Suchergebnis: Katalogdaten im Herbstsemester 2022
Mathematik Bachelor  | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
 Wahlfächer | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Algebra, Zahlentheorie, Topologie, diskrete Mathematik, Logik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-3059-00L | Kombinatorik II Findet dieses Semester nicht statt. | W | 4 KP | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Der Kurs Kombinatorik I und II ist eine Einführung in die abzählende Kombinatorik. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Die Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Inhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3034-00L | Axiomatische Mengenlehre | W | 8 KP | 3V + 1U | L. Halbeisen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Es werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Zudem werden Ultrafilter untersucht und es wird das Martinaxiom eingeführt. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Es werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Insbesondere wird die Kontinuumshypothese behandelt und einige Konsequenzen besprochen. Zudem werden Ultrafilter untersucht und die Existenz gewisser Ultrafilter diskutiert. Im letzten Teil der Vorlesung wird das Martin-Axiom eingeführt, mit dessen Hilfe sich interessante Konsistenzresultate in Topologie und Masstheorie, sowie Resultate über Ultrafilter, beweisen lassen. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | Ich werde mich weitgehend an mein Buch "Combinatorial Set Theory" (2nd ed., 2017) halten. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | "Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2017) https://link.springer.com/book/10.1007/978-3-319-60231-8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Geometrie | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3057-00L | Endliche Geometrien II | W | 4 KP | 2G | N. Hungerbühler | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Endliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Endliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Endliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | - 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-4571-22L | Topology of Manifolds | W | 6 KP | 2V + 1U | D. Cekic | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | This will be an introduction to geometric topology, a field of mathematics concerned with topological properties of manifolds. We will study both topological and smooth manifolds, and prove some fundamental results about them (like the Schoenflies theorem, the generalised Poincaré conjecture, the existence of exotic smooth structures), several of which have been awarded with Fields medals. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | At the end of the course students will be able to differentiate between three types of manifolds, give examples showing various phenomena, and prove some classical results. They will understand what kinds of arguments are used in each of the cases, and where the difficulties arise. Moreover, they will become familiar with many open problems that are guiding current research, especially in the peculiar dimension four. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | There are several notions of a manifold -- namely, topological, piecewise-linear, and smooth -- and only in 1956 did it become clear that these objects are in fact distinct, thanks to the construction by J. Milnor of multiple smooth structures on a single topological manifold. In this course we will start with basic definitions and properties of the three types of manifolds, building our way up to cover some fundamental results. We will first study handle decompositions, transversality and the Whitney trick, the s-cobordism theorem, the Poincaré conjecture, and the Schoenflies theorem. Possible further topics include torus tricks, smoothing theory, exotic spheres, the Rohlin theorem, exotic 4-manifolds. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | • See the lecture notes and a reference list at https://maths.dur.ac.uk/users/mark.a.powell/topological-manifolds.html • Hirsch, M. Differential topology. Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976. • Kosinski, A. Differential manifolds. Pure and Applied Mathematics, 138. Academic Press, Inc., Boston, MA, 1993. • Scorpan, A. The wild world of 4-manifolds. American Mathematical Society, Providence, RI, 2005. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | We will assume familiarity with point-set topology, the fundamental group (as covered in the course Topology), homology (as covered in Algebraic Topology I), and some basics of differential topology and vector bundles (as covered in Differential Geometry I). Some familiarity with cohomology and Poincaré duality would be useful. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Analysis (Noch) kein Angebot in diesem Semester | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Numerische Mathematik (Noch) kein Angebot in diesem Semester | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Wahrscheinlichkeitstheorie, Statistik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3627-00L | High-Dimensional Statistics Findet dieses Semester nicht statt. | 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 Analysis | W | 4 KP | 2G | N. Meinshausen | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-0625-01L | 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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| 401-3628-14L | Bayesian Statistics Findet dieses Semester nicht statt. | W | 4 KP | 2V | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Finanz- und Versicherungsmathematik Im Bachelor-Studiengang 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3922-00L | 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3925-00L | 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 (no semester end exams), and only in person exams (i.e. no remote exams). 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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| 401-3927-00L | Mathematical Modelling in Life Insurance Findet dieses Semester nicht statt. | W | 4 KP | 2V | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3928-00L | Reinsurance Analytics Findet dieses Semester nicht statt. | W | 4 KP | 2V | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| 401-3931-00L | Responsible Machine Learning with Insurance Applications | W | 4 KP | 2G | M. Mayer, C. Lorentzen-Geiser | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | This lecture covers important aspects of applying supervised machine learning models in a responsible way, based on sound statistical theory. The focus is on model interpretability, calibration (bias) assessment, and proper model comparison. The methods are illustrated with actuarial datasets. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | The student is familiar with the main tools of model interpretability, calibration assessment, and model comparison and knows how to apply supervised machine learning in a responsible way. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | • Overview of supervised machine learning (statistical learning theory, GLMs, tree based methods, and neural nets; cross-validation) • Model interpretability methods (partial dependence plots, measures of variable importance, and SHAP) • Bias/calibration assessment with identification functions • Model comparison with consistent scoring functions • Working with dependent observations and further topics | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | 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: Good knowledge in statistics/probability theory, statistical modelling and the R programming language are assumed. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Mathematische Physik, Theoretische Physik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 402-0830-00L | General Relativity Fachstudierende UZH müssen das Modul PHY511 direkt an der UZH buchen. | W | 10 KP | 4V + 2U | L. Senatore | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Mathematische Optimierung, Diskrete Mathematik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3055-64L | Algebraic Methods in Combinatorics Findet dieses Semester nicht statt. | 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3054-14L | Probabilistic Methods in Combinatorics | W | 6 KP | 2V + 1U | B. Sudakov | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | This course provides a gentle introduction to the Probabilistic Method, with an emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | The topics covered in the class will include (but are not limited to): linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities and pseudo-randomness. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | - The Probabilistic Method, by N. Alon and J. H. Spencer, 3rd Edition, Wiley, 2008. - Random Graphs, by B. Bollobás, 2nd Edition, Cambridge University Press, 2001. - Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000. - Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Theoretische Informatik | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 252-0417-00L | Randomized Algorithms and Probabilistic Methods | W | 10 KP | 3V + 2U + 4A | A. Steger | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Las Vegas & Monte Carlo algorithms; inequalities of Markov, Chebyshev, Chernoff; negative correlation; Markov chains: convergence, rapidly mixing; generating functions; Examples include: min cut, median, balls and bins, routing in hypercubes, 3SAT, card shuffling, random walks | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | After this course students will know fundamental techniques from probabilistic combinatorics for designing randomized algorithms and will be able to apply them to solve typical problems in these areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Randomized Algorithms are algorithms that "flip coins" to take certain decisions. This concept extends the classical model of deterministic algorithms and has become very popular and useful within the last twenty years. In many cases, randomized algorithms are faster, simpler or just more elegant than deterministic ones. In the course, we will discuss basic principles and techniques and derive from them a number of randomized methods for problems in different areas. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | Yes. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | - Randomized Algorithms, Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995) - Probability and Computing, Michael Mitzenmacher and Eli Upfal, Cambridge University Press (2005) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 252-1425-00L | Geometry: Combinatorics and Algorithms | W | 8 KP | 3V + 2U + 2A | B. Gärtner, E. Welzl, M. Hoffmann | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Skript | yes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Literatur | Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Auswahl: Weitere Gebiete sowie einige Kurse der UZH | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Nummer | Titel | Typ | ECTS | Umfang | Dozierende | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-0000-00L | Communication in Mathematics Findet dieses Semester nicht statt. | W | 2 KP | 1V | keine Angaben | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Kurzbeschreibung | Don't hide your Next Great Theorem behind bad writing. This course teaches fundamental communication skills in mathematics: how to write clearly and how to structure mathematical content for different audiences, from theses, to preprints, to personal statements in applications. In addition, the course will help you establish a working knowledge of LaTeX. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Lernziel | Knowing how to present written mathematics in a structured and clear manner. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Inhalt | Topics covered include: - Language conventions and common errors. - How to write a thesis (more generally, a mathematics paper). - How to use LaTeX. - How to write a personal statement for Masters and PhD applications. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Voraussetzungen / Besonderes | There are no formal mathematical prerequisites. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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