Search result: Catalogue data in Spring Semester 2021

Mathematics Master Information
Core Courses
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3002-12LAlgebraic Topology II Information W8 credits4GP. Biran
AbstractThis 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
Literature1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

The book can be downloaded for free at:
http://www.math.cornell.edu/~hatcher/AT/ATpage.html


3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeGeneral 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.
401-3226-00LSymmetric Spaces Information W8 credits4GA. 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).
ObjectiveLearn the basics of symmetric spaces
401-3532-08LDifferential Geometry IIW10 credits4V + 1UW. Merry
AbstractThis is a continuation course of Differential Geometry I.

Topics covered include:

- Connections and curvature,
- Riemannian geometry,
- Gauge theory and Chern-Weil theory.
Objective
Lecture notesI will produce full lecture notes, available on my website:

https://www.merry.io/courses/differential-geometry/
LiteratureThere are many excellent textbooks on differential geometry.

A friendly and readable book that contains everything covered in Differential Geometry I is:

John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.

For Differential Geometry II, the textbooks:

- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,

are both excellent. The monograph

- A. L. Besse "Einstein Manifolds", (1987), Springer,

gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.)
Prerequisites / NoticeFamiliarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). Lecture notes for Differential Geometry I can be found on my website.
401-3462-00LFunctional Analysis II Information W10 credits4V + 1UA. Carlotto
AbstractSobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles.
ObjectiveAcquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods for studying elliptic boundary value problems.
LiteratureMichael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018.

David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, 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, Berlin, 2003.
Prerequisites / NoticeFunctional Analysis I plus a solid background in measure theory, Lebesgue integration and L^p spaces.
401-8142-21LAlgebraic Geometry II (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT517

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html
W9 credits4V + 1UUniversity lecturers
AbstractWe continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc.
Objective
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
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NumberTitleTypeECTSHoursLecturers
401-3052-10LGraph Theory Information W10 credits4V + 1UB. Sudakov
AbstractBasics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem
ObjectiveThe students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.
Lecture notesLecture will be only at the blackboard.
LiteratureWest, D.: "Introduction to Graph Theory"
Diestel, R.: "Graph Theory"

Further literature links will be provided in the lecture.
Prerequisites / NoticeStudents are expected to have a mathematical background and should be able to write rigorous proofs.
401-3642-00LBrownian Motion and Stochastic Calculus Information W10 credits4V + 1UW. Werner
AbstractThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
ObjectiveThis course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.
Lecture notesLecture notes will be distributed in class.
Literature- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).
- D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
Prerequisites / NoticeFamiliarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).
401-3632-00LComputational StatisticsW8 credits3V + 1UM. Mächler
AbstractWe discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.
ObjectiveThe student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.
ContentSee the class website
Prerequisites / NoticeAt least one semester of (basic) probability and statistics.

Programming experience is helpful but not required.
401-3602-00LApplied Stochastic Processes Information W8 credits3V + 1UV. Tassion
AbstractPoisson processes; renewal processes; Markov chains in discrete and in continuous time; some applications.
ObjectiveStochastic processes are a way to describe and study the behaviour of systems that evolve in some random way. In this course, the evolution will be with respect to a scalar parameter interpreted as time, so that we discuss the temporal evolution of the system. We present several classes of stochastic processes, analyse their properties and behaviour and show by some examples how they can be used. The main emphasis is on theory; in that sense, "applied" should be understood to mean "applicable".
LiteratureR. N. Bhattacharya and E. C. Waymire, "Stochastic Processes with Applications", SIAM (2009), available online: http://epubs.siam.org/doi/book/10.1137/1.9780898718997
R. Durrett, "Essentials of Stochastic Processes", Springer (2012), available online: http://link.springer.com/book/10.1007/978-1-4614-3615-7/page/1
M. Lefebvre, "Applied Stochastic Processes", Springer (2007), available online: http://link.springer.com/book/10.1007/978-0-387-48976-6/page/1
S. I. Resnick, "Adventures in Stochastic Processes", Birkhäuser (2005)
Prerequisites / NoticePrerequisites are familiarity with (measure-theoretic) probability theory as it is treated in the course "Probability Theory" (401-3601-00L).
401-3652-00LNumerical Methods for Hyperbolic Partial Differential Equations Information W10 credits4V + 1UA. Ruf
AbstractThis 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.
ObjectiveThe 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 notesLecture slides will be made available to participants. However, additional material might be covered in the course.
LiteratureH. 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 / NoticeHaving 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"
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-4116-12LLectures on Drinfeld Modules Information W6 credits3VR. Pink
AbstractDrinfeld modules: Basic theory, analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations, endomorphism rings, etc.
Objective
ContentA central role in the arithmetic of fields of positive characteristic p is played by the Frobenius map x ---> x^p. The theory of Drinfeld modules exploits this map in a systematic fashion. Drinfeld modules of rank 1 can be viewed as analogues of the multiplicative group and are used in the class field theory of global function fields. Drinfeld modules of arbitrary rank possess a rich theory which has many aspects in common with that of elliptic curves, including analytic uniformization, moduli spaces, good/bad/semistable reduction, Tate modules, Galois representations.

A full understanding of Drinfeld modules requires some knowledge in the arithmetic of function fields and, for comparison, the arithmetic of elliptic curves, which cannot all be presented in the framework of this course. Relevant results from these areas will be presented only cursorily when they are needed, but a fair amount of the theory can be developed without them.
LiteratureDrinfeld, V. G.: Elliptic modules (Russian), Mat. Sbornik 94 (1974), 594--627, translated in Math. USSR Sbornik 23 (1974), 561--592.

Deligne, P., Husemöller, D: Survey of Drinfeld modules, Contemp. Math. 67, 1987, 25-91.

Goss, D.: Basic structures in function field arithmetic. Springer-Verlag, 1996.

Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in Alden-Biesen, September 9¿14, 1996. Edited by E.-U. Gekeler, M. van der Put, M. Reversat and J. Van Geel. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.

Thakur, Dinesh S.: Function field arithmetic. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.

Further literature will be indicated during the course
401-3109-65LProbabilistic Number Theory Information W8 credits4GE. Kowalski
AbstractThe course presents some results of probabilistic number theory in a unified manner, including distribution properties of the number of prime divisors of integers, probabilistic properties of the zeta function and statistical distribution of exponential sums.
ObjectiveThe goal of the course is to present some results of probabilistic number theory in a unified manner.
ContentThe main concepts will be presented in parallel with the proof of a few main theorems:
(1) the Erdős-Wintner and Erdős-Kac theorems concerning the distribution of values of arithmetic functions;
(2) the distribution of values of the Riemann zeta function, including Selberg's central limit theorem for the Riemann zeta function on the critical line;
(3) the Chebychev bias for primes in arithmetic progressions;
(4) functional limit theorems for the paths of partial sums of families of exponential sums.
Lecture notesThe lecture notes for the class are available at

https://www.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf
Prerequisites / NoticePrerequisites: Complex analysis, measure and integral, and at least the basic language of probability theory (the main concepts, such as convergence in law, will be recalled).
Some knowledge of number theory is useful but the main results will also be summarized.
401-3362-21LSpectral Theory of Eisenstein Series Information W4 credits2VP. D. Nelson
AbstractWe 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).
Objective
Prerequisites / NoticeSome 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.
401-3058-00LCombinatorics IW4 credits2GN. Hungerbühler
AbstractThe course Combinatorics I and II is an introduction into the field of enumerative combinatorics.
ObjectiveUpon completion of the course, students are able to classify combinatorial problems and to apply adequate techniques to solve them.
ContentContents of the lectures Combinatorics I and II: congruence transformation of the plane, symmetry groups of geometric figures, Euler's function, Cayley graphs, formal power series, permutation groups, cycles, Bunside's lemma, cycle index, Polya's theorems, applications to graph theory and isomers.
Prerequisites / NoticeRecognition of credits as an elective course in the Mathematics Bachelor's or Master's Programmes is only possible if you have not received credits for the course unit 401-3052-00L Combinatorics (which was for the last time taught in the spring semester 2008).
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-4118-21LSpectral Theory of Hyperbolic Surfaces Information W4 credits2VC. Burrin
AbstractThe Laplacian plays a prominent role in many parts of mathematics. On a flat surface like the torus, understanding its spectrum is the topic of Fourier analysis, whose 19th century development allowed to solve the heat and wave equations. On the sphere, one studies spherical harmonics. In this course, we will study the spectrum of hyperbolic surfaces and its Maass forms (eigenfunctions).
ObjectiveWe will start from scratch, with an overview of hyperbolic geometry and harmonic analysis on the hyperbolic plane. The objectives are to prove the spectral theorem and Selberg's trace formula, and explore applications in geometry and number theory.
ContentTentative syllabus:
Hyperbolic geometry (the hyperbolic plane and Fuchsian groups)
Construction of arithmetic hyperbolic surfaces
Harmonic analysis on the hyperbolic plane
The spectral theorem
Selberg's trace formula
Applications in geometry (isoperimetric inequalities, geodesic length spectrum)
and number theory (links to the Riemann zeta function and Riemann hypothesis)

Possible further topics (if time permits):
Eisenstein series
Explicit constructions of Maass forms (after Maass)
A special case of the Jacquet-Langlands correspondence (after the exposition of Bergeron, see references)
LiteratureNicolas Bergeron, The Spectrum of Hyperbolic Surfaces, Springer Universitext 2011.
Armand Borel, Automorphic forms on SL(2,R), Cambridge University Press 1997.
Peter Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser 1992.
Henryk Iwaniec, Spectral methods of automorphic forms. Graduate studies in mathematics, AMS 2002.
Prerequisites / NoticeKnowledge of the material covered in the first two years of bachelor studies is assumed. Prior knowledge of differential geometry, functional analysis, or Riemann surfaces is not required.
401-4206-17LGroups Acting on Trees Information W6 credits3GB. Brück
AbstractAs a main theme, we will see how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure.
ObjectiveLearn basics of Bass-Serre theory; get to know concepts from geometric group theory.
ContentAs a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces.

This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field.

Topics that will be covered in the lecture include:
- Trees and their automorphisms
- Different characterisations of free groups
- Amalgamated products and HNN extensions
- Graphs of groups
- Kurosh's theorem on subgroups of free (amalgamated) products
LiteratureJ.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9

O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8

C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Prerequisites / NoticeBasic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary.
In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient.
401-3056-00LFinite Geometries I
Does not take place this semester.
W4 credits2GN. Hungerbühler
AbstractFinite geometries I, II: Finite geometries combine aspects of geometry, discrete mathematics and the algebra of finite fields. In particular, we will construct models of axioms of incidence and investigate closing theorems. Applications include test design in statistics, block design, and the construction of orthogonal Latin squares.
ObjectiveFinite geometries I, II: Students will be able to construct and analyse models of finite geometries. They are familiar with closing theorems of the axioms of incidence and are able to design statistical tests by using the theory of finite geometries. They are able to construct orthogonal Latin squares and know the basic elements of the theory of block design.
ContentFinite geometries I, II: finite fields, rings of polynomials, finite affine planes, axioms of incidence, Euler's thirty-six officers problem, design of statistical tests, orthogonal Latin squares, transformation of finite planes, closing theorems of Desargues and Pappus-Pascal, hierarchy of closing theorems, finite coordinate planes, division rings, finite projective planes, duality principle, finite Moebius planes, error correcting codes, block design
Literature- Max Jeger, Endliche Geometrien, ETH Skript 1988

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

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

- Dembowski: Finite Geometries.
401-3574-61LIntroduction to Knot Theory Information
Does not take place this semester.
W6 credits3G
AbstractIntroduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
ObjectiveThe aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
ContentDefinition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
LiteratureAn extensive bibliography will be handed out in the course.
Prerequisites / NoticePrerequisites are some elementary knowledge of algebra and topology.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4422-21LAn Introduction to the Calculus of VariationsW4 credits2VA. Figalli
AbstractCalculus of variations is a fundamental tool in mathematical analysis, used to investigate the existence, uniqueness, and properties of minimizers to variational problems.
Classic examples include, for instance, the existence of the shortest curve between two points, the equilibrium shape of an elastic membrane, and so on.
Objective
ContentIn the course, we will study both 1-dimensional and multi-dimensional problems.
Prerequisites / NoticeBasic knowledge of Sobolev spaces is important, so some extra additional readings would be required for those unfamiliar with the topic.
401-3378-19LEntropy in Dynamics Information W8 credits4GM. Einsiedler
AbstractDefinition and basic property of measure theoretic dynamical entropy (elementary and conditionally). Ergodic theorem for entropy. Topological entropy and variational principle. Measures of maximal entropy. Equidistribution of periodic points. Measure rigidity for commuting maps on the circle group.
ObjectiveThe course will lead to a firm understanding of measure theoretic dynamical entropy and its applications within dynamics. We will start with the basic properties of (conditional) entropy, relate it to the question of effective coding techniques, discuss and prove the Shannon-McMillan-Breiman theorem that is also known as the ergodic theorem for entropy. Moreover, we will discuss a topological counter part and relate this topological entropy to the measure theoretic entropy by the variational principle. We will use these methods to classify certain natural homogeneous measures, prove equidistribution of periodic points on compact quotients of hyperbolic surfaces, and establish a measure rigidity theorem for commuting maps on the circle group.
Lecture notesEntropy book under construction, available online under
https://tbward0.wixsite.com/books/entropy
Prerequisites / NoticeNo prior knowledge of dynamical systems will be assumed but measure theory will be assumed and very important. Doctoral students are welcome to attend the course for 2KP.
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