Search result: Catalogue data in Autumn Semester 2020

Mathematics Master Information
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3503-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W3 credits6ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-70LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 credits9ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-0000-00LCommunication in MathematicsW2 credits1VW. Merry
AbstractDon'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.
ObjectiveKnowing how to present written mathematics in a structured and clear manner.
ContentTopics 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.
Lecture notesFull lecture notes will be made available on my website:

Link
Prerequisites / NoticeThere are no formal mathematical prerequisites.
Electives: Applied Mathematics and Further Application-Oriented Fields
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Selection: Numerical Analysis
NumberTitleTypeECTSHoursLecturers
401-4657-00LNumerical Analysis of Stochastic Ordinary Differential Equations Information Restricted registration - show details
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
W6 credits3V + 1UD. Salimova
AbstractCourse 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.
ObjectiveThe 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.
ContentGeneration 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
Lecture notesThere will be English, typed lecture notes for registered participants in the course.
LiteratureP. 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.
Prerequisites / NoticePrerequisites:

Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.

a) mandatory courses:
Elementary Probability,
Probability Theory I.

b) recommended courses:
Stochastic Processes.

Start of lectures: Wednesday, September 16, 2020.
401-4785-00LMathematical and Computational Methods in PhotonicsW8 credits4GH. Ammari
AbstractThe 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
ObjectiveThe 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.
401-4427-70LRepresentation Theory in Signal AnalysisW4 credits2VF. Bartolucci
AbstractThe scope of the course is to give an introduction to the theory of unitary representations of locally compact groups with a particular regard to the applications of this theory in signal analysis.
Objective
ContentThe scope of the course is to give an introduction to the theory of
unitary representations of locally compact groups with a particular regard to
the applications of this theory in signal analysis. The course starts with an
overview of the measure theory on locally compact groups. Then, the fundamental
definitions and results in representation theory are presented (irreducible
unitary representations, Schur’s lemma, voice transforms, square-integrable
representations, reproducing formulae). We conclude the course showing that
some of the most important transforms in applied harmonic analysis such as the
Gabor transform, the wavelet transform and the shearlet transform are related
to square-integrable unitary representations.
Prerequisites / NoticePrerequisites: measure theory, topology, functional analysis, operator
theory, Fourier analysis
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-4607-70LA Medley of Advanced ProbabilityW4 credits2VW. Werner
AbstractWe will review various topics of probability theory, with the goal to provide a short self-contained introduction to each of them, and try to describe the type of ideas and techniques that are used.
Exact topics will include (small bits of) Lévy processes, continuous-state branching processes, large deviation theory, large random matrices.
ObjectiveThe goal is for each of the topics that will be covered to provide:
- A general introduction to the subject
- An example of one of the main statements, and some of the ideas that go into the proof
- A detailed proof of one statement
Prerequisites / NoticePrerequisites: Martingales, Markov chains, Brownian motion, stochastic calculus.
401-3628-14LBayesian Statistics
Does not take place this semester.
W4 credits2V
AbstractIntroduction 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.
ObjectiveStudents understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.
ContentTopics 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)
Lecture notesA script will be available in English.
LiteratureChristian 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.
Prerequisites / NoticeFamiliarity 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.
401-0625-01LApplied Analysis of Variance and Experimental DesignW5 credits2V + 1UL. Meier
AbstractPrinciples 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.
ObjectiveParticipants 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.
ContentPrinciples 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.
LiteratureG. Oehlert: A First Course in Design and Analysis of Experiments, W.H. Freeman and Company, New York, 2000.
Prerequisites / NoticeThe 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.
401-0649-00LApplied Statistical RegressionW5 credits2V + 1UM. Dettling
AbstractThis 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.
ObjectiveThe students acquire advanced practical skills in linear regression analysis and are also familiar with its extensions to generalized linear modeling.
ContentThe 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.
Lecture notesA script will be available.
LiteratureFaraway (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
Prerequisites / NoticeThe 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.
401-4521-70LGeometric Tomography - Uniqueness, Statistical Reconstruction and Algorithms Information Restricted registration - show details W4 credits2VJ. Hörrmann
AbstractSelf-contained course on the theoretical aspects of the reconstruction of geometric objects from tomographic projection and section data.
ObjectiveIntroduction to geometric tomography and understanding of various theoretical aspects of reconstruction problems.
ContentThe problem of reconstruction of an object from geometric information like X-ray data is a classical inverse problem on the overlap between applied mathematics, statistics, computer science and electrical engineering. We focus on various aspects of the problem in the case of prior shape information on the reconstruction object. We will answer questions on uniqueness of the reconstruction and also cover statistical and algorithmic aspects.
LiteratureR. Gardner: Geometric Tomography
F. Natterer: The Mathematics of Computerized Tomography
A. Rieder: Keine Probleme mit inversen Problemen
Prerequisites / NoticeA sound mathematical background in geometry, analysis and probability is required though a repetition of relevant material will be included. The ability to understand and write mathematical proofs is mandatory.
401-4607-59LPercolation Theory Information W4 credits2VV. Tassion
AbstractAn introduction to the percolation theory.
ObjectivePercolation theory has many applications and is one of the most famous model to
describe phase transition phenomena in physics. One reason for this success is
the variety of mathematical tools, which allows for a precise and rigorous
description of the models. The objective of this course is to gain familiarity
with the methods of the percolation theory and to learn some of its important
results. The students will develop their background and intuition in
probability, and the course is particularly recommended to students with
additional interests in physics or graph theory.
ContentDefinition of percolation. Standard tools: FKG, BK inequalities, Mixing
property, Russo's formula. Sharpness of the phase transition. Correlation
length and interpretations. Uniqueness of the infinite cluster. Critical
percolation in dimension 2. Supercritical percolation in dimension d>2,
Grimmett-Marstrand Theorem and consequences.
LiteratureB. Bollobas, O. Riordan: Percolation, CUP 2006
G. Grimmett: Percolation 2ed, Springer 1999
Prerequisites / NoticePreliminaries:
401-2604-00L Probability and Statistics (mandatory)
401-3601-00L Probability Theory (recommended)
401-4619-67LAdvanced Topics in Computational Statistics
Does not take place this semester.
W4 credits2Vnot available
AbstractThis lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.
ObjectiveStudents learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.
ContentThe main focus will be on graphical models in various forms:
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
Prerequisites / NoticeWe assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.
401-3627-00LHigh-Dimensional Statistics
Does not take place this semester.
W4 credits2VP. L. Bühlmann
Abstract"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.
ObjectiveKnowledge of methods and basic theory for high-dimensional statistical inference
ContentLasso 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
LiteraturePeter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag.
ISBN 978-3-642-20191-2.
Prerequisites / NoticeKnowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).
401-4623-00LTime Series AnalysisW6 credits3GF. Balabdaoui
AbstractThe 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.
ObjectiveThe 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.
ContentThis 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
LiteratureThe main reference for this course is the book "Introduction to Time Series and Forecasting", by P. J. Brockwell and R. A. Davis
Prerequisites / NoticeBasic knowledge in probability and statistics
401-3612-00LStochastic SimulationW5 credits3GF. Sigrist
AbstractThis course introduces statistical Monte Carlo methods. This includes applications of stochastic simulation in various fields (statistics, statistical mechanics, operations research, financial mathematics), generating uniform and arbitrary random variables (incl. rejection and importance sampling), the accuracy of methods, variance reduction, quasi-Monte Carlo, and Markov chain Monte Carlo.
ObjectiveStudents know the stochastic simulation methods introduced in this course. Students understand and can explain these methods, show how they are related to each other, know their weaknesses and strengths, apply them in practice, and proof key results.
ContentExamples of simulations in different fields (statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (including rejection sampling and importance sampling), simulation of multivariate normal variables and stochastic differential equations. The accuracy of Monte Carlo methods. Methods for variance reduction and quasi-Monte Carlo. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). Algorithms introduced in the course are illustrated with the statistical software R.
Lecture notesA script will be available in English.
LiteratureP. 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).
Prerequisites / NoticeIt is assumed that students have had an introduction to probability theory and statistics (random variables, joint and conditional distributions, law of large numbers, central limit theorem, basics of measure theory).

The course resources (including script, slides, exercises) will be provided via the Moodle online learning platform.
Selection: Financial and Insurance Mathematics
In the Master's programmes in Mathematics resp. Applied Mathematics 401-3913-01L Mathematical Foundations for Finance is eligible as an elective course, but only if 401-3888-00L Introduction to Mathematical Finance isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (Link) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3925-00LNon-Life Insurance: Mathematics and Statistics Information W8 credits4V + 1UM. V. Wüthrich
AbstractThe 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.
ObjectiveThe 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.
ContentThe 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
Lecture notesM. V. Wüthrich, Non-Life Insurance: Mathematics & Statistics
Link
Prerequisites / NoticeThe 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 Link.

Prerequisites: knowledge of probability theory, statistics and applied stochastic processes.
401-3922-00LLife Insurance MathematicsW4 credits2VM. Koller
AbstractThe 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.
Objective
401-3928-00LReinsurance AnalyticsW4 credits2VP. Antal, P. Arbenz
AbstractThis 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
ObjectiveThis 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
ContentThis 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
Lecture notesSlides and lecture notes will be made available.

An excerpt of last year's lecture notes is available here: Link
Prerequisites / NoticeBasic knowledge in statistics, probability theory, and actuarial techniques
401-3927-00LMathematical Modelling in Life InsuranceW4 credits2VT. J. Peter
AbstractIn 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.
ObjectiveThe 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.
ContentFollowing 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
Lecture notesLectures notes and slides will be provided
Prerequisites / NoticeThe 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.
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