Table of contents

This page was last modified on: 07 February 2023, 17:37, CST

Master theses

Students interested in my research area (see Research) may compile Master theses under my supervision. A basic prerequiste for this purpose is sound knowledge of real analysis; ideally, such a course (see Real Analysis I+II below) could already be taken during BSc studies. Thereafter, successful participation in a reading course of mine (see, for instance, Topics in Geometric Analysis I+II below) is strongly recommended. Most topics in the reading courses will pertain to basic geometric measure theory; some topics will also pertain to functional analysis, elliptic partial differential equations, and differential geometry.

PhD theses

The following list of courses exemplifies the material an ideal candidate would have mastered:

  • Real analysis (General measures, measurable functions, Lebesgue integration, linear functionals on Lebesgue spaces, Riesz's representation theorem),
  • basic geometric measure theory (Grassmann algebra, covering theorems, derivatives of measures, Hausdorff measures, area of Lipschitzian maps, rectifiable sets),
  • functional analysis (Banach and Hilbert spaces, distributions, and Sobolev functions),
  • partial differential equations (linear elliptic partial differential equations: a priori estimates in Hölder and Sobolev spaces), and
  • differential geometry (Riemannian manifolds, submanifolds).

Typically however, completing knowledge of the above material and proceeding to more advanced material will constitute the main part of the first year's work of a PhD student. To obtain an impression of the latter, one may have a look at H. Federer's comprehensive treatise "Geometric Measure Theory" and W. Allard's fundamental paper "On the first variation of a varifold".

Spring term 2023

2023 Geometric Measure Theory II (MSc/PhD) (lecture), NTNU
Topics: alternating forms and duality, interior multiplications, simple m-vectors, inner products, differentials and tangents, Lipschitzian maps.

Previous courses

2022–2023 Geometric Measure Theory I (MSc/PhD) (lecture), NTNU
Topics: covering theorems, derivatives, Carathéodory's construction, graded algebras, the symmetric algebra of a vector space, symmetric forms, polynomial functions.
2022 Real Analysis II (BSc/MSc) (lecture with tutorial), NTNU/TMS
Topics: tensor products, Lebesgue integration (limit theorems and Lebesgue spaces), linear functionals (Daniell integrals, linear functionals on Lebesgue spaces, Riesz's representation theorem, and Riemann-Stieltjes integration), product measures (Fubini's theorem and Lebesgue measure).
2021–2022 Real Analysis I (BSc/MSc) (lecture with tutorial), NTNU/TMS
Topics: some point-set topology, measures and measurable sets, Borel sets and their images, Borel regular measures (approximation by closed sets and Radon measures), measurable functions (Lusin's theorem and Egoroff's theorem).
2021 Topics in Geometric Measure Theory II (MSc/PhD) (seminar), NTNU/TMS
Topics: distribution theory, some elliptic PDEs, first variation of area, radial deformations and the rectifiability theorem, compactness theorem for integral varifolds, isoperimetric inequality, Allard's regularity theorem.
2020–2021 Topics in Geometric Measure Theory I (MSc/PhD) (seminar), NTNU/TMS
Topics: locally convex spaces, Daniell integrals (decomposition and weak convergence), Grassmann manifolds, some structure theory, some elliptic PDEs, curvature of submanifolds, basic properties of varifolds.
2020 Geometric Measure Theory II (MSc/PhD) (lecture), NTNU/TMS
Topics: Carathéodory’s construction, curves of finite length, differentials and tangents, second fundamental form, area of Lipschitzian maps.
2020 Topics in Geometric Analysis II (MSc/PhD) (seminar, joint with Chun-Chi Lin), NTNU
Topics: rectifiability and approximate differentiability of higher order for sets, pointwise differentiability of higher order for distributions, Hall's theorem on perfect matching.
2019–2020 Geometric Measure Theory I (MSc/PhD) (lecture), NTNU/TMS
Topics: Grassmann algebra (tensor products, graded algebras, exterior algebra of a vector space, interior multiplications, simple m-vectors, inner products), Borel regular measures, covering theorems, derivatives of measures.
2019–2020 Topics in Geometric Analysis I (MSc/PhD) (seminar, joint with Chun-Chi Lin), NTNU
Topics: Sobolev functions and functions of bounded variation, topological vector spaces, pointwise differentiability of higher order for sets, Kohn's example concerning approximate differentiation.
2019 Special topics in analysis (lecture), NTNU
Topics: Borel and Suslin sets, symmetric algebra of a vectorspace, polynomial functions, classical, approximate, and pointwise differentiation of higher order, rectifiability of higher order.
2019 Real analysis II (lecture with tutorial), NTNU
Topics: Lebesgue spaces, Jensen's inequality, Daniell integrals, linear functionals on Lebesgue spaces, Riesz's representation theorem, Fubini's theorem, Lebesgue measure.
2018–2019 Real analysis I (lecture with tutorial, joint with Chun-Chi Lin), NTNU
Topics own part: Measures and measurable sets, Borel sets, measurable functions (approximation theorems, spaces of measurable functions), Lebesgue integration (basic properties, limit theorems).
2018 Geometric variational problems (lecture), University of Leipzig (UL)
Topics: Multilinear algebra, basic geometric measure theory (Hausdorff densities, Hausdorff distance, Kirszbraun's theorem, relative differentials for closed sets, area formula, rectifiable sets), varifolds.
2017–2018 Sets of finite perimeter (seminar, in German), UL.
Topics: Whitney's extension theorem, Sobolev functions, functions of bounded variation (isoperimetric inequalities, reduced boundary, criterion for finite perimeter).
2017–2018 Introduction to geometric analysis (lecture, in German), UL.
Topics: submanifolds of Euclidean space (second fundamental form, covariant differentiation, equations of Gauss and Codazzi), Brunn-Minkowski inequality, monotonicity formula, isoperimetric inequalities, subharmonic functions.
2016–2017 Mathematics for business information systems (lecture with tutorial, in German), University of Potsdam (UP).
Topics: basics of logic, set theory, number systems, linear algebra, and analysis.
2016 Surfaces in analysis, geometry and physics (lecture series with tutorial, joint with Christian Bär, Jan Metzger, and Sylvie Paycha), UP.
Topics own part: BV-functions (convolution, Sobolev- and Poincaré-inequalities), geometric variational problems.
2015–2016 Topics in elliptic partial differential equations (lecture with tutorial), UP.
Topics: Whitney's extension theorem, rectifiability of higher order for functions, Sobolev functions, ellipticity, pointwise differentiability of solutions for linear elliptic partial differential equations.
2015 Real analysis (lecture with tutorial, in German), UP.
Topics: covering theorems of Besicovitch and Vitali; differentiation theory of Radon measures: existence, Lebesgue points, densities, approximate continuity; curves of finite length; theorems of Rademacher and Stepanoff.
2014–2015 Borel sets and Suslin sets (seminar, in German), UP.
Topics: Borel sets, spaces of sequences, Suslin sets; measures, measurable sets, regular measures, nonmeasurable sets, Borel regular measures; measurability of Suslin sets.
2013–2014 Partial Differential Equations (lecture with tutorial), UP.
Topics: properties of harmonic functions; multilinear algebra; differentials of higher order; convolution; Sobolev- and Hölder-spaces; $L^2$ theory: Lax Milgram, Gårding-inequality, existence, strong solutions.
2013 Introduction to geometric measure theory (lecture with tutorial, in German), UP.
Topics: Hausdorff measures, Cantor sets, Steiner symmetrisation, Kirszbraun's theorem; submanifolds of Euclidean spaces; rectifiable sets; area formula on rectifiable sets.
2012–2013 Real analysis (lecture with tutorial, in German), UP.
Topics: see the course Real analysis above.
2012 Elements in measure theory (seminar, in German), UP.
Topics: basics of outer measures, Carathéodory's criterion, theorems of Lusin and Egoroff, Daniell integrals, Riesz-Radon representation theorem, duality of Lebesgue spaces.