# Teaching

The information on this page is available in German on the page "Geometric Analysis" at the University of Potsdam. This page contains information concerning the following topics.

- Bachelor and Master theses.
- Information for PhD students.
- Lecture Topics in elliptic partial differential equations in the winter term 2015/2016.
- Lecture Real Analysis in the summer term 2015.

- Seminar Borel- and Suslin sets in the winter term 2014/2015.

- Lecture Partial Differential Equations in the winter term 2013/2014.

- Lecture Introduction to Geometric Measure Theory in the summer term 2013.

- Lecture Real Analysis in the winter term 2012/2013.

- Seminar Elements of Measure Theory in the summer term 2012.

## Bachelor and Master theses

Interested students may compile Bachelor and Master theses. For this purpose successful participation at a seminar (see for example "Elements of Measure Theory").

## PhD theses

An ideal candidate would have mastered material which is covered at Potsdam University in the courses

- Measure Theory (for instance Elements of Measure Theory is sufficient),
- Real Analysis,
- Introduction to Geometric Measure Theory,
- Functional Analysis I (Banach and Hilbert spaces, distributions and Sobolev functions),
- Partial Differential Equations I (linear elliptic partial differential equations),
- Differential Geometry I.

Typically, completing knowledge of the above material and proceeding to more advanced material will be 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".

## Lecture "Topics in elliptic partial differential equations" with tutorial

Lecturer: Prof. Dr. Ulrich Menne

Tutor: Mario Santilli

Place and time of the lecture: in 2.09.0.12 on Mondays 08:15-09:45.

Place and time of the tutorial: in 2.05.0.04 on Mondays 10:15-11:45.

Content: In this lecture pointwise estimates of solutions of elliptic partial differential equations as pioneered by Caldéron and Zygmund in their classical paper [CZ61] shall be studied mainly for the Laplace operator. We investigate the approximability of order k + α of a function at a fixed point by polynomial functions of degree at most k measured in Lebesgue spaces. The associated condition is satisfied uniformly if and only if the function is k times differentiable with its k-th derivative being α Hölder continuous. To put this study into context, Reshetnyak’s theorem on the differentiability of Sobolev functions, see [Reš68], which extends Rademacher’s theorem will be proven. As tools, algebraic properties of polynomial functions and Whitney’s extension theorem will be derived following [Fed69].

References: Probably, lecture notes will be created during the course.

[CZ61] A.-P. Calderón and A. Zygmund. Local properties of solutions of elliptic partial differential equations. *Studia Math.*, 20:171–225, 1961.

[Fed69] Herbert Federer. *Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[Reš68] Ju. G. Rešetnjak. Generalized derivatives and differentiability almost everywhere. *Mat. Sb. (N.S.)*, 75(117):323–334, 1968.

Prerequisites: Knowledge of “L2 theory” for the Laplace operator as obtainable by attending the lecture “Partielle Differentialgleichungen” by Prof. Metzger in parallel.

All further information may be retrieved via moodle2. For this an account at the University of Potsdam is required.

## Lecture "Real Analysis" with tutorial

Lecturer: Prof. Dr. Ulrich Menne

Tutor: Mario Santilli

Place and time of the lecture: in 1.08.0.53 on Fridays 08:15-09:45.

Place and time of the tutorial: in 1.19.1.19 on Fridays 16:15-17:45.

Content: The following topics which are of importance for instance in partial differential equations and geometric measure theory will be treated:

- covering theorems (e.g. those of Vitali and Besicovitch),
- differentiation theory of locally finite measures, Lebesgue points and differentiability Lebesgue almost everywhere of monotone functions,
- characterisation of differentiability almost everywhere for real valued functions (theorems of Rademacher and Stepanoff)

References: There will be lecture notes in German available. Background reading is as follows:

- Lawrence C. Evans and Ronald F. Gariepy.
*Measure theory and fine properties of functions*. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. - Herbert Federer.
*Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

Prerequistes: Basics of measure theory including Lebesgue integration.

All further information may be retrieved via moodle2. For this an account at the University of Potsdam is required.

## Seminar Borel- and Suslin sets

Lecturer: Prof. Dr. Ulrich Menne

Place and time of the seminar: in 1.22.1.28 on Tuesdays 16:00-17:30.

Finding rooms at Potsdam University: A.x.y.z indicates campus A, house x, floor y, room z; here is a plan of campus I (Neues Palais).

Content: Borel sets play a central role in the measure theory on Euclidean spaces or more generally on complete, separable metric spaces. In the seminar we will be guided by two questions.

- Under which conditions is the continuous image of a Borel set again a Borel set?
- Is the continuous image of a Borel set measurable with respect to every Borel measure?

In order to answer these questions, we will present the theory of Suslin sets which was created for this purpose. The required statements from measure theory are mostly of elementary nature and will be developed in the seminar.

Anyone interested is asked to **register by email**. The selection of topics is based on a **first-come first-served basis**.

Prerequistes: Basic analysis and linear algebra.

All further information may be retrieved via moodle2. For this an account at the University of Potsdam is required.

## Lecture "Partial Differential Equations" with tutorial

Lecturer: Prof. Dr. Ulrich Menne

Tutor: Dr. Sławomir Kolasiński

Place and time of the lecture: in 1.09.2.06 on Thursdays 8:15-09:45 and in 1.19.1.19 on Fridays 14:15-15:45.

Place and time of the tutorial: in 1.09.2.06 on Mondays 10:15-11:45.

Finding rooms at Potsdam University: A.x.y.z indicates campus A, house x, floor y, room z; here is a plan of campus I (Neues Palais).

Content: Partial differential equations play a central role in geometric analysis. Also, they play an important role in physics. The lecture will primarily treat linear elliptic systems and equations of second order. Initially, elementary properties of harmonic functions will be studied which are examplary for the further theory. The most important aim of the lecture is the proof of existence and a priori estimates of solutions in Sobolev spaces. To deepen the functional analytic aspects of the lecture it is recommended (though not necessary) to participate in the lecture functional analysis by Prof. Dr. Markus Klein.

Prerequistes: Basic analysis (including measure theory and Lebesgue spaces) and linear algebra.

## Lecture "Introduction to Geometric Measure Theory" with tutorial

**Important**: Note the new day of the week and the new room for the lecture.

Lecturer: Prof. Dr. Ulrich Menne

Tutor: Christian Scharrer

Place and time of the lecture: in I.08.0.59, on Fridays 10:15-11:45.

Place and time of the tutorial: in I.11.1.25, on Fridays 12:15-13:00.

Finding rooms at Potsdam University: A.x.y.z indicates campus A, house x, floor y, room z; here is a plan of campus I (Neues Palais).

Content: To prove existence of geometric variational problems, it turs out to be useful to consider rectifiable sets which generalise the notion of submanifold. In this course some fundamental properties of rectifiable sets shall be treated. In particular, the notions of surface measure, tangent space and relative differential will be replaced by the more general concepts of Hausdorff measure, approximate tangent cone and approximate differential.

References: There will be lecture notes available. The following books serve as background:

- Lawrence C. Evans and Ronald F. Gariepy.
*Measure theory and fine properties of functions*. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. - Herbert Federer.
*Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

Prerequistes: Aufbaumodul Analysis 1 (A standard measure and integration course will do.)

## Lecture "Real Analysis" with tutorial

Lecturer: Prof. Dr. Ulrich Menne

Tutor: Christian Scharrer

Place and time of the lecture: in **I.22.1.27** on **Mondays** at 14:15-15:45

Place and time of the tutorial: in I.08.0.53 on Fridays at 09:00-09:45

Content: The following topics which are of importance for instance in partial differential equations and geometric measure theory will be treated:

- covering theorems (e.g. those of Vitali and Besicovitch),
- differentiation theory of locally finite measures, Lebesgue points and differentiability Lebesgue almost everywhere of monotone functions,
- characterisation of differentiability almost everywhere for real valued functions (theorems of Rademacher and Stepanoff),
- generalisation of the classical transformation formula to area formula and coarea formula for Lipschitzian maps.

References: There will be lecture notes available. The following books serve as background:

- Lawrence C. Evans and Ronald F. Gariepy.
*Measure theory and fine properties of functions*. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. - Herbert Federer.
*Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

Prerequistes: Basics of measure theory.

All further information may be retrieved via moodle. For this an account at the University of Potsdam is required.

## Seminar "Elements of Measure Theory"

Interested students please contact the lecturer.

Lecturer: Prof. Dr. Ulrich Menne

Place and Time: usually in I.09.2.06 on Wednesdays at 10:15-11:45

Content: In the seminar some fundamental theorems in measure theory are treated which, for example, may be applied in partial differential equations, calculus of variations or geometric measure theory. Amongst the theorems covered are the theorems of Lusin and Egoroff, the duality of Lebesgue spaces and the Riesz Radon representation theorem.

Prerequistes: "Analysis", "Linear Algebra und Analytische Geometrie"; some talks also require "Aufbaumodul Analysis 1"

Target audience: BA-M, BA-LG, MA-M, MA-LG, DM

All further information may be retrieved via moodle. For this an account at the University of Potsdam is required.