students are welcome, too!.just to learn more about minimal surfaces □.as Specialization Supplement (Ergänzungsmodul), or.as a type II lecture in the specializations Differential Geometry or Geometric Structures,.You should be familiar with the basic notions of differential geometry (manifolds, Riemannian metrics, geodesics, curvature) and not be afraid of PDEs.two lectures a week, including quizzes, and a few optional exercises every other week.Hypersurface singularities and how to cope with them.Geometric measure theory: Existence and regularity for area-minimizing hypersurfaces.The isoperimetric problem: enclosing volume with least area.All of that in manifolds of arbitrary dimension. Minimal surfaces and their relatives: surfaces of prescribed mean curvature, almost minimizers, horizons in general relativity.Motivation from physics: soap films, bubbles.Tuesday and Friday, 14:15–15:45 in M6, Einsteinstr.
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