### Specialisation Options in the Master’s Programme in Mathematics Minimum Set

In the following, you will find the concrete planning for courses at specialisation level. The Master’s seminars in the respective specialisations will always be scheduled so that they do not clash but are not included here for space reasons. The same is true of the relevant core modules in the Bachelor’s Programme in Mathematics.

Please note that details may change.

A complete overview of course planning can be found on this page.

All the specialisation offers on the English-language Master’s programme up to the Summer Semester 2024 can be found here.

**WiSe 2020/21**

- Complex Multiplication (2+1 en, Li)
- Einführung in das Langlandsprogramm (2+1 ger, Richarz)
- Modulformen (2+1 ger, Bruinier)

**SuSe 2021**

- Algebraische Geometrie (4+2 ger, Bruinier)
- Hilbert modular forms (2+1 en, Li)

**WiSe 2021/22**

- Lie-Groups (2+1 en, Scheithauer)
- Algebraic Geometry (4+2 en, Richarz and Li)

**SuSe 2022**

- Specialisation: Algebra (4+2 en, Wedhorn)
- Algebraische Zahlentheorie (4+2 ger, Richarz)

**Requirements for a specialisation in Algebra:**

- Attending the third year course “Algebra” resp. courses with similar contents, e.g. in former studies
- The prerequisites for a specialisation in algebra are the following:
- Lang, Algebra, GTM 211, Revised third edition, Springer
- Chapter I Groups: Sections 1 to 11
- Chapter II Rings
- Chapter III Modules
- Chapter IV Polynomials: Sections 1 to 6
- Chapter V Algebraic extensions: Sections 1 to 5
- Capter VI Galois theory: Sections 1 to 7
- Chapter VII Extensions of rings: Section 1
- Chapter X Noetherian rings and modules: Section 1
- Chapter XIII Matrices and linear maps: Sections 1 to 6
- Chapter XIV Representation of one endomorphism
- Chapter XV Structure of bilinear forms: Sections 1 to 8
- Chapter XVI The tensor product: Sections 1 and 2

- Lang, Algebra, GTM 211, Revised third edition, Springer

For additional information visit Algebra.

The specialisation in Analysis (focus: PDE) comprises two lectures (4+2 each), a seminar and, potentially, other special classes. It can be completed in three semesters. The Research Group guarantees that it will begin in every Summer Semester as well as every Winter Semester. You must have existing knowledge of functional analysis. (It is thus essential to attend the course in Functional Analysis if you want to specialise in Analysis; the relevant course is NOT included in the three semesters mentioned above).

The specialisation in Analysis (focus: Banach Algebra – Roch) can only be offered sporadically.

**WiSe 2020/21**

- Partial Differential Equations I (4+2 en., Stinner)
- Lecture on Online Seminar (9 CPs, Haller-Dintelmann)
- Mathematical Modelling of Fluid Interfaces II (2+1, Bothe)

**SuSe 2021**

- Partial Differential Equations II (4+2 en, Stinner)
- Introduction to mathematical fluid mechanics (2+1 en, Modena)
- Banach and C*-Algebra (4+2 ger, Roch)
- Funktional Analysis II (2+1 ger., Roch)
- Mathematical Modelling of Fluid Interfaces I (2+1, Bothe)
- Sobolev Spaces (2+1 en, Stinner)
- Reaktions-Diffusions-Syteme (2+1 ger, Bothe)

**WiSe 2021/22**

- Partial Differential Equations I (4+2 ger, Haller-Dintelmann)
- Partial Differential Equations II/2 (2+1 en, NN)
- Mathematical Modelling of Fluid Interfaces II (2+1, Bothe)
- Harmonische Analysis (2+1 ger, Hieber)
- Functional Analysis (4+2 ger, Bothe)
- Lecture on Online Seminar (9 CPs, Haller-Dintelmann)

**SuSe 2022**

- Partial Differential Equations (Evolutionary Equations) (4+2 ger, Bothe)
- Partial Differential Equations II.2 (2+1 en, NN)
- Introduction to incompressible flows (2+1 en, Modena)
- Mathematical Modelling of Fluid Interfaces I (2+1 ger, Bothe)
- Sobolev Spaces (2+1 en, Stinner)
- Topologie (2+1 ger, Roch)

**Requirements for a specialisation in Analysis:**

- Attending the third year course “Funktionalanalysis” resp. courses with similar contents, e.g. in former studies
- Recommended reading:
- Bühler, Salamon: Functional Analysis, Graduate Studies in Mathematics, 191. American Mathematical Society. Providence, RI, 2018. Paragraphs 1.1 – 1.3, 1.5, 2 (w/o 2.4.4), 3.1 – 3.2.1.

For additional information visit Analysis.

**WiSe 2020/21**

- Riemannian Geometry 2 (Lecture Course, 9 CP en, Große-Brauckmann)

**SuSe 2021**

- Mean curvature flow (4+2 en, Mäder-Baumdicker)

**WiSe 2021/22** – no specification

**SuSe 2022**

- Specialisation: Geometry (4+2 ger, Reif)
- Specialisation: Geometry (2+1 ger, Mäder-Baumdicker)

**Requirements for a specialisation in Geometry and Approximation:**

- Attending the third year course “Differentialgeometrie” resp. courses with similar contents, e.g. in former studies
- Recommended reading:
- do Carmo, Manfredo P.: Differential geometry of curves and surfaces, Dover 2016, relevant are Chapter 1 to 4.
- Oprea, John: Differential geometry and its applications, Pearson Education 2007 (or Prentice-Hall 1997), relevant are Chapter 1 to 6.

For additional information visit Geometry and Approximation.

Specialisation lecture courses are offered in four fields of specialisation that can be combined (see this page); specialisation cycles can be started every semester.

**WiSe 2020/21**

- Classical and non-classical model theory (2+1 en, Otto)
- Introduction to Categorial Theory (2+1 en, Streicher)

**SuSe 2021**

- Applied Proof Theory (4+2 en, Kohlenbach)
- Modal Logics (2+1 en, Otto)
- Graph Theory (4+2 en, Eickmeyer)
- Algorithmen und Symmetrie (4+2 ger, Schweitzer)

**WiSe 2021/22**

- Algorihtmic Metatheorems (2+1 en, Eickmeyer)
- Computability Theory (2+1 en, Kohlenbach)
- Kategorielle Logik (2+1 ger, Streicher)

**SuSe 2022**

- Applied Proof Theory (4+2 en, Kohlenbach)

**Requirements for a specialisation in Logic:**

- Attending the third year course “Introduction to Mathematical Logic” resp. courses with similar contents, e.g. in former studies
- Recommended reading:
- Forster, T.: Logic, Induction and Sets. Cambridge University Press, 234pp., 2003

For additional information visit Logic.

The modules can be taken in any order so that over a period of two years you can complete an entire specialisation in numerics (with or without a Master’s dissertation), irrespective of the semester in which you begin your Master’s programme.

**WiSe 2020/21**

- Numerical Methods for PDEs (4+2 en, Egger)
- Modelling and Simulation in Mathematical Biology (2+1 ger, Gerisch)
- Analyse und Numerik singulär gestörter Probleme (2+1 ger, Schmidt)

**SuSe 2021**

- Computational Fluid Dynamics (4+2 en, Egger)
- Modelling and efficient simulation of dynamical systems (2+1 ger, Kiehl)
- Computational Electromagnetics (2+1 ger, Schmidt)

**WiSe 2021/22**

- Numerical Methods for PDEs (4+2 en, Giesselmann)
- Numerics for hyperbolic differential equations (2+1 ger, Lang)
- Computational Inverse Problems (2+1 en, Egger)

**SuSe 2022**

- Kinetische Gleichungen (4+2 ger, Giesselmann)
- Numerical Methods of Differential Algebraic Equations (2+1 en, Egger)
- Stochastic Finite Elements (2+1 en, Lang)

**Requirements for a specialisation in Numerics and Scientific Computing:**

- Attending the third year course “Numerik gewöhnlicher Differentialgleichungen” resp. courses with similar contents, e.g. in former studies
- For specialized courses in numerical analysis of the master program, we require knowledge from the preceeding bachelor programm, that compares to chapter 1-7 and 10-14 of the following textbook:
- Endre Süli, University of Oxford, David F. Mayers, University of Oxford: An Introduction to Numerical Analysis, 2003, Cambridge University Press ISBN: 9780511801181

- The content of these chapters are taught at TU Darmstadt in the two German taught modules
- 04-10-0013/de Einführung in die Numerische Mathematik 9 CP
- 04-10-0393/de Numerik von Differentialgleichungen. 9 CP

For additional information visit Numerics and Scientific Computing.

The specialisation cycle in Optimisation, which consists of Discrete Optimisation and Non-Linear Optimisation, can be started in any semester and completed within a year. Moreover, at irregular intervals, additional classes are offered at specialisation level.

**WiSe 2020/21**

- Nonlinear Optimization (4+2, ger, Ulbrich)
- Mathematical Programs with Equlibrium Constraints (2+1 en, Schwartz)
- Optimization for Transport and Traffic (2+1 ger, Pfetsch)

**SuSe 2021**

- Discrete Optimization (4+2 ger, Paffenholz)
- Non-Smooth Optimization (2+1 ger, Ulbrich)
- Optimization with Partial Differential Equations (2+1 en, Wollner)

**WiSe 2021/22**

- Nonlinear Optimization (4+2 en, Wollner)
- Interior Point Methods for Convex Optimisation (2+1 ger, Ulbrich)
- Optimization Methods for Machine Learning (2+1 ger, Pfetsch)
- Online Optimization (2+1 en, Disser)

**SuSe 2022**

- Discrete Optimization (4+2 en, Pfetsch)
- Non-Smooth Optimization (2+1 ger, Wollner)
- Optimization for Functional Spaces (2+1 en, Ulbrich)
- Combinatorial Optimization (2+1 en, Disser)

**Requirements for a specialisation in Optimisation:**

- Attending the third year course “Einführung in die Optimierung” resp. courses with similar contents, e.g. in former studies
- Recommended reading:
- V. Chvatal, Linear Programming, Freeman and Company (2003) (entirely)
- J. Nocedal and S. Wright, Numerical Optimization, Springer 1999, Chapter 12 to 12.3 (incl), Chapter 16 to 16.4 (incl)
- ADM (Algorithmic Discrete Mathematics)

For additional information visit Optimisation.

A specialisation cycle in Stochastics lasting two semesters is offered every Winter Semester.

**WiSe 2020/21**

- Mathematical Statistics (4+2 ger, Wichelhaus)

**SuSe 2021**

- Curve Estimation (4+2 ger, Kohler)
- Statisches Grundlagen des Deep Learnings (2+1 de, Langer)

**WiSe 2021/22**

- Specialisation: Stochastics (4+2 en, Aurzada)

**SuSe 2022**

- Specialisation: Stochastics (4+2 en, Betz)
- Specialisation: Stochastics (4+2 ger, Betz)

**Requirements for a specialisation in Stochastics:**

- Attending the third year course “Probability Theory” resp. courses with similar contents, e.g. in former studies
- Recommended reading:
- Durret, Rick: "Probability, Theory and Examples”, 4th edition. Please make sure you not only understand the contents of the following chapters but also do most of the exercises in the book.
- Chapter 1: Measure Theory
- Chapter 2: Laws of Large Numbers; sections 2.1 – 2.5.
- Chapter 3: Central Limit Theorems; sections 3.1 – 3.1 and section 3.9
- Chapter 4: Random walks; sections 4.1 and 4.2
- Chapter 5: Martingales
- Chapter 6: Markov Chains; knowledge of this chapter is not essential, but it would be good to have some ideas about Markov chains.

- Lecture Script of the course “Probability Theory”

- Durret, Rick: "Probability, Theory and Examples”, 4th edition. Please make sure you not only understand the contents of the following chapters but also do most of the exercises in the book.
- In the study programme M.Sc. Mathematics a specialisation in Statistics is not possible.

For additional information visit Stochastics.