Search result: Catalogue data in Spring Semester 2016
Doctoral Department of Mathematics More Information at: Link | ||||||
Doctoral and Post-Doctoral Courses | ||||||
Graduate School | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-5002-16L | High Dimensional Expanders | Z | 0 credits | 2V | A. Lubotzky | |
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | Expander graphs in general, and Ramanujan graphs in particular, have played an important role in computer science and pure mathematics in the last 4 decades. In recent years the area of high dimensional expanders (i.e. simplical complexes/hypergraphs with properties generalizing those of expanding graphs) and Ramanujan complexes is starting to emerge. It appears naturally (so far) in 3 topics: a) Linial-Meshulam theory of random complexes generalizing the Erdos-Renyi random graphs; b) Gromov's overlapping properties (these are far reaching extensions of the following result: for every N points set P in the plane, there is a point z which is covered by at least 2/9 of the (N choose 3) triangles determined by P); c) Testability properties in computer science. We will discuss these developments and present some new results and open problems. | |||||
401-5004-16L | Geometric and Topological Aspects of Coxeter Groups and Buildings | Z | 0 credits | 2V | A. Thomas | |
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | We will begin by reviewing the basic theory of Coxeter and reflection groups. We will then study the Davis complex, a cell complex with "good" geometric and topological properties on which the associated Coxeter group acts "nicely". We will prove Moussong's Theorem, which characterises the Coxeter groups which are hyperbolic in the sense of Gromov, and discuss the use of the Davis complex to determine cohomology of Coxeter groups. In the second part of the course we will study buildings. Using the theory of Coxeter groups and the Davis complex already discussed, we will establish the equivalence of the main definitions of a building, and describe the main geometric realisations of a building. We then discuss the use of buildings to study groups which act on them, including algebraic groups over local fields, arithmetic groups, and other lattices. If time permits we will consider the theory of twin buildings, which appears in the study of Kac-Moody groups. | |||||
401-5006-16L | Variational Approach to SPDEs and Corresponding Fokker-Planck-Kolmogorov Equations | Z | 0 credits | 2V | M. Röckner | |
Abstract | Nachdiplom lecture | |||||
Objective | ||||||
Content | The lectures will follow mainly [1]. As prerequisites a course in probability theory and some basic knowledge about Hilbert spaces would be helpful, though e.g. even the notion of a martingale will be recalled in the lectures. The first part will be a self-contained introduction to stochastic integration on Hilbert spaces, followed by a part on stochastic differential equations (SDEs) on finite dimensional state spaces. Then as the core of the lectures, the variational approach to SDEs on Hilbert spaces will be presented, first under global monotonicity conditions on the coefficients and subsequently under merely local monotonicity as well as generalized coercivity conditions. Applications to standard stochastic partial differential equations, including the stochastic versions of the parabolic porous media, p-Laplace, Cahn-Hillard, Burgers and 2D as well as 3D Navier-Stokes equations will be presented. Finally, the connection to the Fokker-Planck-Kolmogorov equations will be discussed and, time permitting, some recent results on the latter explained. This last part will be based on [2]. The detailedness in which the respective parts of the lectures will be presented, will depend on the background knowledge and the interest of the audience. One way to realize this in a sort of individualized manner is to inform the audience about the beginning/end of the next/previous part via e-mail so everyone may decide to skip a part she or he knows about, or join another more advanced part she or he is particularly interested in, respectively. | |||||
Literature | [1] Wei Liu und Michael Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015, pp. 266 [2] Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner und Stanislav V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Russian version: Izhewsk Institute of Computer Science, 2013, English version: AMS-Monographs to appear, pp. 488. | |||||
Prerequisites / Notice | As prerequisites a course in probability theory and some basic knowledge about Hilbert spaces would be helpful, though e.g. even the notion of a martingale will be recalled in the lectures. | |||||
401-3226-01L | Unitary Representations of Lie Groups and Discrete Subgroups of Lie Groups | W | 8 credits | 4G | M. Einsiedler | |
Abstract | This course will contain three parts: * Classification of simple Lie algebras * Introduction to unitary representations of Lie groups * Introduction to the study of discrete subgroups of Lie groups, the quotient space, and some applications. | |||||
Objective | The goal is to acquire familiarity with the basic formalism and results concerning Lie groups and their unitary representations, and to apply these to the study of discrete subgroups, especially lattices, in Lie groups. | |||||
Content | * Classification using Dynkin diagrams * 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 | |||||
Literature | Bekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press. | |||||
Prerequisites / Notice | Functional analysis I and Lie Groups I or Differential geometry I. | |||||
401-4144-16L | Reading Course: Deformation Theory | W | 2 credits | 4A | J. Fresán | |
Abstract | ||||||
Objective | ||||||
401-3108-16L | Topics in Automorphic Forms | W | 6 credits | 2V + 1U | P. D. Nelson | |
Abstract | ||||||
Objective | ||||||
401-3002-12L | Algebraic Topology II | W | 8 credits | 4G | P. Biran | |
Abstract | This 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. | |||||
Objective | ||||||
Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 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. | |||||
Prerequisites / Notice | General 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. | |||||
401-3532-08L | Differential Geometry II | W | 10 credits | 4V + 1U | M. Burger | |
Abstract | The aim of this course is to give an introduction to Riemannian Geometry and modern metric geometry. | |||||
Objective | Riemannian Geometry, metric geometry. | |||||
Content | The aim of this course is to give an introduction to Riemannian Geometry and modern metric geometry. We will present the basics on affine and riemannian connections, discuss existence and properties of geodesics; then we proceed to the central concept of riemannian curvature tensor and its various avatars, like sectional curvature and scalar curvature. We will then move to Topogonov's comparison theorems. This constitutes the bridge with metric geometry and the modern notion of negative curvature, which applies to singular spaces, and constitutes the topic of the second part of this course. | |||||
Lecture notes | Will be made available. | |||||
Literature | M.P. do Carmo, "Riemannian Geometry", Birkhauser, 1992 M. Bridson, A. Haefliger, "Metric Spaces of Non-Positive Curvature", Springer 1999. | |||||
Prerequisites / Notice | Prerequisite are the sections concerning manifolds and tangent bundles of the Differential Geometry I course, Fall Semester 2015. | |||||
401-3462-00L | Functional Analysis II | W | 10 credits | 4V + 1U | D. A. Salamon | |
Abstract | Sobolev spaces, Calderon-Zygmund inequality, elliptic regularity, strongly continuous semigroups, parabolic pde's. | |||||
Objective | The lecture course will begin with an introduction to Sobolev spaces and Sobolev embedding theorems, a proof of the Calderon-Zygmund inequality, and regularity theorems for second order elliptic operators, followed by an introduction to the theory of strongly continuous operator semigroups and some basic results about parabolic regularity. Applications to geometry will be included if time allows. | |||||
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 + 1U | R. Abgrall | |
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" | |||||
401-4766-16L | Topics in Mathematical and Computational Fluid Dynamics | W | 4 credits | 2V | S. Mishra, F. Weber | |
Abstract | The course will cover some essential advanced topics in fluid dynamics, from both a theoretical and numerical point of view. The proposed topics include theory for the incompressible Euler and Navier-Stokes equations and numerical methods to approximate them. Additional topics including theory and numerics for the compressible Euler equations may also be covered. | |||||
Objective | To learn both theoretical aspects of PDEs governing fluid flows as well as numerical methods to approximate them. | |||||
Content | 1. Derivation of the PDEs governing fluid flows from first principles. 2. Theory for incompressible Navier-Stokes equation -- Leray-Hopf weak solutions, global existence. Regularity in two dimensions. 3. Theory for incompressible Euler equations: Well-posedness in two-space dimensions, vortex sheets, blow-up criteria in three dimensions. Non-uniqueness of admissible weak solutions. 4. Spectral and spectral viscosity methods for the Euler and Navier-Stokes equations and their convergence. 5. Finite difference projection methods. 6. Vortex methods for the incompressible Euler equations. 7. Measure valued and Statistical solutions. If time permits, we also cover some topics on the Compressible Euler equations. | |||||
Lecture notes | Last version of lecture notes of the course can be found here: Link | |||||
Prerequisites / Notice | A solid background in functional analysis, PDE and numerical methods for PDE. | |||||
401-4788-16L | Mathematics of Super-Resolution Biomedical Imaging | W | 8 credits | 4G | H. Ammari | |
Abstract | The aim of this course is to review recent mathematical and computational frameworks for super-resolution cell and tissue imaging. | |||||
Objective | The objective is twofold: (i) To exhibit the fundamental models underlying spectroscopic electrical and mechanical tissue properties imaging in order to improve differentiation of tissue pathologies; (ii) To develop new mathematical models for multi-wave tissue property imaging approaches in order to beat the resolution limit. | |||||
401-4653-63L | Inverse Problems | W | 6 credits | 3G | R. Alaifari | |
Abstract | Introduction into the mathematical theory for linear and non-linear inverse problems, and discussion of numerical methods for their numerical solution. | |||||
Objective | Understanding the nature of inverse problems and familiarity with a few important specimens. Grasp, why regularization is needed, and how it can be implemented and controlled. | |||||
Content | 1. Introduction: Examples of inverse problems 2. Ill-posed linear operator equations 3. Regularization operators 4. Continuous regularization methods and parameter choice rules 5. Tikhonov regularization 6. Landweber-type methods 7. The conjugate gradient method 8. Tikhonov regularization of nonlinear problems 9. Nonlinear iterative regularization methods | |||||
Lecture notes | No lecture notes will be made available | |||||
Prerequisites / Notice | Prerequisites: Some familiarity with concepts of Hilbert space theory as covered in an introductory course on functional analysis is expected. | |||||
401-4605-16L | 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 possible subjects are for instance topics in random media, percolation, random walks on graphs, stochastic calculus, stochastic partial differential equations. | |||||
Objective | ||||||
401-4614-16L | Diffusion Processes | W | 4 credits | 2V | R. Rosenthal | |
Abstract | ||||||
Objective | ||||||
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. | |||||
Prerequisites / Notice | The 2009 title of this course unit was "Computational Methods for Quantitative Finance II: Finite Element and Finite Difference Methods". | |||||
401-3629-00L | Quantitative Risk Management | W | 4 credits | 2V | P. Embrechts | |
Abstract | The aim of this course is to present a concise overview of mathematical methods from the areas of probability and statistics that can be used by financial institutions to model market, credit and operational risk. Topics addressed include loss distributions, multivariate models, dependence and copulas, extreme value theory, risk measures, risk aggregation and risk allocation. | |||||
Objective | The aim of this course is to present a concise overview of mathematical methods from the areas of probability and statistics that can be used by financial institutions to model market, credit and operational risk. | |||||
Content | 1. 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 | |||||
Lecture notes | The course material (pdf-slides and further reading material) are available at Link in the section "Course material" (the username and password have been sent by email). The textbook listed under "Literatur" below makes ideal background reading. | |||||
Literature | Quantitative Risk Management: Concepts, Techniques and Tools AJ McNeil, R Frey and P Embrechts Princeton University Press, Princeton, 2015 (Revised Edition) Link (For this course the 2005 first edition also suffices) | |||||
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 | 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. | |||||
401-4920-00L | 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, 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 | |||||
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. | |||||
401-3953-00L | Interest Rate Modeling in Discrete Time | W | 4 credits | 2V | M. V. Wüthrich | |
Abstract | This course gives an introduction to stochastic interest rate modeling in discrete time. Starting from cash flow valuation with state price deflators, we derive the equivalent martingale measures for pricing financial instruments. The lecture is supplemented by several examples such as the Vasicek model, the Heath-Jarrow-Morton framework and the consistent re-calibration approach. | |||||
Objective | The students are familiar with the basic terminology of stochastic interest rate modeling and they are able to transfer their (financial) mathematical knowledge to real world pricing of cash flows and financial instruments. | |||||
Content | The following topics are covered: 1) stochastic discounting with state price deflators 2) equivalent martingale measures 3) pricing of cash flows and primary assets 4) pricing of derivatives, e.g. European put options 5) (multi-factor) Vasicek state price deflator model 6) Heath-Jarrow-Morton interest rate modeling framework 7) consistent re-calibration approach | |||||
Literature | 1) Part I of: Wüthrich, M.V., Merz, M. (2013). Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer. 2) Wüthrich, M.V. (2015). Consistent re-calibration in yield curve modeling: an example. SSRN Manuscript, ID 2630164. For further reading: 1) Brigo, D., Mercurio, F. (2006). Interest Rate Models - Theory and Practice. 2nd Edition, Springer. 2) Filipovic, D. (2009). Term-Structure Models. A Graduate Course. Springer. 3) Harms, P., Stefanovits, D., Teichmann, J., Wüthrich, M.V. (2015). Consistent recalibration of yield curve models. preprint on arXiv.org. | |||||
Prerequisites / Notice | The exams ONLY take place during the official ETH examination period. Prerequisites: knowledge of probability theory and applied stochastic processes. |
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