Autumn Semester 2020 takes place in a mixed form of online and classroom teaching.
Please read the published information on the individual courses carefully.

Search result: Catalogue data in Spring Semester 2017

Mathematics Master Information
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Algebra, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-4142-17LAlgebraic CurvesW6 credits3GR. Pandharipande
AbstractI will discuss the classical theory of algebraic curves. The topics will include:
divisors, Riemann-Roch, linear systems, differentials, Clifford's theorem,
curves on surfaces, singularities, curves in projective space, elliptic curves,
hyperelliptic curves, families of curves, moduli, and enumerative geometry.
There will be many examples and calculations.
Objective
ContentLecture homepage: https://metaphor.ethz.ch/x/2017/fs/401-4142-17L/
LiteratureForster, "Lectures on Riemann Surfaces"

Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves"

Mumford, "Curves and their Jacobians"
Prerequisites / NoticeFor background, a semester course in algebraic geometry should be
sufficient (perhaps even if taken concurrently). You should know the definitions
of algebraic varieties and algebraic morphisms and their basic properties.
401-3106-17LClass Field TheoryW6 credits2V + 1UJ. Fresán
AbstractClass Field Theory aims at describing the Galois group of the maximal abelian extension of global and local fields.
Objective
Literature[1] D. Harari, Cohomologie galoisienne et théorie du corps de classes, EDP Sciences, CNRS Éditions, Paris, 2017.
[2] K. Kato, N. Kurokawa, T. Saito, Number theory 2. Introduction to class field theory, Translations of Mathematical Monographs 240, AMS, 2011.
[3] J. S. Milne, Class Field Theory (available at http://www.jmilne.org/math/CourseNotes/cft.html)
[4] J-P. Serre, Local fields, Grad. Texts Math. 67. Springer-Verlag, 1979.
401-3033-00LGödel's TheoremsW8 credits3V + 1UL. Halbeisen
AbstractDie Vorlesung besteht aus drei Teilen:
Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe.
Teil II behandelt den Gödel'schen Vollständigkeitssatz
Teil III behandelt die Gödel'schen Unvollständigkeitssätze
ObjectiveDas Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln.
ContentSyntax und Semantik der Prädikatenlogik
Gödel'scher Vollständigkeitssatz
Gödel'sche Unvollständigkeitssätze
LiteratureErgänzende Literatur wird in der Vorlesung angegeben.
401-3058-00LCombinatorics IW4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers.
Prerequisites / NoticeRecognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008).
401-3112-17LIntroduction to Number TheoryW4 credits2VC. Busch
AbstractThis course gives an introduction to number theory. The focus will be on algebraic number theory.
Objective
ContentThe following subjects will be covered:
- Euclidean algorithm, greatest common divisor, ...
- Congruences, Chinese Remainder Theorem
- Quadratic residues, Legendre symbol, law of quadratic reciprocity
- Quadratic number fields, integers and primes
- Units of quadratic number fields, Pell's equation, Dirichlet unit theorem
- Continued fractions and quadratic irrationalities, Theorem of Euler Lagrange, relation to units.
Literature- A. Fröhlich, M.J. Taylor, Algebraic number theory, Cambridge studies in advanced mathematics 27, Cambridge University Press, 1991
- S. Lang, Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics, 110, Springer, 1994
- J. Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften 322, Springer 1999
- R. Remmert, P. Ullrich, Elementare Zahlentheorie, Grundstudium Mathematik, Basel Birkhäuser, 2008
- P. Samuel, Algebraic Theory of Numbers, Kershaw Publishing Company LTD, 1972 (Original edition in French at Hermann)
- J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer 1973 (Original edition in French at Presses Universitaires de France)
Prerequisites / NoticeBasic knowledge of Algebra as taught in a course Algebra I + II.
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-4206-17LGroup Actions on TreesW4 credits2VN. Lazarovich
AbstractAs a main theme, we will explain how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. After introducing the general theory, we will cover various topics in this general theme.
ObjectiveIntroduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups.
ContentDepending on time we will cover some of the following topics.
- Free groups and their subgroups.
- The general theory of actions on trees, i.e, Bass-Serre theory.
- Trees as 1-dimensional buildings.
- Stallings' theorem.
- Grushko's and Dunwoody's accessibility results.
- Actions on R-trees and the Rips machine.
LiteratureJ.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9

C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Prerequisites / NoticeFamiliarity with the basics of fundamental group (and covering theory).
401-4148-17LModuli of Maps and Gromov-Witten invariantsW2 credits4AG. Bérczi
AbstractEnumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology.
ObjectiveThe aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position.
ContentTopics covered:
1) Brief survey on moduli spaces: fine and coarse moduli.
2) Stable n-pointed curves
3) Stable maps
4) Enumerative geometry via stable maps
5) Gromov-Witten invariants
6) Quantum cohomology and quantum product
7) Kontsevich's formula
LiteratureThe main reference for the course is:
J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves
www.math.utah.edu/%7eyplee/teaching/gw/Koch.pdf

Background material:
-Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
-Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag
-Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag
Prerequisites / NoticeSome minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves).
Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples.
401-3056-00LFinite Geometries I
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractFinite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.
ObjectiveFinite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.
ContentFinite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design
Literature- Max Jeger, Endliche Geometrien, ETH Skript 1988

- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983

- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press

- Dembowski: Finite Geometries.
401-3574-61LIntroduction to Knot Theory Information
Does not take place this semester.
W6 credits3G
AbstractIntroduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
ObjectiveThe aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
ContentDefinition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
LiteratureAn extensive bibliography will be handed out in the course.
Prerequisites / NoticePrerequisites are some elementary knowledge of algebra and topology.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4832-17LMathematical Themes in General Relativity IIW4 credits2VA. Carlotto
AbstractSecond part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems.
ObjectiveAcquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis.
ContentAnalysis of Jang's equation and application to the proof of the spacetime positive energy theorem; the conformal method for the Einstein constraint equations and links with the Yamabe problem; gluing methods for the Einstein constraint equations: canonical asymptotics, N-body solutions, gravitational shielding.
Lecture notesLecture notes written by the instructor will be provided to all enrolled students.
Prerequisites / NoticeThe content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable.
**This course is the sequel of the one offered during the first semester.**
401-3352-09LAn Introduction to Partial Differential EquationsW6 credits3VF. Da Lio
AbstractThis course aims at being an introduction to first and second order partial differential equations (in short PDEs).
We will present the so called method of characteristics to solve quasilinear PDEs and some basic properties of classical solutions to second order linear PDEs.
Objective
ContentA preliminary plan is the following
- Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem.
- Weak and strong maximum principle for elliptic operators.
- Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity.
- Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties, uniqueness by energy methods.
- The Method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws.
- A brief introduction to viscosity solutions.
Lecture notesThe teacher provides the students with personal notes.
LiteratureBibliography
- L.Evans Partial Differential Equations, AMS 2010 (2nd edition)
- D. Gilbarg, N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 1998.
- E. Di Benedetto Partial Differential Equations, Birkauser, 2010 (2nd edition).
- W. A. Strauss Partial Differential Equations. An Introduction, Wiley, 1992.
Prerequisites / NoticeDifferential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory.
401-3496-17LTopics in the Calculus of VariationsW4 credits2VA. Figalli
Abstract
Objective
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3502-17LReading Course Restricted registration - show details
THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.

Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information:
1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register;
2) in which semester;
3) for which degree programme;
4) your name and first name;
5) your student number;
6) the name and first name of the supervisor of the Reading Course.
W2 credits4AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3503-17LReading Course Restricted registration - show details
THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.

Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information:
1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register;
2) in which semester;
3) for which degree programme;
4) your name and first name;
5) your student number;
6) the name and first name of the supervisor of the Reading Course.
W3 credits6AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-17LReading Course Restricted registration - show details
THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.

Please send an email to Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> including the following pieces of information:
1) which Reading Course (60, 90, 120 hours of work, corresponding to 2, 3, 4 ECTS credits) you wish to register;
2) in which semester;
3) for which degree programme;
4) your name and first name;
5) your student number;
6) the name and first name of the supervisor of the Reading Course.
W4 credits9AProfessors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
Electives: Applied Mathematics and Further Application-Oriented Fields
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Selection: Numerical Analysis
NumberTitleTypeECTSHoursLecturers
401-4606-00LNumerical Analysis of Stochastic Partial Differential Equations
Does not take place this semester.
W8 credits4Gnot available
AbstractIn this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type and some of their numerical approximation methods are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences.
ObjectiveThe aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs), on some numerical approximation methods for such equations and on the functional analytic and probabilistic concepts used to formulate and study such equations.
ContentThe course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs) (e.g., nuclear operators, Hilbert-Schmidt operators, diagonal linear operators on Hilbert spaces, interpolation spaces associated to a diagonal linear operator, semigroups of bounded linear operators, Gronwall-type inequalities), (ii) on the probabilistic concepts used to study SPDEs (e.g., Hilbert space valued random variables, Hilbert space valued stochastic processes, infinite dimensional Wiener processes, stochastic integration with respect to infinite dimensional Wiener processes, infinite dimensional jump processes), (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs), and (iv) on numerical approximations of SPDEs (e.g., spatial and temporal discretizations, strong convergence, weak convergence). Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. They appear, for example, in models from neurobiology for the approximative description of the propagation of electrical impulses along nerve cells, in models from financial engineering for the approximative pricing of financial derivatives, in models from fluid mechanics for the approximative description of velocity fields in fully developed turbulent flows, in models from quantum field theory for describing the temporal dynamics associated to Euclidean quantum field theories, and in models from chemistry for the approximative description of the temporal evolution of the concentration of an undesired chemical contaminant in the groundwater system.
Lecture notesLecture notes will be available as a PDF file.
Literature1. Stochastic Equations in Infinite Dimensions
G. Da Prato and J. Zabczyk
Cambridge Univ. Press (1992)

2. Taylor Approximations for Stochastic Partial Differential Equations
A. Jentzen and P.E. Kloeden
Siam (2011)

3. Numerical Solution of Stochastic Differential Equations
P.E. Kloeden and E. Platen
Springer Verlag (1992)

4. A Concise Course on Stochastic Partial Differential Equations
C. Prévôt and M. Röckner
Springer Verlag (2007)

5. Galerkin Finite Element Methods for Parabolic Problems
V. Thomée
Springer Verlag (2006)
Prerequisites / NoticeMandatory prerequisites: Functional analysis, probability theory;
Recommended prerequisites: stochastic processes;
401-4658-00LComputational Methods for Quantitative Finance: PDE Methods Information Restricted registration - show details W6 credits3V + 1UC. Schwab
AbstractIntroduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming
and knowledge of numerical mathematics at ETH BSc level.
ObjectiveIntroduce the main methods for efficient numerical valuation of derivative contracts in a
Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility
models. Develop implementation of pricing methods in MATLAB.
Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Content1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic
volatility models.
2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees.
European contracts.
3. Finite Difference methods for Asian, American and Barrier type contracts.
4. Finite element methods for European and American style contracts.
5. Pricing under local and stochastic volatility in Black-Scholes Markets.
6. Finite Element Methods for option pricing under Levy processes. Treatment of
integrodifferential operators.
7. Stochastic volatility models for Levy processes.
8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and
stochastic volatility models in Black Scholes and Levy markets.
9. Introduction to sparse grid option pricing techniques.
Lecture notesThere will be english, typed lecture notes as well as MATLAB software for registered participants in the course.
LiteratureR. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004.

Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005.

D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008.

J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.

N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013.
Prerequisites / NoticeStart of the lecture: Wednesday, March 1, 2017 (second week of the semester).
401-4788-16LMathematics of (Super-Resolution) Biomedical ImagingW8 credits4GH. Ammari
AbstractThe aim of this course is to review different methods used to address challenging problems in biomedical imaging. The emphasis will be on scale separation techniques, hybrid imaging, spectroscopic techniques, and nanoparticle imaging. These approaches allow one to overcome the ill-posedness character of imaging reconstruction in biomedical applications and to achieve super-resolution imaging.
ObjectiveSuper-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other.

In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold:
(i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence
of small anomalies;
(ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies;
(iii) To design efficient inversion algorithms in multi-wave modalities;
(iv) to develop inversion techniques using multi-frequency measurements;
(v) to develop a mathematical and numerical framework for nanoparticle imaging.

In this course we shall consider both analytical and computational
matters in biomedical imaging. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, inverse problems, mathematical imaging, optimal control, stochastic modelling, and analysis of physical phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in biomedical imaging, requires a deep understanding of the different scales in the physical models, an accurate mathematical modelling of the imaging techniques, and fine analysis of complex physical phenomena.

An emphasis is put on mathematically analyzing acoustic-electric imaging, thermo-elastic imaging, Lorentz force based imaging, elastography, multifrequency electrical impedance tomography, and plasmonic resonant nanoparticles.
252-0504-00LNumerical Methods for Solving Large Scale Eigenvalue Problems Information
Does not take place this semester.
W4 credits3GP. Arbenz
AbstractIn this lecture algorithms are investigated for solving eigenvalue problems
with large sparse matrices. Some of these eigensolvers have been developed
only in the last few years. They will be analyzed in theory and practice (by means
of MATLAB exercises).
ObjectiveKnowing the modern algorithms for solving large scale eigenvalue problems, their numerical behavior, their strengths and weaknesses.
ContentThe lecture starts with providing examples for applications in which
eigenvalue problems play an important role. After an introduction
into the linear algebra of eigenvalue problems, an overview of
methods (such as the classical QR algorithm) for solving small to
medium-sized eigenvalue problems is given.

Afterwards, the most important algorithms for solving large scale,
typically sparse matrix eigenvalue problems are introduced and
analyzed. The lecture will cover a choice of the following topics:

* vector and subspace iteration
* trace minimization algorithm
* Arnoldi and Lanczos algorithms (including restarting variants)
* Davidson and Jacobi-Davidson Algorithm
* preconditioned inverse iteration and LOBPCG
* methods for nonlinear eigenvalue problems

In the exercises, these algorithm will be implemented (in simplified forms)
and analysed in MATLAB.
Lecture notesLecture notes,
Copies of slides
LiteratureZ. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000.

Y. Saad: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, 1994.

G. H. Golub and Ch. van Loan: Matrix Computations, 3rd ed. Johns Hopkins University Press, Baltimore 1996.
Prerequisites / NoticePrerequisite: linear agebra
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-3919-60LAn Introduction to the Modelling of ExtremesW4 credits2VP. Embrechts
AbstractThis course yields an introduction into the MATHEMATICAL THEORY of one-dimensional extremes, and this mainly from a more probabilistic point of view.
ObjectiveIn this course, students learn to distinguish between so-called normal models, i.e. models based on the normal or Gaussian distribution, and so-called heavy-tailed or power-tail models.
They learn to do probabilistic modelling of extremes in one-dimensional data. The probabilistic key theorems are the Fisher-Tippett Theorem and the Balkema-de Haan-Pickands Theorem. These lead to the statistical techniques for the analysis of extremes or rare events known as the Block Method, and Peaks Over Threshold method, respectively.
Content- Introduction to rare or extreme events
- Regular Variation
- The Convergence to Types Theorem
- The Fisher-Tippett Theorem
- The Method of Block Maxima
- The Maximal Domain of Attraction
- The Fre'chet, Gumbel and Weibull distributions
- The POT method
- The Point Process Method: a first introduction
- The Pickands-Balkema-de Haan Theorem and its applications
- Some extensions and outlook
Lecture notesThere will be no script available, students are required to take notes from the blackboard lectures. The course follows closely Extreme Value Theory as developed in:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
LiteratureThe main text on which the course is based is:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
Further relevant literature is:
S. I. Resnick (2007) Heavy-Tail Phenomena. Probabilistic and
Statistical Modeling. Springer.
S. I. Resnick (1987) Extreme Values, Regular Variation,
and Point Processes. Springer.
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