Search result: Catalogue data in Spring Semester 2020

Mathematics Bachelor Information
Electives
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-3201-00LAlgebraic Groups Information W8 credits4GP. D. Nelson
AbstractIntroduction to the theory of linear algebraic groups. Lie algebras, the Jordan Chevalley decomposition, semisimple and reductive groups, root systems, Borel subgroups, classification of reductive groups and their representations.
Objective
LiteratureA. L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups
Prerequisites / NoticeAbstract algebra: groups, rings, fields, tensor product, etc.

Some familiarity with the basics of Lie groups and their Lie algebras would be helpful, but is not absolutely necessary.

We will develop what we need from algebraic geometry, without assuming prior knowledge.
401-3109-65LProbabilistic Number Theory Information
Does not take place this semester.
W8 credits4GE. Kowalski
AbstractThe course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.
ObjectiveThe goal of the course is to present some results of probabilistic number theory in a unified manner.
ContentThe main concepts will be presented in parallel with the proof of a few main theorems:
(1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions;
(2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line;
(3) the Chebychev bias for primes in arithmetic progressions;
(4) functional limit theorems for the paths of partial sums of families of exponential sums.
Lecture notesThe lecture notes for the class are available at

Link
Prerequisites / NoticePrerequisites: Complex analysis, measure and integral; some probability theory is useful but the main concepts needed will be recalled.
Some knowledge of number theory is useful but the main results will be summarized.
401-3202-09LThe Representation Theory of the Finite Symmetric Groups Information
NOTICE: No physical class for the next few weeks until further notice. Instead a video recording will be offered.
W4 credits2VL. Wu
AbstractThis course is an Introduction to the Representation Theory of the Groups.
ObjectiveOur goal is to give an introduction of the Representation Theory using the examples of the Finite Symmetry Groups.
Literature* Jean-Pierre Serre: Linear Representations of Finite Groups, Graduate Texts in Mathematics, Springer.

* William Fulton and Joe Harris: Representation Theory A First Course, Graduate Texts in Mathematics, Springer.

* G. D. James: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Springer.

* Bruce E. Sagan: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, Springer.
Prerequisites / NoticeSome basic knowledge of the Group Theory and Linear Algebra.
401-8112-20LGeometry of Numbers (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT548

Mind the enrolment deadlines at UZH:
Link
W9 credits4V + 1UUniversity lecturers
AbstractThe Geometry of Numbers studies distribution of lattice points in the n dimensional space, for instance, existence of lattice points in various domains and existence of integral solutions of polynomial inequalities.
This subject is also closely related to the Theory of Diophantine Approximation, which seeks good rational approximations for real vectors.
ObjectiveLearn basic techniques in the Geometry of Numbers
Literature1. Cassels, An introduction to Diophantine Approximation
2. Cassels, An introduction to the Geometry of Numbers
3. Schmidt, Diophantine approximation
4. Siegel, Lectures on the Geometry of Numbers
401-3058-00LCombinatorics I
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers.
Prerequisites / NoticeRecognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008).
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-3056-00LFinite Geometries IW4 credits2GN. Hungerbühler
AbstractFinite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.
ObjectiveFinite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.
ContentFinite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design
Literature- Max Jeger, Endliche Geometrien, ETH Skript 1988

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

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

- Dembowski: Finite Geometries.
401-3556-20LTopics in Symplectic TopologyW6 credits3VP. Biran
AbstractThis will be an introductory course in symplectic geometry and topology.
We will cover the simplest instances of symplectic rigidity phenomena, and techniques to detect and study them. The last part of the course will be devoted to more advanced techniques such as Floer theory.
ObjectiveGet acquainted with the basics of symplectic topology and phenomena
of symplectic rigidity.
Literature1) Book: "Introduction to Symplectic Topology", 3'rd edition, by McDuff and Salamon.
Oxford Graduate Texts in Mathematics

2) Some published articles that will be announced during the semester.
401-3574-61LIntroduction to Knot Theory Information
Does not take place this semester.
W6 credits3G
AbstractIntroduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
ObjectiveThe aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
ContentDefinition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
LiteratureAn extensive bibliography will be handed out in the course.
Prerequisites / NoticePrerequisites are some elementary knowledge of algebra and topology.
Selection: Analysis
No offering in this semester yet
Selection: Numerical Analysis
No offering in this semester yet
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-6102-00LMultivariate Statistics
Does not take place this semester.
W4 credits2Gnot available
AbstractMultivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications.
ObjectiveAfter the course, you should be able to:
- describe the various methods and the concepts and theory behind them
- identify adequate methods for a given statistical problem
- use the statistical software "R" to efficiently apply these methods
- interpret the output of these methods
ContentVisualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Lecture notesNone
LiteratureThe course will be based on class notes and books that are available electronically via the ETH library.
Prerequisites / NoticeTarget audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background.

Prerequisite: A basic course in probability and statistics.

Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units.
401-4626-00LAdvanced Statistical Modelling: Mixed ModelsW4 credits2VM. Mächler
AbstractMixed Models = (*| generalized| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms.

In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.
Objective- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models.

- Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis:
"fit -> interpret & diagnose -> modify the fit -> interpret & ...."

- Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.
ContentThe lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online.

Inference (significance of factors, confidence intervals) will focus on the more realistic *un*balanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.
Lecture notesWe will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples.

These lecture notes and all R scripts are made available from
Link
Literature(see web page and lecture notes)
Prerequisites / Notice- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..).

Typically this means at least two classes of (math based) statistics, say
1. Intro to probability and statistics
2. (Applied) regression including Matrix-Vector notation Y = X b + E

- Basic (1 semester) "Matrix calculus" / linear algebra is also assumed.

- If familiarity with [R](Link) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00LEmpirical Process Theory and ApplicationsW4 credits2VS. van de Geer
AbstractEmpirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc.
Objective
ContentIn this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
Selection: Financial and Insurance Mathematics
NumberTitleTypeECTSHoursLecturers
401-3888-00LIntroduction to Mathematical Finance Information
A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.
W10 credits4V + 1UC. Czichowsky
AbstractThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.
ObjectiveThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and also study convex duality in utility maximization.
ContentThis course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
Lecture notesThe course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.
LiteratureLiterature:

Michael U. Dothan, "Prices in Financial Markets", Oxford University Press

Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter

Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press

Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer
Prerequisites / NoticeA related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In other words, it is also not possible to earn credit points with one for the Bachelor and with the other for the Master degree.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
401-3629-00LQuantitative Risk Management Information W4 credits2V + 1UP. Cheridito
AbstractThis course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, risk measures, extreme value theory, multivariate models, copulas, dependence structures and operational risk.
ObjectiveThe goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.
Content1. Introduction
2. Basic Concepts in Risk Management
3. Empirical Properties of Financial Data
4. Financial Time Series
5. Extreme Value Theory
6. Multivariate Models
7. Copulas and Dependence
8. Operational Risk
Lecture notesCourse material is available on Link
LiteratureQuantitative Risk Management: Concepts, Techniques and Tools
AJ McNeil, R Frey and P Embrechts
Princeton University Press, Princeton, 2015 (Revised Edition)
Link
Prerequisites / NoticeThe course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-3923-00LSelected Topics in Life Insurance MathematicsW4 credits2VM. Koller
AbstractStochastic Models for Life insurance
1) Markov chains
2) Stochastic Processes for demography and interest rates
3) Cash flow streams and reserves
4) Mathematical Reserves and Thiele's differential equation
5) Theorem of Hattendorff
6) Unit linked policies
Objective
401-3917-00LStochastic Loss Reserving MethodsW4 credits2VR. Dahms
AbstractLoss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.
ObjectiveOur goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.
ContentWe will present the following stochastic claims reserving methods/models:
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, with and without inflation
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios
LiteratureM. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination periods.

This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under Link.

Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-3956-00LEconomic Theory of Financial Markets
Does not take place this semester.
W4 credits2VM. V. Wüthrich
AbstractThis lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
ObjectiveThis lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.
ContentWe treat the following topics:
- Fundamental concepts in economics
- Portfolio theory
- Mean variance analysis, capital asset pricing model
- Arbitrage pricing theory
- Cash flow theory
- Valuation principles
- Stochastic discounting, deflator techniques
- Interest rate modeling
- Utility theory
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.

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3936-00LData Analytics for Non-Life Insurance PricingW4 credits2VC. M. Buser, M. V. Wüthrich
AbstractWe study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.
ObjectiveThe student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.
ContentWe present the following chapters:
- generalized linear models (GLMs)
- generalized additive models (GAMs)
- neural networks
- credibility theory
- classification and regression trees (CARTs)
- bagging, random forests and boosting
Lecture notesThe lecture notes are available from:
Link
Prerequisites / NoticeThis course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under Link

Good knowledge in probability theory, stochastic processes and statistics is assumed.
401-4920-00LMarket-Consistent Actuarial ValuationW4 credits2VM. V. Wüthrich, H. Furrer
AbstractIntroduction to market-consistent actuarial valuation.
Topics: Stochastic discounting, full balance sheet approach, valuation portfolio in life and non-life insurance, technical and financial risks, risk management for insurance companies.
ObjectiveGoal is to give the basic mathematical tools for describing insurance products within a financial market and economic environment and provide the basics of solvency considerations.
ContentIn this lecture we give a full balance sheet approach to the task of actuarial valuation of an insurance company. Therefore we introduce a multidimensional valuation portfolio (VaPo) on the liability side of the balance sheet. The basis of this multidimensional VaPo is a set of financial instruments. This approach makes the liability side of the balance sheet directly comparable to its asset side.

The lecture is based on four sections:
1) Stochastic discounting
2) Construction of a multidimensional Valuation Portfolio for life insurance products (with guarantees)
3) Construction of a multidimensional Valuation Portfolio for a run-off portfolio of a non-life insurance company
4) Measuring financial risks in a full balance sheet approach (ALM risks)
LiteratureMarket-Consistent Actuarial Valuation, 3rd edition.
Wüthrich, M.V.
EAA Series, Springer 2016.
ISBN: 978-3-319-46635-4

Wüthrich, M.V., Merz, M.
Claims run-off uncertainty: the full picture.
SSRN Manuscript ID 2524352 (2015).

England, P.D, Verrall, R.J., Wüthrich, M.V.
On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins.
Insurance: Mathematics and Economics 85 (2019), 74-88.

Wüthrich, M.V., Embrechts, P., Tsanakas, A.
Risk margin for a non-life insurance run-off.
Statistics & Risk Modeling 28 (2011), no. 4, 299--317.

Financial Modeling, Actuarial Valuation and Solvency in Insurance.
Wüthrich, M.V., Merz, M.
Springer Finance 2013.
ISBN: 978-3-642-31391-2

Cheridito, P., Ery, J., Wüthrich, M.V.
Assessing asset-liability risk with neural networks.
Risks 8/1 (2020), article 16.
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.

Knowledge in probability theory, stochastic processes and statistics is assumed.
Selection: Mathematical Physics, Theoretical Physics
In the Bachelor's programme in Mathematics 402-0204-00L Electrodynamics is eligible as an elective course, but only if 402-0224-00L Theoretical Physics 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
401-3814-00LQuantum Mechanics for Mathematicians
NOTICE: The class scheduled for 5 March 2020 has been cancelled.
W4 credits2VJ. Wisniewska
AbstractIntroduction to quantum mechanics aimed at mathematics students
ObjectiveThe course begins with the fundamentals of classical mechanics and its mathematical description i.e. Hamiltonian dynamics. We will introduce the notion of states and observables in the classical setting and further on its counter parts in the quantum setting. We then will discuss quantisation and the mathematical formulation of quantum mechanics. Further on we will study the Heisenberg’s uncertainty relations and quantum entanglement. The course then goes on to study the dynamics of quantum systems described by the Schrödinger’s equation.
Content1. Hamiltonian mechanics and fundamentals of symplectic geometry
2. Classical observables and Poisson bracket
3. Basic principles of quantum mechanics and quantisation
4. Heisenberg’s uncertainty relations
5. Quantum entanglement and EPR paradox
6. Schrödinger’s equation
LiteratureTakhtajan, Leon A.
Quantum mechanics for mathematicians.
Graduate Studies in Mathematics, 95. American Mathematical Society, Providence, RI, 2008. xvi+387 pp. ISBN: 978-0-8218-4630-8
Prerequisites / NoticePrerequisites:
Essential: Differential Geometry 1
Recommended: basic Functional Analysis and Algebraic Topology
401-2334-00LMethods of Mathematical Physics II Information Restricted registration - show details W6 credits3V + 2UG. Felder
AbstractGroup theory: groups, representation of groups, unitary and orthogonal groups, Lorentz group. Lie theory: Lie algebras and Lie groups. Representation theory: representation theory of finite groups, representations of Lie algebras and Lie groups, physical applications (eigenvalue problems with symmetry).
Objective
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