Suchergebnis: Katalogdaten im Herbstsemester 2017

Mathematik Master Information
Wahlfächer
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Algebra, Topologie, diskrete Mathematik, Logik
NummerTitelTypECTSUmfangDozierende
401-3034-00LAxiomatische MengenlehreW8 KP3V + 1UL. Halbeisen
KurzbeschreibungEs werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Zudem werden Ultrafilter untersucht und es wird das Martinaxiom eingeführt.
Lernziel
InhaltEs werden ausführlich die Axiome der Mengenlehre besprochen und parallel dazu wird die Theorie der Ordinal- und Kardinalzahlen aufgebaut. Insbesondere wird die Kontinuumshypothese behandelt und einige Konsequenzen besprochen. Zudem werden Ultrafilter untersucht und die Existenz gewisser Ultrafilter diskutiert. Im letzten Teil der Vorlesung wird das Martin-Axiom eingeführt, mit dessen Hilfe sich interessante Konsistenzresultate in Topologie und Masstheorie, sowie Resultate über Ultrafilter, beweisen lassen.
SkriptIch werde mich weitgehend an mein Buch "Combinatorial Set Theory" (2nd ed., erscheint im Herbst 2017) halten.
Literatur"Combinatorial Set Theory: with a gentle introduction to forcing" (Springer-Verlag 2012)

http://www.springer.com/mathematics/book/978-1-4471-2172-5
401-3118-67LClassical Modular FormsW8 KP4GI. N. Petrow
Kurzbeschreibung
Lernziel
401-3129-67LDifferential Galois TheoryW4 KP2VP. S. Jossen
KurzbeschreibungAlgebraic theory of linear differential equations, Picard-Vessiot theory, Differential Galois groups, local theory of differential equations, the Frobenius method, Newton polygons, Connections and local systems, Riemann-Hilbert correspondence on ℙ¹.
LernzielWe introduce differential Galois theory and mathematical concepts surrounding it. We formulate and prove an important case of the Riemann-Hilbert correspondence.
InhaltWe study linear differential equations from an algebraic perspective, introducing differential rings, fields and differential modules (so-called Picard-Vessiot theory), and very soon the Galois group of a differential equation. We relate then the algebraic theory with the analytic theory, which leads us to the classical Riemann-Hilbert correspondence. In particular we will prove that differential equations on the complex projective line ℙ¹ with regular singularities in a finite set S correspond to representations of the fundamental group of ℙ¹∖S. If time permits, we have a look at differential equations in positive characteristic.
LiteraturM. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. Vol 328, Springer 2003
401-3203-67LSmall Cancellation TheoryW4 KP2VD. Gruber
KurzbeschreibungSmall cancellation theory studies groups given by presentations in which defining relations have small common subwords. By translating group theoretic questions into geometric objects and applying concepts of negative curvature, it produces a variety of theorems on infinite groups. We will give an introduction to the theory, discuss important results, and touch on more recent developments.
LernzielFamiliarity with the fundamental methods of small cancellation theory and its main applications; ability to apply the methods to create new examples of infinite groups with prescribed properties; basic understanding of connections with Gromov hyperbolicity.
InhaltWe plan to cover a selection (depending on time) of the following topics:
- Methods of classical small cancellation theory (e.g. van Kampen diagrams, van Kampen's lemma, Greendlinger's lemma)
- Fundamental properties of small cancellation groups (e.g. Torsion Theorem, asphericity, linear/quadratic Dehn function)
- Connections with algorithmic decision problems in groups (e.g. Dehn's algorithm for solving the word problem in surface groups, solvability of word and conjugacy problems in small cancellation groups)
- Easy examples of small cancellation monsters (e.g. Pride's example, Rips construction)
- Graphical generalization of small cancellation theory and applications (e.g. groups with expander graphs embedded in their Cayley graphs)
- Connections with Gromov hyperbolicity
LiteraturV. Guirardel, Geometric small cancellation. Geometric group theory, 55-90, IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, RI, 2014.

R. C. Lyndon, P. E. Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ISBN: 3-540-41158-5.

A. Yu. Olshanskii, Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. ISBN: 0-7923-1394-1.

R. Strebel, Appendix. Small cancellation groups. In: Sur les groupes hyperbolic d'après Mikhael Gromov (Bern, 1988), 227-273, Progr. Math. 83, Birkhäuser Boston, Boston, MA, 1990.
Voraussetzungen / BesonderesFamiliarity with very basic notions of group theory, definitions of free groups, group presentations, and graphs.
401-3177-67LIntroduction to Vertex Operator Algebras Information W4 KP2VC. A. Keller
KurzbeschreibungA first introduction to the theory of vertex operator algebras.
LernzielUnderstand the basic concepts of vertex operator algebras and their most important examples.
InhaltTentative plan:

1) Formal power series, local fields
2) Vertex Algebras
3) Conformal symmetry
4) Vertex Operator Algebras
5) Correlation functions
6) VOAs from lattices
7) Connection to modular forms: Zhu's Theorem
8) Connection to Monstrous Moonshine
LiteraturVictor Kac: Vertex Algebras for Beginners

James Lepowksy, Haisheng Li: Introduction to Vertex Operator Algebras and Their Representations
Voraussetzungen / BesonderesBasic algebra and linear algebra. Some background in quantum mechanics is helpful, but not necessary.
401-3059-00LKombinatorik IIW4 KP2GN. Hungerbühler
KurzbeschreibungDer Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende Kombinatorik.
LernzielDie Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
InhaltInhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.
Auswahl: Geometrie
NummerTitelTypECTSUmfangDozierende
401-3375-67LHomogeneous Dynamics and ApplicationsW8 KP4GM. Einsiedler, M. Akka Ginosar, Ç. Sert
KurzbeschreibungThe aim is to reach a few of the applications of homogeneous dynamics to number theory, e.g. counting results concerning quadratic forms, but also develop the theory from scratch. The first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume.
LernzielThe aim is to reach a few of the applications of homogeneous dynamics to number theory, e.g. counting results concerning quadratic forms, but also develop the theory from scratch. The first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume.
InhaltThe first part of the course will be based on the book "Ergodic Theory with a view towards number theory" by Einsiedler and Ward, but several topics go beyond this volume. Some of the aims of the course are:
-) Pointwise ergodic theorem for a certain class of amenable groups
-) Dynamics on hyperbolic surfaces, equidistribution of periodic horocycle orbits
-) Applications to counting
-) Some cases of Ratner theorems in Unipotent dynamics


Course website: https://metaphor.ethz.ch/x/2017/hs/401-3375-67L/
401-3301-67LIntroduction to Hyperbolic GeometryW4 KP2VQ. Chen
KurzbeschreibungHyperbolic geometry and ideal tetrahedra, decomposition of the Figure-8 knot, The Rigidity Theorem (Compact Case), hyperbolic structures on knot complements, the Space of Hyperbolic Manifolds and the Volume Function
Lernziel
Literatur"Low dimensional geometry: from euclidean surfaces to hyperbolic knots" by Bonahon
"Hyperbolic Knot Theory" by Purcell
"Lectures on Hyperbolic Geometry" by Benedetti and Petronio
"The Geometry and Topology of Three-Manifolds" by Thurston
401-3057-00LEndliche Geometrien II
Findet dieses Semester nicht statt.
W4 KP2GN. Hungerbühler
KurzbeschreibungEndliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate.
LernzielEndliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne.
InhaltEndliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne
Literatur- 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.
Auswahl: Analysis
NummerTitelTypECTSUmfangDozierende
401-4355-67LNonlinear Evolution ProblemsW8 KP4GM. Struwe
KurzbeschreibungWe discuss short-time and global existence, regularity, and uniqueness of solutions to various evolution problems in geometric analysis and physics, including the harmonic map heat flow, Yamabe flow, and the Navier-Stokes equations. In some cases, solutions or their derivatives may blow up in finite or infinite time, and we analyze the structure of these singularities.
LernzielMethods for proving local existence for the Cauchy problem for nonlinear parabolic equations, use of structural properties to establish global existence, blow-up techniques for the analysis of singularities.
InhaltWe discuss short-time and global existence, regularity, and uniqueness of solutions to various evolution problems in geometric analysis and physics, including the harmonic map heat flow, Yamabe flow, and the Navier-Stokes equations. In some cases, solutions or their derivatives may blow up in finite or infinite time, and we analyze the structure of these singularities.
Voraussetzungen / BesonderesPrerequisites are a working knowledge of Sobolev spaces, functional analysis, and measure theory.
401-4589-63LCalculus of Variations and Conformal InvarianceW6 KP3VT. Rivière
KurzbeschreibungIn this course we will present the classical theory as well as more recent developments of the calculus of variation of surfaces. We will expose method mixing functional analysis and differential geometry in order to produce and describe global and local minimizers or saddle points to two dimensional Lagrangians.
Lernziel
InhaltIn the first part of the class we shall consider the area functional whose critical points are minimal surfaces and study the so called Plateau problem. Introduced originally by Lagrange in the 18th century. Then we will move to the systematic study of 2-dimensional conformally invariant Lagrangians and explain how they are all related to a generalized Plateau problem of prescribed mean curvature surfaces into submanifolds. In the last part of the class we will present a theory merging minimal surface theory and conformal invariance. This theory has been introduced in the early 20th century by Wilhelm Blaschke and is presently a very active field of research in geometric analysis due in particular to numerous applications in many fields of sciences such as general relativity, elasticity theory, cell biology etc.
Voraussetzungen / BesonderesRequirements:
Fundamental knowledge in functional analysis, Fourier analysis and differential geometry (FAI and DGI)
Auswahl: Weitere Gebiete
NummerTitelTypECTSUmfangDozierende
401-3502-66LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W2 KP4AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-66LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W3 KP6AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-66LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <studiensekretariat@math.ethz.ch> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W4 KP9AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
Wahlfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel:
Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Auswahl: Numerische Mathematik
NummerTitelTypECTSUmfangDozierende
401-4657-00LNumerical Analysis of Stochastic Ordinary Differential Equations Information
Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods"
W6 KP3V + 1UA. Jentzen
KurzbeschreibungCourse on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables.
LernzielThe aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues.
InhaltGeneration of random numbers
Monte Carlo methods for the numerical integration of random variables
Stochastic processes and Brownian motion
Stochastic ordinary differential equations (SODEs)
Numerical approximations of SODEs
Multilevel Monte Carlo methods for SODEs
Applications to computational finance: Option valuation
SkriptLecture Notes are available in the lecture homepage (please follow the link in the Learning materials section).
LiteraturP. Glassermann:
Monte Carlo Methods in Financial Engineering.
Springer-Verlag, New York, 2004.

P. E. Kloeden and E. Platen:
Numerical Solution of Stochastic Differential Equations.
Springer-Verlag, Berlin, 1992.
Voraussetzungen / BesonderesPrerequisites:

Mandatory: Probability and measure theory,
basic numerical analysis and
basics of MATLAB programming.

a) mandatory courses:
Elementary Probability,
Probability Theory I.

b) recommended courses:
Stochastic Processes.

Start of lectures: Wednesday, September 20, 2017

Date of the End-of-Semester examination: Wednesday, December 20, 2017, 13:00-15:00; students must arrive before 12:30 at ETH HG E 19.
Room for the End-of-Semester examination: ETH HG E 19.

Exam inspection: Monday, March 5, 2018,
13:00-14:00 at HG D 5.1
Please bring your legi.
401-4785-00LMathematical and Computational Methods in PhotonicsW8 KP4GH. Ammari
KurzbeschreibungThe aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces
LernzielThe field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications.

The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength.

Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures.

The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions.

In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials.
401-4645-67LNumerics for Computational Uncertainty Quantification Information W10 KP3V + 2UC. Schwab
KurzbeschreibungThe course presents the mathematical foundation of various numerical methods
for the efficient quantification of uncertainty in partial differential equations.
Mathematical foundations include high dimensional polynomial approximation,
sparse grid approximations, generalized polynomial chaos expansions and their
summability properties, as well the computer implementation in model problems.
LernzielThe course will provide a survey of the mathematical properties and
the computational realization of the most widely used numerical methods
for uncertainy quantification in PDEs from engineering and the sciences.
In particular, Monte-Carlo, Quasi-Monte Carlo and their multilevel extensions
for PDEs, Sparse grid and Smolyak approximations, stochastic collocation
and Galerkin discretizations will be discussed.
SkriptThere will be typed lecture notes.
LiteraturLecture Notes.
Voraussetzungen / BesonderesCompleted BSc MATH or equivalent.
Auswahl: Wahrscheinlichkeitstheorie, Statistik
NummerTitelTypECTSUmfangDozierende
401-4607-67LSchramm-Loewner EvolutionsW4 KP2VW. Werner
KurzbeschreibungThis course will be an introduction to Schramm-Loewner Evolutions which are natural random planar curve that arise in a number of contexts in probability theory and statistical theory in two dimensions.
LernzielThe goal of the course is to provide an overview of the definition and the main properties of Schramm-Loewner Evolutions (SLE).
InhaltMost of the following items will be covered in the lectures:
- Introduction to SLE
- Definition of SLE
- Phases of SLE, hitting probabilities
- How does one prove that an SLE is actually a curve?
- Restriction, locality
- Relation to loop-soups and the Gaussian Free Field
- Some SLEs as scaling limit of lattice models
401-4597-67LProbability on Transitive GraphsW4 KP2VV. Tassion
KurzbeschreibungIn this course, we will present modern topics at the interface between probability and geometric group theory. We will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group.
LernzielPresent in an original framework important tools in the study of
- random walks: spectral gap, harmonic functions, entropy,...
- percolation: uniqueness of the infinite cluster, mass-transport principle,...
InhaltIn this course, we will present modern topics at the interface between probability and geometric group theory. To every group with a finite generating set, one can associate a graph, called Cayley graph. (For example, the d-dimensional grid is a Cayley graph associated to the group Z^d.) Then, we will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group. The focus will be on the random processes and their properties, and we will use very few notions of geometric group theory.
LiteraturProbability on trees and network (R. Lyons, Y. Peres)
Voraussetzungen / Besonderes- Probability Theory
- No prerequisite on group theory, all the background will be introduced in class.
401-4619-67LAdvanced Topics in Computational StatisticsW4 KP2VN. Meinshausen
KurzbeschreibungThis lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.
LernzielStudents learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.
InhaltThe main focus will be on graphical models in various forms:
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
Voraussetzungen / BesonderesWe assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.
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