# Suchergebnis: Katalogdaten im Herbstsemester 2021

Mathematik Master | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Wahlfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Wahlfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Numerische Mathematik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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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 | A. Stein | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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/Python programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday September 22, 2021. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-5003-71L | At the Interface Between Semiclassical Analysis and Numerical Analysis of Wave-Scattering Problems | W | 4 KP | 2V | E. Spence | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Postgraduate degree lecture | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Semiclassical analysis (SCA) is a branch of microlocal analysis concerned with rigorously analysing PDEs with large (or small) parameters. On the other hand, numerical analysis (NA) seeks to design numerical methods that are accurate, efficient, and robust, with theorems guaranteeing these properties. In the context of high-frequency wave scattering, both SCA and NA share the same goal – that of understanding the behaviour of the scattered wave – but these two fields have operated largely in isolation, mainly because the tools and techniques of the two fields are somewhat disjoint. This by-and-large self-contained course focuses on the Helmholtz equation, which is arguably the simplest possible model of wave propagation. Our first goal will be to show how even relatively-simple tools from semiclassical analysis can be used to prove fundamental results about the numerical analysis of finite-element method applied to the high-frequency Helmholtz equation. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The course will aim at being accessible both to students coming from a numerical-analysis/applied-maths background and to students coming from an analysis background. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4607-67L | Schramm-Loewner Evolutions | W | 4 KP | 2V | W. Werner | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This advanced course will be an introduction to SLE (Schramm-Loewner Evolutions), which are a class of conformally invariant random curves in the plane. We will discuss their construction and some of their main properties. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Knowledge of Brownian motion and stochastic calculus and basic knowledge of complex analysis (Riemann's mapping theorem). Familiarity of lattice models such as percolation or the Ising model can be useful but not necessary. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3822-17L | Ising Model | W | 4 KP | 2V | V. Tassion | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | - Probability Theory. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-0625-01L | Applied Analysis of Variance and Experimental Design | W | 5 KP | 2V + 1U | L. Meier | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Participants will be able to plan and analyze efficient experiments in the fields of natural sciences. They will gain practical experience by using the software R. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Principles of experimental design, one-way analysis of variance, contrasts and multiple comparisons, multi-factor designs and analysis of variance, complete block designs, Latin square designs, random effects and mixed effects models, split-plot designs, incomplete block designs, two-series factorials and fractional designs, power. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | G. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software R, for which an introduction will be held. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-0649-00L | Applied Statistical Regression | W | 5 KP | 2V + 1U | M. Dettling | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This course offers a practically oriented introduction into regression modeling methods. The basic concepts and some mathematical background are included, with the emphasis lying in learning "good practice" that can be applied in every student's own projects and daily work life. A special focus will be laid in the use of the statistical software package R for regression analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | The course starts with the basics of linear modeling, and then proceeds to parameter estimation, tests, confidence intervals, residual analysis, model choice, and prediction. More rarely touched but practically relevant topics that will be covered include variable transformations, multicollinearity problems and model interpretation, as well as general modeling strategies. The last third of the course is dedicated to an introduction to generalized linear models: this includes the generalized additive model, logistic regression for binary response variables, binomial regression for grouped data and poisson regression for count data. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | A script will be available. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Faraway (2005): Linear Models with R Faraway (2006): Extending the Linear Model with R Draper & Smith (1998): Applied Regression Analysis Fox (2008): Applied Regression Analysis and GLMs Montgomery et al. (2006): Introduction to Linear Regression Analysis | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The exercises, but also the classes will be based on procedures from the freely available, open-source statistical software package R, for which an introduction will be held. In the Mathematics Bachelor and Master programmes, the two course units 401-0649-00L "Applied Statistical Regression" and 401-3622-00L "Statistical Modelling" are mutually exclusive. Registration for the examination of one of these two course units is only allowed if you have not registered for the examination of the other course unit. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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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. ISBN 978-3-642-20191-2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Knowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-4623-00L | Time Series AnalysisFindet dieses Semester nicht statt. | W | 6 KP | 3G | F. Balabdaoui | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The course offers an introduction into analyzing times series, that is observations which occur in time. The material will cover Stationary Models, ARMA processes, Spectral Analysis, Forecasting, Nonstationary Models, ARIMA Models and an introduction to GARCH models. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The goal of the course is to have a a good overview of the different types of time series and the approaches used in their statistical analysis. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | This course treats modeling and analysis of time series, that is random variables which change in time. As opposed to the i.i.d. framework, the main feature exibited by time series is the dependence between successive observations. The key topics which will be covered as: Stationarity Autocorrelation Trend estimation Elimination of seasonality Spectral analysis, spectral densities Forecasting ARMA, ARIMA, Introduction into GARCH models | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | The main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Basic knowledge in probability and statistics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3612-00L | Stochastic SimulationFindet dieses Semester nicht statt. | W | 5 KP | 3G | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This course provides an introduction to statistical Monte Carlo methods. This includes applications of simulations in various fields (Bayesian statistics, statistical mechanics, operations research, financial mathematics), algorithms for the generation of random variables (accept-reject, importance sampling), estimating the precision, variance reduction, introduction to Markov chain Monte Carlo. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Stochastic simulation (also called Monte Carlo method) is the experimental analysis of a stochastic model by implementing it on a computer. Probabilities and expected values can be approximated by averaging simulated values, and the central limit theorem gives an estimate of the error of this approximation. The course shows examples of the many applications of stochastic simulation and explains different algorithms used for simulation. These algorithms are illustrated with the statistical software R. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Examples of simulations in different fields (computer science, statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (quantile transform, accept-reject, importance sampling), simulation of Gaussian processes and diffusions. The precision of simulations, methods for variance reduction. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | A script will be available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer 2004. B. D. Ripley. Stochastic Simulation. Wiley, 1987. Ch. Robert, G. Casella. Monte Carlo Statistical Methods. Springer 2004 (2nd edition). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Familiarity with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Finanz- und Versicherungsmathematik In den Master-Studiengängen Mathematik bzw. Angewandte Mathematik ist auch 401-3913-01L Mathematical Foundations for Finance als Wahlfach anrechenbar, aber nur unter der Bedingung, dass 401-3888-00L Introduction to Mathematical Finance nicht angerechnet wird (weder im Bachelor- noch im Master-Studiengang). Wenden Sie sich für die Kategoriezuordnung nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (www.math.ethz.ch/studiensekretariat). | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3925-00L | Non-Life Insurance: Mathematics and Statistics | W | 8 KP | 4V + 1U | M. V. Wüthrich | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The lecture aims at providing a basis in non-life insurance mathematics which forms a core subject of actuarial science. It discusses collective risk modeling, individual claim size modeling, approximations for compound distributions, ruin theory, premium calculation principles, tariffication with generalized linear models and neural networks, credibility theory, claims reserving and solvency. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The student is familiar with the basics in non-life insurance mathematics and statistics. This includes the basic mathematical models for insurance liability modeling, pricing concepts, stochastic claims reserving models and ruin and solvency considerations. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | The following topics are treated: Collective Risk Modeling Individual Claim Size Modeling Approximations for Compound Distributions Ruin Theory in Discrete Time Premium Calculation Principles Tariffication Generalized Linear Models and Neural Networks Bayesian Models and Credibility Theory Claims Reserving Solvency Considerations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | M.V. Wüthrich, Non-Life Insurance: Mathematics & Statistics http://ssrn.com/abstract=2319328 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | M.V. Wüthrich, M. Merz. Statistical Foundations of Actuarial Learning and its Applications http://ssrn.com/abstract=3822407 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch. Prerequisites: knowledge of probability theory, statistics and applied stochastic processes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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401-3922-00L | Life Insurance Mathematics | W | 4 KP | 2V | M. Koller | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | The classical life insurance model is presented together with the important insurance types (insurance on one and two lives, term and endowment insurance and disability). Besides that the most important terms such as mathematical reserves are introduced and calculated. The profit and loss account and the balance sheet of a life insurance company is explained and illustrated. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3928-00L | Reinsurance Analytics | W | 4 KP | 2V | P. Antal, P. Arbenz | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and models for extreme events such as natural or man-made catastrophes. The lecture covers reinsurance contracts, Experience and Exposure pricing, natural catastrophe modelling, solvency regulation, and insurance linked securities | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Insurance linked securities: Alternative risk transfer techniques such as catastrophe bonds | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | This course provides an introduction to reinsurance from an actuarial perspective. The objective is to understand the fundamentals of risk transfer through reinsurance and the mathematical approaches associated with low frequency high severity events such as natural or man-made catastrophes. Topics covered include: - Reinsurance Contracts and Markets: Different forms of reinsurance, their mathematical representation, history of reinsurance, and lines of business. - Experience Pricing: Modelling of low frequency high severity losses based on historical data, and analytical tools to describe and understand these models - Exposure Pricing: Loss modelling based on exposure or risk profile information, for both property and casualty risks - Natural Catastrophe Modelling: History, relevance, structure, and analytical tools used to model natural catastrophes in an insurance context - Solvency Regulation: Regulatory capital requirements in relation to risks, effects of reinsurance thereon, and differences between the Swiss Solvency Test and Solvency 2 - Insurance linked securities: Alternative risk transfer techniques such as catastrophe bonds | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Slides and lecture notes will be made available. An excerpt of last year's lecture notes is available here: https://sites.google.com/site/philipparbenz/reinsuranceanalytics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Basic knowledge in statistics, probability theory, and actuarial techniques | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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401-3927-00L | Mathematical Modelling in Life Insurance | W | 4 KP | 2V | T. J. Peter | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | In life insurance, it is essential to have adequate mortality tables, be it for reserving or pricing purposes. The course provides the tools necessary to create mortality tables from scratch. Additionally, we study various guarantees embedded in life insurance products and learn to price them with the help of stochastic models. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The course's objective is to provide the students with the understanding and the tools to create mortality tables on their own. Additionally, students should learn to price embedded options in life insurance. Aside of the mere application of specific models, they should develop an intuition for the various drivers of the value of these options. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Following main topics are covered: 1. Guarantees and options embedded in life insurance products. - Stochastic valuation of participating contracts - Stochastic valuation of Unit Linked contracts 2. Mortality Tables: - Determining raw mortality rates - Smoothing techniques: Whittaker-Henderson, smoothing splines,... - Trends in mortality rates - Stochastic mortality model due to Lee and Carter - Neural Network extension of the Lee-Carter model - Integration of safety margins | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Lectures notes and slides will be provided | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | The exams ONLY take place during the official ETH examination period. The course counts towards the diploma of "Aktuar SAV". Good knowledge in probability theory and stochastic processes is assumed. Some knowledge in financial mathematics is useful. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Mathematische Physik, Theoretische Physik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0843-00L | Quantum Field Theory IFachstudierende UZH müssen das Modul PHY551 direkt an der UZH buchen. | W | 10 KP | 4V + 2U | G. M. Graf | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This course discusses the quantisation of fields in order to introduce a coherent formalism for the combination of quantum mechanics and special relativity. Topics include: - Relativistic quantum mechanics - Quantisation of bosonic and fermionic fields - Interactions in perturbation theory - Scattering processes and decays - Elementary processes in QED - Radiative corrections | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The goal of this course is to provide a solid introduction to the formalism, the techniques, and important physical applications of quantum field theory. Furthermore it prepares students for the advanced course in quantum field theory (Quantum Field Theory II), and for work on research projects in theoretical physics, particle physics, and condensed-matter physics. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Will be provided as the course progresses | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kompetenzen |
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402-0861-00L | Statistical Physics | W | 10 KP | 4V + 2U | M. Sigrist | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | This lecture covers the concepts of classical and quantum statistical physics. Several techniques such as second quantization formalism for fermions, bosons, photons and phonons as well as mean field theory and self-consistent field approximation. These are used to discuss phase transitions, critical phenomena and superfluidity. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | This lecture gives an introduction in 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 | Kinetic approach to statistical physics: H-theorem, detailed balance and equilibirium conditions. Classical statistical physics: microcanonical ensembles, canonical ensembles and grandcanonical ensembles, applications to simple systems. Quantum statistical physics: density matrix, ensembles, Fermi gas, Bose gas (Bose-Einstein condensation), photons and phonons. Identical quantum particles: many body wave functions, second quantization formalism, equation of motion, correlation functions, selected applications, e.g. Bose-Einstein condensate and coherent state, phonons in elastic media and melting. One-dimensional interacting systems. Phase transitions: mean field approach to Ising model, Gaussian transformation, Ginzburg-Landau theory (Ginzburg criterion), self-consistent field approach, critical phenomena, Peierls' arguments on long-range order. Superfluidity: Quantum liquid Helium: Bogolyubov theory and collective excitations, Gross-Pitaevskii equations, Berezinskii-Kosterlitz-Thouless transition. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Lecture notes available in English. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | No specific book is used for the course. Relevant literature will be given in the course. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0830-00L | General Relativity Fachstudierende UZH müssen das Modul PHY511 direkt an der UZH buchen. | W | 10 KP | 4V + 2U | C. Anastasiou | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Introduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations of the theory as well as the underlying physical principles and concepts. It covers selected applications, such as the Schwarzschild solution and gravitational waves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Basic understanding of general relativity, its mathematical foundations (in particular the relevant aspects of differential geometry), and some of the phenomena it predicts (with a focus on black holes). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Introduction to the theory of general relativity. The course puts a strong focus on the mathematical foundations, such as differentiable manifolds, the Riemannian and Lorentzian metric, connections, and curvature. It discusses the underlying physical principles, e.g., the equivalence principle, and concepts, such as curved spacetime and the energy-momentum tensor. The course covers some basic applications and special cases, including the Newtonian limit, post-Newtonian expansions, the Schwarzschild solution, light deflection, and gravitational waves. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | Suggested textbooks: C. Misner, K, Thorne and J. Wheeler: Gravitation S. Carroll - Spacetime and Geometry: An Introduction to General Relativity R. Wald - General Relativity S. Weinberg - Gravitation and Cosmology | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

402-0897-00L | Introduction to String Theory | W | 6 KP | 2V + 1U | J. Brödel | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | String theory is an attempt to quantise gravity and unite it with the other fundamental forces of nature. It is related to numerous interesting topics and questions in quantum field theory. In this course, an introduction to the basics of string theory is provided. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | Within this course, a basic understanding and overview of the concepts and notions employed in string theory shall be given. More advanced topics will be touched upon towards the end of the course briefly in order to foster further research. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | - mechanics of point particles and extended objects - string modes and their quantisation; higher dimensions, supersymmetry - D-branes, T-duality - supergravity as a low-energy effective theory, strings on curved backgrounds - two-dimensional field theories (classical/quantum, conformal/non-conformal) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Literatur | D. Lust, S. Theisen, Lectures on String Theory, Lecture Notes in Physics, Springer (1989). M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory I, CUP (1987). B. Zwiebach, A First Course in String Theory, CUP (2004). J. Polchinski, String Theory I & II, CUP (1998). | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Recommended: Quantum Field Theory I (in parallel) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Auswahl: Mathematische Optimierung, Diskrete Mathematik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

401-3055-64L | Algebraic Methods in Combinatorics | W | 6 KP | 2V + 1U | B. Sudakov | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Kurzbeschreibung | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Lernziel | The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inhalt | Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, well-developed tools. One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. The most fruitful such tool is the dimension argument. Roughly speaking, the method can be described as follows. In order to bound the cardinality of of a discrete structure A one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors. It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include (but are not limited to): Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at https://moodle-app2.let.ethz.ch/course/view.php?id=15757 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Skript | Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Voraussetzungen / Besonderes | Students are expected to have a mathematical background and should be able to write rigorous proofs. |

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