Search result: Catalogue data in Spring Semester 2020

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
401-3146-12LAlgebraic Geometry Information W10 credits4V + 1UD. Johnson
AbstractThis course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).
ObjectiveLearning Algebraic Geometry.
LiteraturePrimary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.

Secondary reference:
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013).

Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry, http://math.stanford.edu/~vakil/216blog/
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html

"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/CourseNotes/AG.pdf

Further readings:
* Günter Harder: Algebraic Geometry 1 & 2
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Prerequisites / NoticeRequirement: Some knowledge of Commutative Algebra.
401-3002-12LAlgebraic Topology II Information W8 credits4GA. Sisto
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) 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

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

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 credits4GM. 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).
ObjectiveLearn the basics of symmetric spaces
401-3372-00LDynamical Systems IIW10 credits4V + 1UW. Merry
AbstractThis course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.
ObjectiveMastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.
ContentTopics covered include:

- Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem.
- Hyperbolic sets, Anosov diffeomorphisms.
- The (Un)stable Manifold Theorem.
- Shadowing Lemmas and stability.
- The Lambda Lemma.
- Transverse homoclinic points, horseshoes, and chaos.
- Complex dynamics of rational maps on the Riemann sphere
- Julia sets and Fatou sets.
- Fractals and the Mandelbrot set.
Lecture notesI will provide full lecture notes, available here:

https://www.merry.io/courses/dynamical-systems/
LiteratureThe most useful textbook is

- Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
Prerequisites / NoticeIt will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:

https://www.merry.io/courses/dynamical-systems/

However we will only really use material covered in the first 10 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 10 lectures.

In addition, it would be useful to have some familiarity with basic differential geometry and complex analysis.
401-3532-08LDifferential Geometry II Information W10 credits4V + 1UU. Lang
AbstractIntroduction 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.
ObjectiveLearn 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 / NoticePrerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms.
401-3462-00LFunctional Analysis II Information W10 credits4V + 1UM. Struwe
AbstractSobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity
ObjectiveAcquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates
Lecture notesFunktionalanalysis II, Michael Struwe
LiteratureFunktionalanalysis II, Michael Struwe

Functional Analysis, Spectral Theory and Applications.
Manfred Einsiedler and Thomas Ward, GTM Springer 2017
Prerequisites / NoticeFunctional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.
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. H. Maathuis
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
Does not take place this semester.
W8 credits3V + 1Unot available
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 (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
W10 credits4V + 2UUniversity lecturers
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
401-3201-00LAlgebraic Groups Information W8 credits4GP. D. Nelson
AbstractIntroduction to the theory of linear algebraic groups. Lie algebras, the Jordan Chevalley decomposition, semisimple and reductive groups, root systems, Borel subgroups, classification of reductive groups and their representations.
Objective
LiteratureA. L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups
Prerequisites / NoticeAbstract algebra: groups, rings, fields, tensor product, etc.

Some familiarity with the basics of Lie groups and their Lie algebras would be helpful, but is not absolutely necessary.

We will develop what we need from algebraic geometry, without assuming prior knowledge.
401-3109-65LProbabilistic Number Theory Information
Does not take place this semester.
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; 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.
401-3202-09LThe Representation Theory of the Finite Symmetric Groups Information
NOTICE: No physical class for the next few weeks until further notice. Instead a video recording will be offered.
W4 credits2VL. Wu
AbstractThis course is an Introduction to the Representation Theory of the Groups.
ObjectiveOur goal is to give an introduction of the Representation Theory using the examples of the Finite Symmetry Groups.
Literature* Jean-Pierre Serre: Linear Representations of Finite Groups, Graduate Texts in Mathematics, Springer.

* William Fulton and Joe Harris: Representation Theory A First Course, Graduate Texts in Mathematics, Springer.

* G. D. James: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, Springer.

* Bruce E. Sagan: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics, Springer.
Prerequisites / NoticeSome basic knowledge of the Group Theory and Linear Algebra.
401-8112-20LGeometry of Numbers (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT548

Mind the enrolment deadlines at UZH:
https://www.uzh.ch/cmsssl/en/studies/application/mobilitaet.html
W9 credits4V + 1UUniversity lecturers
AbstractThe Geometry of Numbers studies distribution of lattice points in the n dimensional space, for instance, existence of lattice points in various domains and existence of integral solutions of polynomial inequalities.
This subject is also closely related to the Theory of Diophantine Approximation, which seeks good rational approximations for real vectors.
ObjectiveLearn basic techniques in the Geometry of Numbers
Literature1. Cassels, An introduction to Diophantine Approximation
2. Cassels, An introduction to the Geometry of Numbers
3. Schmidt, Diophantine approximation
4. Siegel, Lectures on the Geometry of Numbers
401-3058-00LCombinatorics I
Does not take place this semester.
W4 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
401-3556-20LTopics in Symplectic TopologyW6 credits3VP. Biran
AbstractThis will be an introductory course in symplectic geometry and topology.
We will cover the simplest instances of symplectic rigidity phenomena, and techniques to detect and study them. The last part of the course will be devoted to more advanced techniques such as Floer theory.
ObjectiveGet acquainted with the basics of symplectic topology and phenomena
of symplectic rigidity.
Literature1) Book: "Introduction to Symplectic Topology", 3'rd edition, by McDuff and Salamon.
Oxford Graduate Texts in Mathematics

2) Some published articles that will be announced during the semester.
401-3056-00LFinite Geometries IW4 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.
401-4532-20LIntroduction to 3-ManifoldsW4 credits2VM. Nagel
AbstractThis course provides an introduction to the basic notions and tools of geometric topology with a special focus on three dimensional manifolds.
ObjectiveIn this course, we become familiar with the basic notions and tools of geometric topology, which concerns low-dimensional manifolds and their embeddings. We will focus on 3–dimensional manifolds. While this class of manifolds is very rich, it still allows for many structural results.
An important goal of the lecture is to learn how to manipulate these manifolds: build them from simple pieces, cut them apart, isotope and simplify submanifolds etc. These techniques from differential topology are combined with invariants from algebraic topology, which are incredibly powerful in encoding properties of a 3–manifold. We discuss applications, which give new intuition for these invariants, and answer many questions about manifolds of dimension three or less.
There are many synergies with Algebraic Topology II, which I encourage you to take in parallel.
ContentBackground in differential topology
Foundational results on the topology of 3–manifolds
Knots and concordance
LiteratureKnots and links by D. Rolfsen
3–Manifolds by J. Hempel
Differential topology by T. Bröcker and K. Jänich
Prerequisites / NoticeAlgebraic Topology I
Differential Geometry I
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.
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