Autumn Semester 2020 takes place in a mixed form of online and classroom teaching.
Number Title Type ECTS Hours Lecturers Mathematics Master  ElectivesFor 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: Geometry 401-3574-61L Introduction to Knot Theory Does not take place this semester. W 6 credits 3G Abstract Introduction 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. Objective The 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. Content Definition 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.....) Literature An extensive bibliography will be handed out in the course. Prerequisites / Notice Prerequisites are some elementary knowledge of algebra and topology.   Selection: Analysis 401-4832-17L Mathematical Themes in General Relativity II W 4 credits 2V A. Carlotto Abstract Second 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. Objective Acquisition 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. Content Analysis 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 notes Lecture notes written by the instructor will be provided to all enrolled students. Prerequisites / Notice The 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.** 401-3352-09L An Introduction to Partial Differential Equations W 6 credits 3V F. Da Lio Abstract This 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 Content A 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 notes The teacher provides the students with personal notes. Literature Bibliography- 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 / Notice Differential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory. 401-3496-17L Topics in the Calculus of Variations W 4 credits 2V A. Figalli Abstract Objective   Selection: Further Realms 401-3502-17L Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.Please send an email to Studiensekretariat D-MATH 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. W 2 credits 4A Professors Abstract For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. Objective 401-3503-17L Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.Please send an email to Studiensekretariat D-MATH 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. W 3 credits 6A Professors Abstract For this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study. Objective 401-3504-17L Reading Course THE ENROLMENT IS DONE BY THE STUDY ADMINISTRATION.Please send an email to Studiensekretariat D-MATH 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. W 4 credits 9A Professors Abstract For 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¬   Selection: Numerical Analysis 401-4606-00L Numerical Analysis of Stochastic Partial Differential EquationsDoes not take place this semester. W 8 credits 4G not available Abstract In 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. Objective The 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. Content The 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 notes Lecture notes will be available as a PDF file. Literature 1. 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 / Notice Mandatory prerequisites: Functional analysis, probability theory;Recommended prerequisites: stochastic processes; 401-4658-00L Computational Methods for Quantitative Finance: PDE Methods  W 6 credits 3V + 1U C. Schwab Abstract Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programmingand knowledge of numerical mathematics at ETH BSc level. Objective Introduce the main methods for efficient numerical valuation of derivative contracts in aBlack Scholes as well as in incomplete markets due Levy processes or due to stochastic volatilitymodels. Develop implementation of pricing methods in MATLAB. Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation. Content 1. 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 notes There will be english, typed lecture notes as well as MATLAB software for registered participants in the course. Literature R. 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 / Notice Start of the lecture: Wednesday, March 1, 2017 (second week of the semester). 401-4788-16L Mathematics of (Super-Resolution) Biomedical Imaging W 8 credits 4G H. Ammari Abstract The 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. Objective Super-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. 252-0504-00L Numerical Methods for Solving Large Scale Eigenvalue Problems Does not take place this semester. W 4 credits 3G P. Arbenz Abstract In this lecture algorithms are investigated for solving eigenvalue problemswith large sparse matrices. Some of these eigensolvers have been developedonly in the last few years. They will be analyzed in theory and practice (by meansof MATLAB exercises). Objective Knowing the modern algorithms for solving large scale eigenvalue problems, their numerical behavior, their strengths and weaknesses. Content The lecture starts with providing examples for applications in whicheigenvalue problems play an important role. After an introductioninto the linear algebra of eigenvalue problems, an overview ofmethods (such as the classical QR algorithm) for solving small tomedium-sized eigenvalue problems is given.Afterwards, the most important algorithms for solving large scale,typically sparse matrix eigenvalue problems are introduced andanalyzed. 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 problemsIn the exercises, these algorithm will be implemented (in simplified forms)and analysed in MATLAB. Lecture notes Lecture notes, Copies of slides Literature Z. 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 / Notice Prerequisite: linear agebra   Selection: Probability Theory, Statistics 401-3919-60L An Introduction to the Modelling of Extremes W 4 credits 2V P. Embrechts Abstract This course yields an introduction into the MATHEMATICAL THEORY of one-dimensional extremes, and this mainly from a more probabilistic point of view. Objective In 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 notes There 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. Literature The 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. 401-4632-15L Causality W 4 credits 2G M. H. Maathuis Abstract In statistics, we are used to search for the best predictors of some random variable. In many situations, however, we are interested in predicting a system's behavior under manipulations. For such an analysis, we require knowledge about the underlying causal structure of the system. In this course, we study concepts and theory behind causal inference. Objective After this course, you should be able to- understand the language and concepts of causal inference- know the assumptions under which one can infer causal relations from observational and/or interventional data- describe and apply different methods for causal structure learning- given data and a causal structure, derive causal effects and predictions of interventional experiments Prerequisites / Notice Prerequisites: basic knowledge of probability theory and regression 401-6102-00L Multivariate Statistics W 4 credits 2G N. Meinshausen Abstract Multivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications. Objective After the course, you should be able to:- describe the various methods and the concepts and theory behind them- identify adequate methods for a given statistical problem- use the statistical software "R" to efficiently apply these methods- interpret the output of these methods Content Visualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis Lecture notes None Literature The course will be based on class notes and books that are available electronically via the ETH library. Prerequisites / Notice Target audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background. Prerequisite: A basic course in probability and statistics. Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units. 401-3822-17L Percolation and Ising Model W 4 credits 2V V. Tassion Abstract In this course we will provide a general introduction to the Ising model on the hypercubic lattice Z^d (main properties, standard mathematical tools).In order to answer important questions regarding the Ising model, we will exploit a deep connection of the model with two (dependent) percolation processes, the Fortuin-Kastelen percolation and the random-current model. Objective - Discover important models of statistical mechanics: the Ising model (and its random current representation) and FK percolation.- Learn some important techniques in statistical mechanics (e.g. coupling methods, monotonicity properties, the use of differential inequalities, to name few). Prerequisites / Notice - Probability Theory.- The course "Recent Development in Percolation Theory" (Autumn 2017) of Pierre Nolin is advised but not necessary (the overlap with Nolin's course will be minimal). 401-3597-64L Concentration of Measure W 4 credits 2V J. Aru, T. Lupu Abstract Objective Content The basic examples of the concentration of measure phenomena are the following:1) The visual distance of a N-dimensional unit sphere is only of order N^{-0.5}. In other words, more than 99% of the measure on the sphere lies at distance of at most O(N^{-0.5}) of a fixed hyperplane through the origin.2) The suprema of a centred Gaussian process G(t) even with a possibility infinite index set T is always concentrated around its expected value with a Gaussian tail that only depends on the highest variance among the Gaussians G(t).In this course we will try to understand these two slightly puzzling examples and related phenomena. We try to approach and understand the concentration of measure phenomena from different directions: through elementary martingale inequalities like Azuma-Hoeffding or McDiarmid inequality; through exact isoperimetry; through Poincaré and log-Sobolev inequalities. On the way we aim to discuss several applications and connections to different topics, including Dvoretzky's theorem, convergence of Markov chains to their stationary measure, entropy, threshold phenomena, empirical processes etc... 401-3616-17L An Introduction to Stochastic Partial Differential Equations W 8 credits 4G A. Jentzen Abstract In this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. Objective The aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs) and on the functional analytic and probabilistic concepts used to formulate and study such equations. Content The course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs), (ii) on the probabilistic concepts used to study SPDEs, and (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs). 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 notes The current version of the lecture notes is available as a PDF file here:https://polybox.ethz.ch/index.php/s/bI884u6tz9mO9Vz/download Literature 1. Stochastic Equations in Infinite DimensionsG. Da Prato and J. ZabczykCambridge Univ. Press (1992)2. Taylor Approximations for Stochastic Partial Differential EquationsA. Jentzen and P.E. KloedenSiam (2011)3. Numerical Solution of Stochastic Differential EquationsP.E. Kloeden and E. PlatenSpringer Verlag (1992)4. A Concise Course on Stochastic Partial Differential EquationsC. Prévôt and M. RöcknerSpringer Verlag (2007)5. Galerkin Finite Element Methods for Parabolic ProblemsV. ThoméeSpringer Verlag (2006) Prerequisites / Notice Mandatory prerequisites: Functional analysis, probability theory;Recommended prerequisites: stochastic processes;   Selection: Financial and Insurance Mathematics 401-3629-00L Quantitative Risk Management W 4 credits 2V P. Cheridito Abstract This course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, multivariate models, copulas and dependence structures, extreme value theory, risk measures, aggregation of risk, and risk allocation. Objective The goal is to learn the most important methods from probability theory and statistics used to model financial risks. Content 1. Risk in Perspective2. Basic Concepts 3. Multivariate Models4. Copulas and Dependence5. Aggregate Risk6. Extreme Value Theory7. Operational Risk and Insurance Analytics Lecture notes Course material is available on https://people.math.ethz.ch/~patrickc/qrm Literature Quantitative Risk Management: Concepts, Techniques and ToolsAJ McNeil, R Frey and P EmbrechtsPrinceton University Press, Princeton, 2015 (Revised Edition)http://press.princeton.edu/titles/10496.html Prerequisites / Notice The course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance. 401-4938-14L Stochastic Optimal Control W 4 credits 2V M. Soner Abstract Dynamic programming approach to stochastic optimal control problems will be developed. In addition to the general theory, detailed analysis of several important control problems will be given. Objective Goals are to achieve a deep understanding of1. Dynamic programming approach to optimal control;2. Several classes of important optimal control problems and their solutions.3. To be able to use this models in engineering and economic modeling. Content In this course, we develop the dynamic programming approach for the stochastic optimal control problems. The general approach will be described and several subclasses of problems will also be discussed in including:1. Standard exit time problems;2. Finite and infinite horizon problems;3. Optimal stoping problems;4. Singular problems;5. Impulse control problems. After the general theory is developed, it will be applied to several classical problems including:1. Linear quadratic regulator;2. Merton problem for optimal investment and consumption;3. Optimal dividend problem of (Jeanblanc and Shiryayev);4. Finite fuel problem;5. Utility maximization with transaction costs;6. A deterministic differential game related to geometric flows.Textbook will be Controlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).And lecture notes will be provided. Literature Controlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).And lecture notes will be provided. Prerequisites / Notice Basic knowledge of Brownian motion, stochastic differential equations and probability theory is needed. 401-3923-00L Selected Topics in Life Insurance Mathematics W 4 credits 2V M. Koller Abstract Stochastic Models for Life insurance 1) Markov chains2) Stochastic Processes for demography and interest rates3) Cash flow streams and reserves4) Mathematical Reserves and Thiele's differential equation5) Theorem of Hattendorff 6) Unit linked policies Objective