Module Handbook

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Module MAT-71-11-M-7

Introduction to the Theory of Dirichlet Forms (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-71-11-M-7 Introduction to the Theory of Dirichlet Forms 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-SPAS] Analysis and Stochastics
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-71-11-K-7
Introduction to the Theory of Dirichlet Forms
P 84 h 186 h - - PL1 9.0 irreg.
  • About [MAT-71-11-K-7]: Title: "Introduction to the Theory of Dirichlet Forms"; Presence-Time: 84 h; Self-Study: 186 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86255 ("Introduction to the Theory of Dirichlet Forms")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • resolvents, semigroups, generators (Theorem of Hille and Yosida),
  • coercive bilinear forms (Stampacchia theorem, characterisation by resolvents, semigroups, generators),
  • closed bilinear form,
  • contraction properties (Sub-Markov property, Dirichlet operators, Dirichlet forms).

Competencies / intended learning achievements

Upon successful completion of this module, the students have gained advanced knowledge a subfield of Functional Analysis with applications in current research topics (e.g. in the fields of differential equations and mathematical physics). They are able to name the main propositions of the lecture, classify and explain the illustrated connections. They understand the proofs and are able to reproduce and explain them. In particular, they can critically assess, what conditions are necessary for the validity of the statements.

By solving the given exercise problems, they have gained a precise and independent handling of the terms, propositions and methods of the lecture. In addition, they have learned to apply the methods to new problems, analyze them and develop solution strategies independently or by team work.

Literature

  • Z.-M. Ma, M. Röckner: Introduction to the theory of (non-symmetric) Dirichlet forms,
  • M. Fukushima: Dirichlet Forms and Markov Processes,
  • M. Reed, B. Simon: Methods of modern mathematical physics I.

Requirements for attendance (informal)

Modules:

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-71-11-M-7]

Module-Pool Name
[MAT-70-MPOOL-7] Specialisation Stochastic Analysis (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)