Suchergebnis: Katalogdaten im Frühjahrssemester 2021

Physik Master Information
Wahlfächer
Physikalische und mathematische Wahlfächer
Auswahl: Mathematik
NummerTitelTypECTSUmfangDozierende
401-3532-08LDifferential Geometry IIW10 KP4V + 1UW. Merry
KurzbeschreibungThis is a continuation course of Differential Geometry I.

Topics covered include:

- Connections and curvature,
- Riemannian geometry,
- Gauge theory and Chern-Weil theory.
Lernziel
SkriptI will produce full lecture notes, available on my website:

https://www.merry.io/courses/differential-geometry/
LiteraturThere are many excellent textbooks on differential geometry.

A friendly and readable book that contains everything covered in Differential Geometry I is:

John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.

For Differential Geometry II, the textbooks:

- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,

are both excellent. The monograph

- A. L. Besse "Einstein Manifolds", (1987), Springer,

gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.)
Voraussetzungen / BesonderesFamiliarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). Lecture notes for Differential Geometry I can be found on my website.
401-3462-00LFunctional Analysis IIW10 KP4V + 1UA. Carlotto
KurzbeschreibungSobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles.
LernzielAcquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods for studying elliptic boundary value problems.
LiteraturMichael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018.

David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001.

Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.

Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.
Voraussetzungen / BesonderesFunctional Analysis I plus a solid background in measure theory, Lebesgue integration and L^p spaces.
401-0674-00LNumerical Methods for Partial Differential Equations
Nicht für Studierende BSc/MSc Mathematik
W10 KP2G + 2U + 2P + 4AR. Hiptmair
KurzbeschreibungDerivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in C++ based on a finite element library.
LernzielMain skills to be acquired in this course:
* Ability to implement fundamental numerical methods for the solution of partial differential equations efficiently.
* Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations.
* Ability to select and assess numerical methods in light of the predictions of theory
* Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm.
* Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations.
* Skills in the efficient implementation of finite element methods on unstructured meshes.

This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages.
Inhalt1 Second-Order Scalar Elliptic Boundary Value Problems
1.2 Equilibrium Models: Examples
1.3 Sobolev spaces
1.4 Linear Variational Problems
1.5 Equilibrium Models: Boundary Value Problems
1.6 Diffusion Models (Stationary Heat Conduction)
1.7 Boundary Conditions
1.8 Second-Order Elliptic Variational Problems
1.9 Essential and Natural Boundary Conditions
2 Finite Element Methods (FEM)
2.2 Principles of Galerkin Discretization
2.3 Case Study: Linear FEM for Two-Point Boundary Value Problems
2.4 Case Study: Triangular Linear FEM in Two Dimensions
2.5 Building Blocks of General Finite Element Methods
2.6 Lagrangian Finite Element Methods
2.7 Implementation of Finite Element Methods
2.7.1 Mesh Generation and Mesh File Format
2.7.2 Mesh Information and Mesh Data Structures
2.7.2.1 L EHR FEM++ Mesh: Container Layer
2.7.2.2 L EHR FEM++ Mesh: Topology Layer
2.7.2.3 L EHR FEM++ Mesh: Geometry Layer
2.7.3 Vectors and Matrices
2.7.4 Assembly Algorithms
2.7.4.1 Assembly: Localization
2.7.4.2 Assembly: Index Mappings
2.7.4.3 Distribute Assembly Schemes
2.7.4.4 Assembly: Linear Algebra Perspective
2.7.5 Local Computations
2.7.5.1 Analytic Formulas for Entries of Element Matrices
2.7.5.2 Local Quadrature
2.7.6 Treatment of Essential Boundary Conditions
2.8 Parametric Finite Element Methods
3 FEM: Convergence and Accuracy
3.1 Abstract Galerkin Error Estimates
3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM
3.3 A Priori (Asymptotic) Finite Element Error Estimates
3.4 Elliptic Regularity Theory
3.5 Variational Crimes
3.6 FEM: Duality Techniques for Error Estimation
3.7 Discrete Maximum Principle
3.8 Validation and Debugging of Finite Element Codes
4 Beyond FEM: Alternative Discretizations [dropped]
5 Non-Linear Elliptic Boundary Value Problems [dropped]
6 Second-Order Linear Evolution Problems
6.1 Time-Dependent Boundary Value Problems
6.2 Parabolic Initial-Boundary Value Problems
6.3 Linear Wave Equations
7 Convection-Diffusion Problems [dropped]
8 Numerical Methods for Conservation Laws
8.1 Conservation Laws: Examples
8.2 Scalar Conservation Laws in 1D
8.3 Conservative Finite Volume (FV) Discretization
8.4 Timestepping for Finite-Volume Methods
8.5 Higher-Order Conservative Finite-Volume Schemes
SkriptThe lecture will be taught in flipped classroom format:
- Video tutorials for all thematic units will be published online.
- Tablet notes accompanying the videos will be made available to the audience as PDF.
- A comprehensive lecture document will cover all aspects of the course.
LiteraturChapters of the following books provide supplementary reading
(detailed references in course material):

* D. Braess: Finite Elemente,
Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer 2007 (available online).
* S. Brenner and R. Scott. Mathematical theory of finite element methods, Springer 2008 (available online).
* A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer, New York, 2004.
* Ch. Großmann and H.-G. Roos: Numerical Treatment of Partial Differential Equations, Springer 2007.
* W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.
* P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
* S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.
* R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2002.

However, study of supplementary literature is not important for for following the course.
Voraussetzungen / BesonderesMastery of basic calculus and linear algebra is taken for granted.
Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential.

Important: Coding skills and experience in C++ are essential.

Homework assignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.
401-4816-21LMathematical Aspects of Classical and Quantum Field TheoryW8 KP4VM. Schiavina, Uni-Dozierende
KurzbeschreibungThe course will cover foundational topics in classical and quantum field theory from a mathematical standpoint.
Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided.
LernzielThe objective of this course is to expose master and graduate students in mathematics and physics to the mathematical foundations of classical and quantum field theory.
The course will provide a solid mathematical foundation to essential topics in classical and quantum field theories, both useful to mathematics master and graduate students with an interest but no previous background in QFT, as well as for physics master and graduate students who want to focus on more formal aspects of field theory.
InhaltAbstract (long version)
The course will cover foundational topics in classical and quantum field theory from a mathematical standpoint.
Starting from the example of classical mechanics, the relevant mathematical foundations that are necessary for a rigorous approach to field theory will be provided.
The course will feature relevant instances of field theories and sigma models, and it will provide a first introduction the the concepts of quantisation, from mechanics to field theory.
Using scalar field theory and quantum electrodynamics as guideline, the course will present an overview of quantum field theory, focusing on its more mathematical aspects, including, if time permits, a modern approach to gauge theories and the renormalisation group.

Content
The course will start with an overview of geometric concepts that will be used throughout, such as graded differential geometry, as well as fiber and vector bundles.
After brief review of classical mechanics, interpreted as a first example of a field theory, a thorough discussion of classical, local, Lagrangian field theory will follow, covering topics such as Noether’s Theorems, local and global symmetries. We will then present and discuss a number of examples from gauge theory.
In the second part of the course, quantisation will be discussed. The main examples of the scalar field and electrodynamics will be used as a guideline for more general considerations on the quantisation of more general and involved field theories.
In the last part of the course, we plan the discussion of modern approaches to quantisation of field theories with symmetries, renormalisation, and of the open challenges that arise.
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