Search result: Catalogue data in Spring Semester 2020
Doctoral Department of Mathematics | ||||||
Graduate School Official website of the Zurich Graduate School in Mathematics: Link | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-5004-20L | Limit Shape Phenomenon in Integrable Models in Statistical Mechanics | W | 0 credits | 2V | N. Reshetikhin | |
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | The limit shape phenomenon for large lattice domains is the formation of the most probable state, such that all states that macroscopically differ from it are exponentially improbable. This is a rather general phenomenon of similar nature to the central limit theorems and to the large deviation principle in probability theory. Integrable models in statistical mechanics in many cases admit rather explicit solutions. This not a definition of integrability, but one of the important conse- quences. This allows us to describe many features of the limit shape phenome- non quite explicitly and to prove some important facts about them. The course will consist roughly of three parts: one is on limit shapes, the other is on integrability and the third one is about limit shapes in integrable models. The course is aimed at both mathematics and physics students. | |||||
401-5006-20L | Rough Analysis and Applications | W | 0 credits | 2V | P. Friz | |
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | We plan to discuss: the algebra of iterated integrals, signatures and expec- ted signatures, rough paths and models; Kolmogorov type criteria; Schau- der estimates; p-variation, abstract Riemann-Stieltjes integration (Sewing); rough integration and reconstruction; a paracontrolled view; rough differenti- al equations and flows; rough transport equations; second order rough partial differential equations; rough volatility; a regularity structure view; large devi- ations and precise asymptotics for rough volatility. A good part of the lecture will follow F-Hairer (2014,2020). | |||||
401-5002-20L | The Value Distribution of L-Functions and Multiplicative Number Theory (CANCELLED) | W | 0 credits | K. Soundararajan | ||
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | The broad theme for the lecture series is the value distribution of zeta and L-func- tions. This is related to several important questions concerning (i) the maximal size of L-functions, (ii) asymptotics for moments of L-values, (iii) the distributi- on of zeros, and non-vanishing of L-functions at special points. Further, some of the techniques used in the study of moments have close counterparts in the understanding of other problems in multiplicative number theory --- for instance, the recent results of Harper on random multiplicative functions, and the break- throughs of Matomäki and Radziwiłł on multiplicative functions in short intervals. Much of this work has a strong probabilistic flavor, and in particular we shall dis- cuss connections with random matrix theory, branching Brownian motion, etc. | |||||
401-3109-65L | Probabilistic Number Theory Does not take place this semester. | W | 8 credits | 4G | E. Kowalski | |
Abstract | The 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. | |||||
Objective | The goal of the course is to present some results of probabilistic number theory in a unified manner. | |||||
Content | The 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 notes | The lecture notes for the class are available at Link | |||||
Prerequisites / Notice | Prerequisites: 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. | |||||
401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | A. Sisto | |
Abstract | This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality. | |||||
Objective | ||||||
Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. The book can be downloaded for free at: Link 2) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||
Prerequisites / Notice | General topology, linear algebra, singular homology 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. | |||||
401-3226-00L | Symmetric Spaces | W | 8 credits | 4G | M. Burger | |
Abstract | * Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples * Symmetric spaces of non-compact type: flats and rank, roots and root spaces * Iwasawa decomposition, Weyl group, Cartan decomposition * Hints of the geometry at infinity of SL(n,R)/SO(n). | |||||
Objective | Learn the basics of symmetric spaces | |||||
401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | U. Lang | |
Abstract | Introduction 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, relations between curvature and topology, spaces of Riemannian manifolds. | |||||
Objective | Learn the basics of Riemannian geometry and some elements of modern metric geometry. | |||||
Literature | - 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 | |||||
Prerequisites / Notice | Prerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms. | |||||
401-3462-00L | Functional Analysis II | W | 10 credits | 4V + 1U | M. Struwe | |
Abstract | Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity | |||||
Objective | Acquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates | |||||
Lecture notes | Funktionalanalysis II, Michael Struwe | |||||
Literature | Funktionalanalysis II, Michael Struwe Functional Analysis, Spectral Theory and Applications. Manfred Einsiedler and Thomas Ward, GTM Springer 2017 | |||||
Prerequisites / Notice | Functional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces. | |||||
401-4788-16L | Mathematics of (Super-Resolution) Biomedical Imaging NOTICE: The exercise class scheduled for 5 March has been cancelled | W | 8 credits | 4G | H. Ammari | |
Abstract | The 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. | |||||
Objective | Super-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. | |||||
401-4605-20L | Selected Topics in Probability | W | 4 credits | 2V | A.‑S. Sznitman | |
Abstract | This course will discuss some questions of current interest in probability theory. Among examples of possible subjects are for instance topics in random media, large deviations, random walks on graphs, branching random walks, random trees, percolation, concentration of measures, large random matrices, stochastic calculus, stochastic partial differential equations. | |||||
Objective | This course will discuss some questions of current interest in probability theory. Among examples of possible subjects are for instance topics in random media, large deviations, random walks on graphs, branching random walks, random trees, percolation, concentration of measures, large random matrices, stochastic calculus, stochastic partial differential equations. | |||||
Prerequisites / Notice | Lecture Probability Theory. | |||||
401-4632-15L | Causality | W | 4 credits | 2G | C. Heinze-Deml | |
Abstract | In 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. | |||||
Objective | After 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 | |||||
Prerequisites / Notice | Prerequisites: basic knowledge of probability theory and regression | |||||
401-4658-00L | Computational Methods for Quantitative Finance: PDE Methods | W | 6 credits | 3V + 1U | C. Schwab | |
Abstract | Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming and knowledge of numerical mathematics at ETH BSc level. | |||||
Objective | Introduce 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. | |||||
Content | 1. 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. | |||||
Lecture notes | There will be english, typed lecture notes as well as MATLAB software for registered participants in the course. | |||||
Literature | R. 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. | |||||
401-3629-00L | Quantitative Risk Management | W | 4 credits | 2V + 1U | P. Cheridito | |
Abstract | This 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. | |||||
Objective | The goal is to learn the most important methods from probability theory and statistics used in financial risk modeling. | |||||
Content | 1. 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 notes | Course material is available on Link | |||||
Literature | Quantitative Risk Management: Concepts, Techniques and Tools AJ McNeil, R Frey and P Embrechts Princeton University Press, Princeton, 2015 (Revised Edition) Link | |||||
Prerequisites / Notice | The 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. | |||||
401-3917-00L | Stochastic Loss Reserving Methods | W | 4 credits | 2V | R. Dahms | |
Abstract | Loss 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. | |||||
Objective | Our 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. | |||||
Content | We 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 | |||||
Literature | M. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008. | |||||
Prerequisites / Notice | The 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. | |||||
401-3936-00L | Data Analytics for Non-Life Insurance Pricing | W | 4 credits | 2V | C. M. Buser, M. V. Wüthrich | |
Abstract | We 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. | |||||
Objective | The student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction. | |||||
Content | We 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 notes | The lecture notes are available from: Link | |||||
Prerequisites / Notice | This 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. | |||||
401-4920-00L | Market-Consistent Actuarial Valuation | W | 4 credits | 2V | M. V. Wüthrich, H. Furrer | |
Abstract | Introduction 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. | |||||
Objective | Goal 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. | |||||
Content | In 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) | |||||
Literature | Market-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 / Notice | The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under Link. Knowledge in probability theory, stochastic processes and statistics is assumed. | |||||
401-3956-00L | Economic Theory of Financial Markets Does not take place this semester. | W | 4 credits | 2V | M. V. Wüthrich | |
Abstract | This lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries. | |||||
Objective | This 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. | |||||
Content | We 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 / Notice | The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under Link. Knowledge in probability theory, stochastic processes and statistics is assumed. | |||||
151-0530-00L | Nonlinear Dynamics and Chaos II | W | 4 credits | 4G | G. Haller | |
Abstract | The internal structure of chaos; Hamiltonian dynamical systems; Normally hyperbolic invariant manifolds; Geometric singular perturbation theory; Finite-time dynamical systems | |||||
Objective | The course introduces the student to advanced, comtemporary concepts of nonlinear dynamical systems analysis. | |||||
Content | I. The internal structure of chaos: symbolic dynamics, Bernoulli shift map, sub-shifts of finite type; chaos is numerical iterations. II.Hamiltonian dynamical systems: conservation and recurrence, stability of fixed points, integrable systems, invariant tori, Liouville-Arnold-Jost Theorem, KAM theory. III. Normally hyperbolic invariant manifolds: Crash course on differentiable manifolds, existence, persistence, and smoothness, applications. IV. Geometric singular perturbation theory: slow manifolds and their stability, physical examples. V. Finite-time dynamical system; detecting Invariant manifolds and coherent structures in finite-time flows | |||||
Lecture notes | Students have to prepare their own lecture notes | |||||
Literature | Books will be recommended in class | |||||
Prerequisites / Notice | Nonlinear Dynamics I (151-0532-00) or equivalent | |||||
401-3652-00L | Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich) No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH. UZH Module Code: MAT827 Mind the enrolment deadlines at UZH: Link | W | 10 credits | 4V + 2U | University lecturers | |
Abstract | This 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. | |||||
Objective | The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations. | |||||
Content | * 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. | |||||
Lecture notes | Lecture slides will be made available to participants. However, additional material might be covered in the course. | |||||
Literature | H. 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. | |||||
Prerequisites / Notice | Having 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" | |||||
401-3903-11L | Geometric Integer Programming | W | 6 credits | 2V + 1U | J. Paat | |
Abstract | Integer 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. | |||||
Objective | The purpose of the lecture is to provide a geometric treatment of the theory of integer optimization. | |||||
Content | Key topics are: - Lattice theory and the polynomial time solvability of integer optimization problems in fixed dimension. - Structural properties of integer sets that reveal other parameters affecting the complexity of integer problems - Duality theory for integer optimization problems from the vantage point of lattice free sets. | |||||
Lecture notes | not available, blackboard presentation | |||||
Literature | Lecture notes will be provided. Other helpful materials include Bertsimas, Weismantel: Optimization over Integers, 2005 and Schrijver: Theory of linear and integer programming, 1986. | |||||
Prerequisites / Notice | "Mathematical Optimization" (401-3901-00L) |
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