Suchergebnis: Katalogdaten im Frühjahrssemester 2017

Doktorat Departement Mathematik Information
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Die Liste der Lehrveranstaltungen (samt der zugehörigen Anzahl Kreditpunkte) für Doktoratsstudentinnen und Doktoratsstudenten wird jedes Semester im Newsletter der ZGSM veröffentlicht.
ACHTUNG: Kreditpunkte fürs Doktoratsstudium sind nicht mit ECTS-Kreditpunkten zu verwechseln!
Graduate School / Graduiertenkolleg
Offizielle Website der Zurich Graduate School in Mathematics:
401-5002-17L<ND lecture postponed to HS 2017>
Findet dieses Semester nicht statt.
KurzbeschreibungNachdiplom lecture
401-5004-17LSelf-Organized Dynamics: From Emerging Consensus to Hydrodynamic FlockingW0 KP2VE. Tadmor
KurzbeschreibungNachdiplom lecture
InhaltWe study the dynamics of systems driven by the 'social engagement' of agents with their neighbors. Prototype models are found in opinion dynamics, flocking, self-organization of biological organisms and rendezvous of mobile systems.
Two natural questions arise in this context, namely, what is the large time behavior of such systems, and what is the effective dynamics for large crowds of agents.
The underlying issue of the first question is how different rules of engagement influence the formation of clusters, and in particular, the emergence of 'consensus'. Different paradigms of engagement yield different patterns. The tendency 'to move ahead' leads to the emergence of leaders, and 'to move around' may lead to synchronization. Different descriptions of collective dynamics arise in the context of the second question when the number of agents tends to infinity, with the formation of Dirac masses at the kinetic level of description, and critical thresholds at the level of flocking hydrodynamics.
In recent years there has been extensive activity in analyzing the self-organization of these systems. We will discuss recent developments and present open problems.
401-5006-17LTopics in Scalar CurvatureW0 KP1VR. Schoen
KurzbeschreibungNachdiplom lecture
InhaltThe scalar curvature plays an important role in geometry and physics. Understanding the geometry of manifolds of positive scalar curvature is intimately connected with understanding gravitational mass in general relativity. There are two general methods to deal with such problems, the Dirac operator and the theory of minimal hypersurfaces. A third method, inverse mean curvature flow, has been used effectively in three dimensions. This lecture series will focus mainly on the minimal hypersurface approach and will discuss attempts at local characterizations of manifolds of positive scalar curvature (which are related to quasilocal mass). We will then discuss approaches to proving theorems in the presence of minimal hypersurface singularities which can occur in dimension greater than seven.
401-4142-17LAlgebraic CurvesW6 KP3GR. Pandharipande
KurzbeschreibungI will discuss the classical theory of algebraic curves. The topics will include:
divisors, Riemann-Roch, linear systems, differentials, Clifford's theorem,
curves on surfaces, singularities, curves in projective space, elliptic curves,
hyperelliptic curves, families of curves, moduli, and enumerative geometry.
There will be many examples and calculations.
InhaltLecture homepage:
LiteraturForster, "Lectures on Riemann Surfaces"

Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves"

Mumford, "Curves and their Jacobians"
Voraussetzungen / BesonderesFor background, a semester course in algebraic geometry should be
sufficient (perhaps even if taken concurrently). You should know the definitions
of algebraic varieties and algebraic morphisms and their basic properties.
401-3002-12LAlgebraic Topology IIW8 KP4GP. S. Jossen
KurzbeschreibungThis is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology such as: products, duality, cohomology operations, characteristic classes, spectral sequences etc.
Literatur1) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:

See also:

2) E. Spanier, "Algebraic topology", Springer-Verlag

3) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

4) R. Bott & L. Tu, "Differential forms in algebraic topology",
Graduate Texts in Mathematics, 82. Springer-Verlag, 1982.

5) J. Milnor & J. Stasheff, "Characteristic classes",
Annals of Mathematics Studies, No. 76.
Princeton University Press, 1974.
Voraussetzungen / BesonderesGeneral topology, linear algebra.
Basic knowledge of singular homolgoy and cohomology of topological spaces (e.g. as taught in "Algebraic topology I").

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.
401-3226-01LRepresentation Theory of Lie GroupsW8 KP4GE. Kowalski
KurzbeschreibungThis course will contain two parts:
* Introduction to unitary representations of Lie groups
* Introduction to the study of discrete subgroups of Lie groups and some applications.
LernzielThe goal is to acquire familiarity with the basic formalism and results concerning unitary representations of Lie groups, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups.
Inhalt* Unitary representations of compact Lie groups: Peter-Weyl theory, weights, Weyl character formula
* Introduction to unitary representations of non-compact Lie groups: the examples of SL(2,R), SL(2,C)
* Example: Property (T) for SL(n,R)
* Discrete subgroups of Lie groups: examples and some applications
LiteraturBekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press.
Voraussetzungen / BesonderesDifferential geometry, Functional analysis, Introduction to Lie Groups (or equivalent).
Notice that this course has a large overlap with 401-3226-01L Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups taught in FS 2016. Therefore it is not possible to acquire credits for both courses.
401-3532-08LDifferential Geometry IIW10 KP4V + 1UU. Lang
KurzbeschreibungIntroduction to Riemannian Geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, curvature and topology, spaces of riemannian manifolds.
LernzielThe aim of this course is to give an introduction to Riemannian Geometry in combination with some elements of modern metric geometry.
InhaltRiemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form of submanifolds, riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of riemannian manifolds.
LiteraturRiemannian Geometry:
- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983
Metric Geometry:
- M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999
- D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Amer. Math. Soc. 2001
Voraussetzungen / BesonderesPrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, tangent and tensor bundles, and differential forms.
401-3462-00LFunctional Analysis IIW10 KP4V + 1UM. Struwe
KurzbeschreibungSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity theory, Schauder estimates
LernzielThe lecture course will focus on weak solutions of elliptic boundary value problems in Sobolev spaces and discuss their regularity properties, possibly followed by a proof of the Calderon-Zygmund
inequality and some basic results on parabolic regularity, with
applications to geometry, if time allows.
401-4832-17LMathematical Themes in General Relativity IIW4 KP2VA. Carlotto
KurzbeschreibungSecond part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems.
LernzielAcquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis.
InhaltAnalysis of Jang's equation and application to the proof of the spacetime positive energy theorem; the conformal method for the Einstein constraint equations and links with the Yamabe problem; gluing methods for the Einstein constraint equations: canonical asymptotics, N-body solutions, gravitational shielding.
SkriptLecture notes written by the instructor will be provided to all enrolled students.
Voraussetzungen / BesonderesThe content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable.
**This course is the sequel of the one offered during the first semester.**
401-3496-17LTopics in the Calculus of VariationsW4 KP2VA. Figalli
401-3652-00LNumerical Methods for Hyperbolic Partial Differential Equations Information W10 KP4V + 1UU. S. Fjordholm
KurzbeschreibungThis course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. Exercises involve implementation of numerical methods in MATLAB.
LernzielThe goal of this course is familiarity with the fundamental ideas and mathematical
consideration underlying modern numerical methods for conservation laws and wave equations.
Inhalt* Introduction to hyperbolic problems: Conservation, flux modeling, examples and significance in physics and engineering.

* Linear Advection equations in one dimension: Characteristics, energy estimates, upwind schemes.

* Scalar conservation laws: shocks, rarefactions, solutions of the Riemann problem, weak and entropy solutions, some existence and uniqueness results, finite volume schemes of the Godunov, Engquist-Osher and Lax-Friedrichs type. Convergence for monotone methods and E-schemes.

* Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods.

* Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes.

* Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.
SkriptLecture slides will be made available to participants. However, additional material might be covered in the course.
LiteraturH. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online.

R. J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002. Available online.

E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.
Voraussetzungen / BesonderesHaving attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite.

Programming exercises in MATLAB

Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"
401-4788-16LMathematics of (Super-Resolution) Biomedical ImagingW8 KP4GH. Ammari
KurzbeschreibungThe aim of this course is to review different methods used to address challenging problems in biomedical imaging. The emphasis will be on scale separation techniques, hybrid imaging, spectroscopic techniques, and nanoparticle imaging. These approaches allow one to overcome the ill-posedness character of imaging reconstruction in biomedical applications and to achieve super-resolution imaging.
LernzielSuper-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other.

In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold:
(i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence
of small anomalies;
(ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies;
(iii) To design efficient inversion algorithms in multi-wave modalities;
(iv) to develop inversion techniques using multi-frequency measurements;
(v) to develop a mathematical and numerical framework for nanoparticle imaging.

In this course we shall consider both analytical and computational
matters in biomedical imaging. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, inverse problems, mathematical imaging, optimal control, stochastic modelling, and analysis of physical phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in biomedical imaging, requires a deep understanding of the different scales in the physical models, an accurate mathematical modelling of the imaging techniques, and fine analysis of complex physical phenomena.

An emphasis is put on mathematically analyzing acoustic-electric imaging, thermo-elastic imaging, Lorentz force based imaging, elastography, multifrequency electrical impedance tomography, and plasmonic resonant nanoparticles.
401-4632-15LCausalityW4 KP2GM. H. Maathuis
KurzbeschreibungIn statistics, we are used to search for the best predictors of some random variable. In many situations, however, we are interested in predicting a system's behavior under manipulations. For such an analysis, we require knowledge about the underlying causal structure of the system. In this course, we study concepts and theory behind causal inference.
LernzielAfter this course, you should be able to
- understand the language and concepts of causal inference
- know the assumptions under which one can infer causal relations from observational and/or interventional data
- describe and apply different methods for causal structure learning
- given data and a causal structure, derive causal effects and predictions of interventional experiments
Voraussetzungen / BesonderesPrerequisites: basic knowledge of probability theory and regression
401-4658-00LComputational Methods for Quantitative Finance: PDE Methods Information Belegung eingeschränkt - Details anzeigen W6 KP3V + 1UC. Schwab
KurzbeschreibungIntroduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming
and knowledge of numerical mathematics at ETH BSc level.
LernzielIntroduce the main methods for efficient numerical valuation of derivative contracts in a
Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility
models. Develop implementation of pricing methods in MATLAB.
Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Inhalt1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic
volatility models.
2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees.
European contracts.
3. Finite Difference methods for Asian, American and Barrier type contracts.
4. Finite element methods for European and American style contracts.
5. Pricing under local and stochastic volatility in Black-Scholes Markets.
6. Finite Element Methods for option pricing under Levy processes. Treatment of
integrodifferential operators.
7. Stochastic volatility models for Levy processes.
8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and
stochastic volatility models in Black Scholes and Levy markets.
9. Introduction to sparse grid option pricing techniques.
SkriptThere will be english, typed lecture notes as well as MATLAB software for registered participants in the course.
LiteraturR. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004.

Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005.

D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008.

J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.

N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013.
Voraussetzungen / BesonderesStart of the lecture: Wednesday, March 1, 2017 (second week of the semester).
401-3629-00LQuantitative Risk Management Information W4 KP2VP. Cheridito
KurzbeschreibungThis course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, multivariate models, copulas and dependence structures, extreme value theory, risk measures, aggregation of risk, and risk allocation.
LernzielThe goal is to learn the most important methods from probability theory and statistics used to model financial risks.
Inhalt1. Risk in Perspective
2. Basic Concepts
3. Multivariate Models
4. Copulas and Dependence
5. Aggregate Risk
6. Extreme Value Theory
7. Operational Risk and Insurance Analytics
SkriptCourse material is available on
LiteraturQuantitative Risk Management: Concepts, Techniques and Tools
AJ McNeil, R Frey and P Embrechts
Princeton University Press, Princeton, 2015 (Revised Edition)
Voraussetzungen / BesonderesThe course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-4938-14LStochastic Optimal Control Information W4 KP2VM. Soner
KurzbeschreibungDynamic programming approach to stochastic optimal control problems will be developed. In addition to the general theory, detailed analysis of several important control problems will be given.
LernzielGoals are to achieve a deep understanding of

1. Dynamic programming approach to optimal control;
2. Several classes of important optimal control problems and their solutions.
3. To be able to use this models in engineering and economic modeling.
InhaltIn this course, we develop the dynamic programming approach for the stochastic optimal control problems. The general approach will be described and several subclasses of problems will also be discussed in including:
1. Standard exit time problems;
2. Finite and infinite horizon problems;
3. Optimal stoping problems;
4. Singular problems;
5. Impulse control problems.

After the general theory is developed, it will be applied to several classical problems including:
1. Linear quadratic regulator;
2. Merton problem for optimal investment and consumption;
3. Optimal dividend problem of (Jeanblanc and Shiryayev);
4. Finite fuel problem;
5. Utility maximization with transaction costs;
6. A deterministic differential game related to geometric flows.

Textbook will be

Controlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).

And lecture notes will be provided.
LiteraturControlled Markov Processes and Viscosity Solutions, 2nd edition, (W.H. Fleming and H.M. Soner) Springer-Verlag, (2005).

And lecture notes will be provided.
Voraussetzungen / BesonderesBasic knowledge of Brownian motion, stochastic differential equations and probability theory is needed.
401-3917-00LStochastic Loss Reserving MethodsW4 KP2VR. Dahms
KurzbeschreibungLoss 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.
LernzielOur 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.
InhaltWe 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
LiteraturM. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Voraussetzungen / BesonderesThe 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

Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-4920-00LMarket-Consistent Actuarial Valuation
Findet dieses Semester nicht statt.
W4 KP2VM. V. Wüthrich
KurzbeschreibungIntroduction 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.
LernzielGoal 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.
InhaltIn 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)
LiteraturMarket-Consistent Actuarial Valuation, 2nd edition.
Wüthrich, M.V., Bühlmann, H., Furrer, H.
EAA Series Textbook, Springer, 2010.
ISBN: 978-3-642-14851-4

Wüthrich, M.V., Merz, M.
Claims Run-Off Uncertainty: The Full Picture
SSRN Manuscript ID 2524352 (2015).

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
Voraussetzungen / BesonderesThe 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

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3956-00LEconomic Theory of Financial MarketsW4 KP2VM. V. Wüthrich
KurzbeschreibungThis lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
LernzielThis 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.
InhaltWe 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
Voraussetzungen / BesonderesThe 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

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3903-11LGeometric Integer Programming
Findet dieses Semester nicht statt.
W6 KP2V + 1Ukeine Angaben
KurzbeschreibungInteger programming is the task of minimizing a linear function over all the integer points in a polyhedron. This lecture introduces the key concepts of an algorithmic theory for solving such problems.
LernzielThe purpose of the lecture is to provide a geometric treatment of the theory of integer optimization.
InhaltKey topics are:
- lattice theory and the polynomial time solvability of integer optimization problems in fixed dimension,
- the theory of integral generating sets and its connection to totally dual integral systems,
- finite cutting plane algorithms based on lattices and integral generating sets.
Skriptnot available, blackboard presentation
LiteraturBertsimas, Weismantel: Optimization over Integers, Dynamic Ideas 2005.
Schrijver: Theory of linear and integer programming, Wiley, 1986.
Voraussetzungen / Besonderes"Mathematical Optimization" (401-3901-00L)
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