Suchergebnis: Katalogdaten im Herbstsemester 2019
|Doktorat Departement Mathematik |
Mehr Informationen unter: https://www.ethz.ch/de/doktorat.html
Die Liste der Lehrveranstaltungen (samt der zugehörigen Anzahl Kreditpunkte) für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht.
ACHTUNG: Kreditpunkte fürs Doktoratsstudium sind nicht mit ECTS-Kreditpunkten zu verwechseln!
| Graduate School / Graduiertenkolleg|
Offizielle Website der Zurich Graduate School in Mathematics:
|401-5001-69L||Topics in Modern Analytic Number Theory||W||0 KP||2V||W. D. Duke|
|Inhalt||This series of lectures will treat a number of special topics all of which involve some aspect of the modular group SL(2,ℤ).|
Many have as their natural starting point the class number formula of Dirichlet for binary quadratic forms or, equivalently, ideal classes of quadratic number fields. Most involve L-functions of various kinds and modular forms and their generalizations.
These topics include:
1. New class number formulas, both for quadratic forms and higher degree forms, for example binary cubic forms.
2. Properties of geometric objects associated to the modular group, including cm points, closed geodesics and infinite volume surfaces having closed geodesics as boundaries.
3. The interaction between flat metrics and the hyperbolic metric, for instance the curvature of closed geodesics with respect to a flat metric.
4. Quasimorphisms of the modular group and modular cocycles. Linking numbers of modular knots.
5. Markov forms and modular billiards.
6. Connections with Diophantine approximation and the geometry of numbers.
I will attempt to provide enough background and historical context for each topic to make them accessible to non-experts, but hope to give enough “new” material to stimulate the interest of the experts as well. I will also give suggestions about areas for further research.
|401-5003-69L||Weak Convergence Methods for Nonlinear Partial Differential Equations||W||0 KP||2V||C. Evans|
|Inhalt||This course will be an ambitious survey of rigorous methods for understanding solutions u^ε of various nonlinear PDE in various asymptotic limits. Assuming in particular that the u^ε converge weakly as ε → 0, we want to identify what PDE the weak limit u solves. This can be a hugely complicated problem, since the weak convergence is usually incompatible with the nonlinearities; but a rich variety of modern techniques can handle these issues for many interesting cases.|
The lectures will discuss recent developments concerning (i) asymptotics for ODE, (ii) maximum principle methods, (iii) convexity and monotonicity, (iv) oscillations and compensated compactness, and (v) defect measures, with many examples and applications.
|401-5005-69L||Machine Learning of Dynamic Processes with Applications to Forecasting||W||0 KP||2V||J. P. Ortega Lahuerta|
|Inhalt||The last years have seen the emergence of significant interplays between machine learning, dynamical systems, and stochastic processes with interesting applications in time series analysis and forecasting. The resulting techniques have revolutionized the way in which we learn and path-continue complex and high-dimensional deterministic dynamical systems and preliminary results show that the hope for similar success is justified in the stochastic context.|
In these lectures, we will present in a self-contained manner how these techniques are built and will analyze in detail their connections with classical results in systems and approximation theories, control, filtering, and dynamical systems. Some previous background on any of these topics would help in understanding the material but it is by no means necessary. In order to adopt a point of view as close as possible to the applications, we will work in semi-infinite discrete-time input/output setups. This allows us to adequately model the non-markovianity associated to the observations and the subsystems of large dimensional systems and, additionally, provides us with the necessary tools for the development of both finite-sample and asymptotic results in estimation theory.
Several sessions will be dedicated to explaining the implementation of these techniques in the identification and path continuation of deterministic systems (learning of chaotic attractors) and forecasting of stochastic processes (realized financial covariances). We shall see how these novel techniques outperform all the benchmarks available in the literature to accomplish those tasks.
|401-3225-00L||Introduction to Lie Groups||W||8 KP||4G||P. D. Nelson|
|Kurzbeschreibung||Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.|
|Lernziel||The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.|
|Literatur||A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)|
A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
H. Samelson: "Notes on Lie algebras" (Springer, '90)
S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)
|Voraussetzungen / Besonderes||Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.|
Course webpage: https://metaphor.ethz.ch/x/2018/hs/401-3225-00L/
|401-4117-69L||p-Adic Galois Representations||W||4 KP||2V||M. Mornev|
|Kurzbeschreibung||This course covers the structure theory of Galois groups of local fields, the rings of Witt vectors, the classification of p-adic representations via phi-modules, the tilting construction from the theory of perfectoid spaces, the ring of de Rham periods and the notion of a de Rham representation.|
|Lernziel||Understanding the construction of the ring of de Rham periods.|
|Inhalt||In addition to the subjects mentioned in the abstract the course included the basic theory of local fields, l-adic local Galois representations, an oveview of perfectoid fields, the statements of the theorems of Fontaine-Winterberger and Faltings-Tsuji.|
|Literatur||J.-M. Fontaine, Y. Ouyang. Theory of p-adic Galois representations.|
O. Brinon, B. Conrad. CMI summer school notes on p-adic Hodge theory.
|Voraussetzungen / Besonderes||General topology, linear algebra, Galois theory.|
|401-3001-61L||Algebraic Topology I||W||8 KP||4G||A. Sisto|
|Kurzbeschreibung||This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include:|
singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms.
|Literatur||1) A. Hatcher, "Algebraic topology",|
Cambridge University Press, Cambridge, 2002.
Book can be downloaded for free at:
2) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.
3) E. Spanier, "Algebraic topology", Springer-Verlag
|Voraussetzungen / Besonderes||You should know the basics of point-set topology.|
Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology").
Some knowledge of differential geometry and differential topology is useful but not strictly necessary.
Some (elementary) group theory and algebra will also be needed.
|401-4351-69L||Optimal Transport||W||4 KP||2V||A. Figalli|
|Kurzbeschreibung||In this course I plan to give an introduction to optimal transport: I'll first introduce the optimal transport problem and explain how to solve it in some important cases of interest. Then I'll show a series of applications to geometry and to gradient flows.|
|Lernziel||The aim of the course is to provide a self contained introduction to optimal transport. The students are expected to know the basic concepts of measure theory. Although not strictly required, some basic knowledge of Riemannian geometry may be useful.|
|Literatur||Topics in Optimal Transportation (Graduate Studies in Mathematics, Vol. 58), by Cédric Villani|
Optimal Transport for Applied Mathematicians (Calculus of Variations, PDEs, and Modeling), by Filippo Santambrogio
Optimal transport and curvature, available at
|401-4657-00L||Numerical Analysis of Stochastic Ordinary Differential Equations |
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
|W||6 KP||3V + 1U||K. Kirchner|
|Kurzbeschreibung||Course on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables.|
|Lernziel||The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.|
|Inhalt||Generation of random numbers|
Monte Carlo methods for the numerical integration of random variables
Stochastic processes and Brownian motion
Stochastic ordinary differential equations (SODEs)
Numerical approximations of SODEs
Applications to computational finance: Option valuation
|Skript||There will be English, typed lecture notes for registered participants in the course.|
|Literatur||P. Glassermann: |
Monte Carlo Methods in Financial Engineering.
Springer-Verlag, New York, 2004.
P. E. Kloeden and E. Platen:
Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 1992.
|Voraussetzungen / Besonderes||Prerequisites:|
Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.
a) mandatory courses:
Probability Theory I.
b) recommended courses:
Start of lectures: Wednesday, September 18, 2019.
|401-3651-00L||Numerical Analysis for Elliptic and Parabolic Partial Differential Equations |
Course audience at ETH:
3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course
"Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.
|W||10 KP||4V + 1U||C. Schwab|
|Kurzbeschreibung||This course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.|
|Lernziel||Participants of the course should become familiar with|
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
|Inhalt||The course will address the mathematical analysis of numerical solution methods|
for linear and nonlinear elliptic and parabolic partial differential equations.
Functional analytic and algebraic (De Rham complex) tools will be provided.
Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed.
Particular attention will be placed on developing mathematical foundations
(Regularity, Approximation theory) for a-priori convergence rate analysis.
A-posteriori error analysis and mathematical proofs of adaptivity and optimality
will be covered.
Implementations for model problems in MATLAB and python will illustrate the
A selection of the following topics will be covered:
* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
|Literatur||Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp.|
A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer,
1st Ed. 2004, 2nd Ed. 2015.
R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)
Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp.
D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]
V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).
|Voraussetzungen / Besonderes||Practical exercises based on MATLAB|
Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations
|401-4785-00L||Mathematical and Computational Methods in Photonics||W||8 KP||4G||H. Ammari|
|Kurzbeschreibung||The aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces|
|Lernziel||The field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications. |
The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength.
Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures.
The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions.
In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials.
|401-4597-67L||Random Walks on Transitive Graphs||W||4 KP||2V||V. Tassion|
|Kurzbeschreibung||In this course, we will present modern topics at the interface between probability and geometric group theory. We will be mainly focused on the random walk, and discuss its behavior depending on the geometric properties of the underlying graph.|
|Voraussetzungen / Besonderes||- Probability Theory.|
- Basic properties of Markov Chains.
- No prerequisite on group theory, all the background will be introduced in class.
|401-4619-67L||Advanced Topics in Computational Statistics|
Findet dieses Semester nicht statt.
|W||4 KP||2V||keine Angaben|
|Kurzbeschreibung||This lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.|
|Lernziel||Students learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.|
|Inhalt||The main focus will be on graphical models in various forms: |
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
|Voraussetzungen / Besonderes||We assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.|
|401-4623-00L||Time Series Analysis|
Findet dieses Semester nicht statt.
|W||6 KP||3G||N. Meinshausen|
|Kurzbeschreibung||Statistical analysis and modeling of observations in temporal order, which exhibit dependence. Stationarity, trend estimation, seasonal decomposition, autocorrelations,|
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. Implementations in the software R.
|Lernziel||Understanding of the basic models and techniques used in time series analysis and their implementation in the statistical software R.|
|Inhalt||This course deals with modeling and analysis of variables which change randomly in time. Their essential feature is the dependence between successive observations.|
Applications occur in geophysics, engineering, economics and finance. Topics covered: Stationarity, trend estimation, seasonal decomposition, autocorrelations,
spectral and wavelet analysis, ARIMA-, GARCH- and state space models. The models and techniques are illustrated using the statistical software R.
|Literatur||A list of references will be distributed during the course.|
|Voraussetzungen / Besonderes||Basic knowledge in probability and statistics|
|401-3627-00L||High-Dimensional Statistics||W||4 KP||2V||P. L. Bühlmann|
|Kurzbeschreibung||"High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed.|
|Lernziel||Knowledge of methods and basic theory for high-dimensional statistical inference|
|Inhalt||Lasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling|
|Literatur||Peter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag. |
|Voraussetzungen / Besonderes||Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).|
|401-3619-69L||Mathematics Tools in Machine Learning||W||4 KP||2G||F. Balabdaoui|
|Kurzbeschreibung||The course reviews many essential mathematical tools used in statistical learning. The lectures will cover the notions of hypotheses classes, sample complexity, PAC learnability, model validation and selection as well as results on several well-known algorithms and their convergence.|
|Lernziel||In the exploding world of artifical intelligence and automated learning, there is an urgent need to go back to the basis of what is driving many of the well-establsihed methods in statistical learning. The students attending the lectures will get acquainted with the main theoretical results needed to establish the theory of statistical learning. We start with defining what is meant by learning a task, a training sample, the trade-off between choosing a big class of functions (hypotheses) to learn the task and the difficulty of estimating the unknown function (generating the observed sample). The course will also cover the notion of learnability and the conditions under which it is possible to learn a task. In a second part, the lectures will cover algoritmic apsects where some well-known algorithms will be described and their convergence proved. |
Through the exerices classes, the students will deepen their understanding using their knowledge of the learned theory on some new situations, examples or some counterexamples.
|Inhalt||The course will cover the following subjects:|
(*) Definition of Learning and Formal Learning Models
(*) Uniform Convergence
(*) Linear Predictors
(*) The Bias-Complexity Trade-off
(*) VC-classes and the VC dimension
(*) Model Selection and Validation
(*) Convex Learning Problems
(*) Regularization and Stability
(*) Stochastic Gradient Descent
(*) Support Vector Machines
|Literatur||The course will be based on the book|
"Understanding Machine Learning: From Theory to Algorithms"
by S. Shalev-Shwartz and S. Ben-David, which is available online through the ETH electronic library.
Other good sources can be also read. This includes
(*) the book "Neural Network Learning: Theoretical Foundations" de Martin Anthony and Peter L. Bartlett. This book can be borrowed from the ETH library.
(*) the lectures notes on "Mathematics of Machine Learning" taught by Philippe Rigollet available through the OpenCourseWare website of MIT
|Voraussetzungen / Besonderes||Being able to follow the lectures requires a solid background in Probability Theory and Mathematical Statistical. Notions in computations, convergence of algorithms can be helpful but are not required.|
|401-3628-14L||Bayesian Statistics||W||4 KP||2V||F. Sigrist|
|Kurzbeschreibung||Introduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.|
|Lernziel||Students understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.|
|Inhalt||Topics that we will discuss are:|
Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
|Skript||A script will be available in English.|
|Literatur||Christian Robert, The Bayesian Choice, 2nd edition, Springer 2007.|
A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).
Additional references will be given in the course.
|Voraussetzungen / Besonderes||Familiarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.|
|401-4889-00L||Mathematical Finance||W||11 KP||4V + 2U||J. Teichmann|
|Kurzbeschreibung||Advanced course on mathematical finance:|
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- additional topics
|Lernziel||Advanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes)|
|Inhalt||This is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.|
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- and probably others
|Skript||The course is based on different parts from different books as well as on original research literature. |
Lecture notes will not be available.
|Literatur||(will be updated later)|
|Voraussetzungen / Besonderes||Prerequisites are the standard courses|
- Probability Theory (for which lecture notes are available)
- Brownian Motion and Stochastic Calculus (for which lecture notes are available)
Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts.
This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II.
For an overview of courses offered in the area of mathematical finance, see Link.
|402-0861-00L||Statistical Physics||W||10 KP||4V + 2U||G. M. Graf|
|Kurzbeschreibung||The lecture focuses on classical and quantum statistical physics. Various techniques, cumulant expansion, path integrals, and specific systems are discussed: Fermions, photons/phonons, Bosons, magnetism, van der Waals gas. Phase transitions are studied in mean field theory (Weiss, Landau). Including fluctuations leads to critical phenomena, scaling, and the renormalization group.|
|Lernziel||This lecture gives an introduction into the the basic concepts and applications of statistical physics for the general use in physics and, in particular, as a preparation for the theoretical solid state physics education.|
|Inhalt||Thermodynamics, three laws of thermodynamics, thermodynamic potentials, phenomenology of phase transitions.|
Classical statistical physics: micro-canonical-, canonical-, and grandcanonical ensembles, applications to simple systems.
Quantum statistical physics: single particle, ideal quantum gases, fermions and bosons, statistical interaction.
Techniques: variational approach, cumulant expansion, path integral formulation.
Degenerate fermions: Fermi gas, electrons in magnetic field.
Bosons: photons and phonons, Bose-Einstein condensation.
Magnetism: Ising-, XY-, Heisenberg models, Weiss mean-field theory.
Van der Waals gas-liquid transition.
Landau theory of phase transitions, first- and second order, tricritical.
Fluctuations: field theory approach, Gauss theory, self-consistent field, Ginzburg criterion.
Critical phenomena: scaling theory, universality.
Renormalization group: general theory and applications to spin models (real space RG), phi^4 theory (k-space RG), Kosterlitz-Thouless theory.
|Skript||Lecture notes available in English.|
|Literatur||No specific book is used for the course. Relevant literature will be given in the course.|
|401-3059-00L||Kombinatorik II||W||4 KP||2G||N. Hungerbühler|
|Kurzbeschreibung||Der Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik.|
|Lernziel||Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.|
|Inhalt||Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.|
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