Search result: Catalogue data in Spring Semester 2018

Number	Title	Type	ECTS	Hours	Lecturers
Doctoral Department of Mathematics More Information at: https://www.ethz.ch/en/doctorate.html The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. www.zgsm.ch/index.php?id=260&type=2 WARNING: Do not mistake ECTS credits for credit points for doctoral studies!
Graduate School Official website of the Zurich Graduate School in Mathematics: www.zurich-graduate-school-math.ch
401-5002-18L	A_∞ Structures and Moduli Spaces	W	0 credits	2V	A. Polishchuk
Abstract	Nachdiplom lecture
Learning objective
Content	The concept of an A_∞ algebra, originally motivated by homotopy theory (as a more flexible version of Massey products), more recently features in symplectic geometry and algebraic geometry, due to groundbreaking ideas of Fukaya and Kontsevich's homological mirror symmetry program. In my lectures I will start with basics of A_∞ algebras. In particular, I will discuss their deformation theory and explain how to construct A_∞ enhancements of derived categories using homological perturbations. I will then consider some examples arising from algebraic geometry. From the point of view of establishing equivalences of A_∞ algebras, needed for homological mirror symmetry, it is important to study all possible A_∞ structures extending a given graded associative algebra. I will introduce the corresponding moduli problem and will show that in some cases there exists a fine moduli space parametrizing A_∞ structures. I will consider in detail examples of moduli spaces of A_∞ structures related to moduli spaces of curves.
401-5004-18L	Hyperbolic Flows	W	0 credits	2V	B. Hasselblatt
Abstract	Nachdiplom lecture
Learning objective
Content	The first nontrivial example of an ergodic mechanical system was the motion of a free particle in a negatively curved space. The underlying mechanism is hyperbolicity. Smooth ergodic theory has since broadened its scope well beyond uniformly hyperbolic systems, but these provide rich context for investigations connecting dynamical systems to geometry and topology. The lectures present a panorama of such work for uniformly hyperbolic flows. The course begins with a brisk introduction to topological dynamics and ergodic theory of flows and the study of these both for hyperbolic flows. The emphasis will be on continuous-time systems and sometimes their interplay with discrete-time systems. The first core component of the course then surveys a broad array of mathematics centered on the invariant foliations central to hyperbolic behavior. This centers on their regularity as well as smooth and geometric rigidity. This has a rich interface with geometry. The second core component studies constructions of Anosov flows, some of them quite recent, which showcases deep interactions of dynamics with low-dimensional topology.
401-5006-18L	Dependence, Risk Bounds and Optimal Portfolios	W	0 credits	2V	L. Rüschendorf
Abstract	Nachdiplom lecture
Learning objective
Content	In this lecture series a main focus is on the description of the influence of dependence on price or risk functionals in multivariate or continuous time stochastic models. In particular we are interested in the description of the impact of dependence information on the formulation of risk bounds, on the range of portfolio risk measures and on the size of pricing intervals. On the other hand dependence is a useful tool for the construction of optimal claims and portfolios. We discuss applications in the frame of Lévy and more general semi-martingale models. We will point out general methodological tools for dependence modeling and analysis. In particular we discuss extensions of the classical Hoeffding-Fréchet bounds, the use of stochastic dependence orderings and describe the development and wide range of applications of results from mass transportation, which is a main instrument for this kind of problems.
401-4356-18L	Geometric Wave Equations	W	8 credits	4G	M. Struwe
Abstract	We study nonlinear wave equations arising in mathematical physics and geometry, including nonlinear field equations and wave maps.
Learning objective	The aim of the course is to introduce various methods for studying nonlinear wave equations, including energy methods, Strichartz estimates, and optimal gauges.
Content	We study nonlinear wave equations arising in mathematical physics and geometry, including nonlinear field equations and wave maps.
Literature	Shatah, Jalal; Struwe, Michael: Geometric wave equations. Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998.
Prerequisites / Notice	Measure theory, Sobolev spaces, basic functional analysis and pde.
401-3376-18L	Homogeneous Dynamics II	W	6 credits	3V	M. Einsiedler, M. Akka Ginosar, Ç. Sert
Abstract
Learning objective
Content	We will continue developing homogeneous dynamics with the aim to see several cases of Ratner’s theorems, further measure rigidity results for diagonalisable flows (Einsiedler-Katok-Lindenstrauss) and measure rigidity results for random walks on homogeneous spaces (Benoit-Quint). For this some more tools that were not developed in the first semester have to be introduced along the way, in particular conditional measures, entropy theory, and leafwise measures. We will also discuss a few of the applications of these theorems to Diophantine approximation or other areas of number theory.
Prerequisites / Notice	Doctoral students may receive 3 credits for the course and should be able to follow even without the first semester. For receiving the credits a presentation is required.
401-3002-12L	Algebraic Topology II	W	8 credits	4G	W. Merry
Abstract	This is a continuation course to Algebraic Topology I. Topics covered include: - Universal coefficients, - The Eilenberg-Zilber Theorem and the Künneth Formula), - The cohomology ring, - Fibre bundles, the Leray-Hirsch Theorem, and the Gysin sequence, - Topological manifolds and Poincaré duality, - Higher homotopy groups and fibrations.
Learning objective
Lecture notes	I will produce full lecture notes, available on my website at www.merry.io/algebraic-topology
Literature	"Algebraic Topology" (CUP, 2002) by Hatcher is excellent and covers all the material from both Algebraic Topology I and Algebraic Topology II. You can also download it (legally!) for free from Hatcher's webpage: www.math.cornell.edu/%7ehatcher/AT/ATpage.html Another classic book is Spanier's "Algebraic Topology" (Springer, 1963). This book is very dense and somewhat old-fashioned, but again covers everything you could possibly want to know on the subject.
Prerequisites / Notice	Familiarity with all the material from Algebraic Topology I will be assumed (the fundamental group, singular homology, cell complexes, the Eilenberg-Steenrod axioms, the basics of homological algebra and category theory). Full lecture notes for Algebraic Topology I can be found on my webpage.
401-3226-00L	Symmetric Spaces	W	8 credits	4G	A. Iozzi
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).
Learning objective	Learn the basics of symmetric spaces
401-4226-18L	Spectral theory of Eisenstein series	W	6 credits	2V + 1U	P. D. Nelson
Abstract	We plan to discuss the basic theory of Eisenstein series and the spectral decomposition of the space of automorphic forms, with focus on the groups GL(2) and GL(n).
Learning objective
Prerequisites / Notice	Some familiarity with basics on Lie groups and functional analysis would be helpful, and some prior exposure to modular forms or homogeneous spaces may provide useful motivation.
401-3104-18L	Diophantine Equations and Hilbert's 10th Problem Does not take place this semester.	W	8 credits	4G	E. Kowalski
Abstract	The course presents the solution by Davis, Putnam, Robinson and Matijasevich of Hilbert's Tenth Problem, concerning the non-existence of an algorithm to determine the solubility of a general diophantine equation. All necessary ingredients from logic and number theory will be covered in the class. We will then discuss similar questions in other contexts, such as the word problem in group theory.
Learning objective
Content	The course will present in full details the solution by Davis, Putnam, Robinson and Matijasevich of Hilbert's Tenth Problem, concerning the non-existence of an algorithm to determine the solubility in integers of a general diophantine equation. All necessary ingredients from logic, computability theory and number theory will be covered in the class. If time allows, we will then discuss some similar questions in other contexts, for instance the word problem in group theory.
Literature	J. Robinson, Collected Works, especially papers 9, 10, 19, 24. M. Davis, "Hilbert's Tenth Problem is Unsolvable", The American Mathematical Monthly, Vol. 80, 233-269. J. Rotman, "An introduction to the theory of groups", chapter 12 (Springer, 1995).
Prerequisites / Notice	Prerequisites: Algebra I+II
401-3532-08L	Differential Geometry II	W	10 credits	4V + 1U	D. A. Salamon
Abstract	Introduction to Differential Topology, including degree theory and intersection theory; Differential forms, including deRham cohomology and Poincare duality; Vector bundles, including Thom isomorphism and Euler number.
Learning objective	The aim of this course is to give an introduction to Differential Topology including the degree of a mapping and intersection theory, differential forms including deRham cohomology and Poincare duality, and vector bundles including the Thom isomorphism theorem.
Content	Introduction to Differential Topology, including the mod-2 degree, orientation and the Brouwer degree, Poincare-Hopf Theorem, the Pontryagin construction, Hopf Degree Theorem., intersection theory, Lefschetz numbers; Differential forms, Stokes, Cartan's formula, deRham cohomology, Mayer-Vietoris, Poincare duality, Euler characteristic, Degree Theorem, Gauss-Bonnet, Moser isotopy, Cech-DeRham complex and finite-dimensionality; Vector bundles, Thom isomorphism, Euler number.
Literature	- J. Milnor, Topology from the Differential Viewpoint. Univ Virginia Press, 1969. - V. Guillemin, A. Pollack, Differential Topology. Prentice-Hall, 1974. - R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. - J. Robbin, D. Salamon, Introduction to Differential Topology, in preparation. https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf
Prerequisites / Notice	Prerequisite is a working knowledge of the introductory material in Differential Geometry I, including smooth manifolds, tangent bundles, vector fields and flows. see https://people.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf
401-3534-18L	Complex Singularities and Picard-Lefschetz Theory	W	6 credits	3V	P. Biran
Abstract	We will mainly concentrate on isolated singularities of complex hypersurfaces. Topics covered include: invariants of complex singularities, topological aspects of complex singularities, the Milnor fibration, vanishing cycles, monodromies, Picard-Lefschetz theory. If time permits we will also go into the symplectic topology counterpart of the story.
Learning objective	This course is dedicated to the topology and geometry of singularities of complex varieties, concentrating mainly on isolated singularities of complex hypersurfaces and Picard-Lefschetz theory.
Literature	1) J. Milnor, "Singular points of complex hypersurfaces". Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, N.J. 2) A. Dimca, "Singularities and topology of hypersurfaces". Universitext. Springer-Verlag, New York, 1992. xvi+263 pp. ISBN: 0-387-97709-0 3) V. Arnold, V. Goryunov, O. Lyashko, V. Vasilʹev, "Singularity theory. I". Springer-Verlag, Berlin, 1998. iv+245 pp. ISBN: 3-540-63711-7 4) V. Arnold, S. Gusein-Zade, A. Varchenko, "Singularities of differentiable maps. Volume 1. Classification of critical points, caustics and wave fronts". Birkhäuser/Springer, New York, 2012. xii+382 pp. ISBN: 978-0-8176-8339-9 5) V. Arnold, S. Gusein-Zade, A. Varchenko, "Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals". Monographs in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1988. viii+492 pp. ISBN: 0-8176-3185-2 6) E. Looijenga, "Isolated singular points on complete intersections". London Mathematical Society Lecture Note Series, 77. Cambridge University Press, Cambridge, 1984. xi+200 pp. ISBN: 0-521-28674-3
Prerequisites / Notice	Complex analysis, differential geometry, algebraic topology. Basic knowledge of Morse theory and of symplectic geometry are useful but not absolutely necessary.
401-3462-00L	Functional Analysis II	W	10 credits	4V + 1U	A. Carlotto
Abstract	Fundamentals of the theory of distributions, Sobolev spaces, weak solutions of elliptic boundary value problems (solvability results both via linear methods and via direct variational methods), elliptic regularity theory, Schauder estimates, selected applications coming from physics and differential geometry.
Learning objective	Acquiring the language and methods of the theory of distributions in order to study differential operators and their fundamental solutions; mastering the notion of weak solutions of elliptic problems both for scalar and vector-valued maps, proving existence of weak solutions in various contexts and under various classes of assumptions; learning the basic tools and ideas of elliptic regularity theory and gaining the ability to apply these methods in important instances of contemporary mathematics.
Lecture notes	Lecture notes "Funktionalanalysis II" by Michael Struwe.
Literature	Useful references for the course are the following textbooks: Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011. Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003.
Prerequisites / Notice	Functional Analysis I plus a solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces).
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: https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html	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.
Learning 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"
401-4788-16L	Mathematics of (Super-Resolution) Biomedical Imaging	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.
Learning 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.
401-3618-18L	A Mathematical Introduction to Machine Learning Approximation Algorithms	W	4 credits	3G	A. Jentzen
Abstract	This course provides a mathematical introduction to some machine learning approximation algorithms. This course includes content (i) on convergence proofs and implementations for stochastic gradient descent (SGD) optimization algorithms, (ii) on artificial neural networks, (iii) on deep learning, and (iv) on some applications of such algorithms and such concepts.
Learning objective	The aim of this course is to teach the students a basic knowledge on stochastic gradient descent (SGD) optimization algorithms, on artificial neural networks, on the analytical concepts used to study such algorithms and tools, and on some applications of such algorithms and tools.
Content	This course includes content (i) on convergence proofs and implementations for stochastic gradient descent (SGD) optimization algorithms, (ii) on artificial neural networks, (iii) on deep learning, and (iv) on some applications of such algorithms and concepts.
Lecture notes	Lecture notes are available as a PDF file: see Learning materials. Please contact Timo Welti (timo.welti@sam.math.ethz.ch) in case you are enrolled in the lecture but do not have access to the lecture notes.
401-4605-18L	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.
Learning 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.
401-4607-59L	Percolation Theory	W	4 credits	2V	V. Tassion
Abstract	An introduction to the percolation theory.
Learning objective	The objective is to gain familiarity with the methods of the percolation theory and to learn some of its important results.
Content	Definition of percolation, FKG and BK inequalities, Harris-Kesten Theorem, Menshikov's Theorem, uniqueness of the infinite cluster and possibly Smirnov's Theorem on the conformal invariance of the critical percolation.
Literature	B. Bollobas, O. Riordan: Percolation, CUP 2006 G. Grimmett: Percolation 2ed, Springer 1999
Prerequisites / Notice	Preliminaries: 401-2604-00L Probability and Statistics (mandatory) 401-3601-00L Probability Theory (recommended)
401-4632-15L	Causality	W	4 credits	2G	N. Meinshausen
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.
Learning 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.
Learning 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	Start of the lecture: WED, 28 Feb. 2018 (second week of the semester).
401-3629-00L	Quantitative Risk Management	W	4 credits	2V	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 and dependence structures as well as operational risk.
Learning 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 https://people.math.ethz.ch/~patrickc/qrm
Literature	Quantitative Risk Management: Concepts, Techniques and Tools AJ McNeil, R Frey and P Embrechts Princeton University Press, Princeton, 2015 (Revised Edition) http://press.princeton.edu/titles/10496.html
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.

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