Search result: Catalogue data in Spring Semester 2020

Mathematics Bachelor Information
First Year
» First Year Compulsory Courses
» GESS Science in Perspective
» Minor Courses
First Year Compulsory Courses
First Year Examination Block 1
Offered in the Autumn Semester
First Year Examination Block 2
NumberTitleTypeECTSHoursLecturers
401-1262-07LAnalysis II Information O10 credits6V + 3UP. S. Jossen
AbstractIntroduction to differential and integral calculus in several real variables, vector calculus: differential, partial derivative, implicit functions, inverse function theorem, minima with constraints; Riemann integral, vector fields, differential forms, path integrals, surface integrals, divergence theorem, Stokes' theorem.
Objective
ContentCalculus in several variables; curves and surfaces in R^n; extrema with constraints; integration in n dimensions; vector calculus.
LiteratureH. Amann, J. Escher: Analysis II
https://link.springer.com/book/10.1007/3-7643-7402-0

J. Appell: Analysis in Beispielen und Gegenbeispielen
https://link.springer.com/book/10.1007/978-3-540-88903-8

R. Courant: Vorlesungen über Differential- und Integralrechnung
https://link.springer.com/book/10.1007/978-3-642-61973-1

O. Forster: Analysis 2
https://link.springer.com/book/10.1007/978-3-658-02357-7

H. Heuser: Lehrbuch der Analysis
https://link.springer.com/book/10.1007/978-3-322-96826-5

K. Königsberger: Analysis 2
https://link.springer.com/book/10.1007/3-540-35077-2

W. Walter: Analysis 2
https://link.springer.com/book/10.1007/978-3-642-97614-8

V. Zorich: Mathematical Analysis II (englisch)
https://link.springer.com/book/10.1007/978-3-662-48993-2
401-1152-02LLinear Algebra II Information Restricted registration - show details O7 credits4V + 2UT. H. Willwacher
AbstractEigenvalues and eigenvectors, Jordan normal form, bilinear forms, euclidean and unitary vector spaces, selected applications.
ObjectiveBasic knowledge of the fundamentals of linear algebra.
LiteratureSiehe Lineare Algebra I
Prerequisites / NoticeLinear Algebra I
401-1652-10LNumerical Analysis I Information Restricted registration - show details O6 credits3V + 2UC. Schwab
AbstractThis course will give an introduction to numerical methods, aimed at mathematics majors. It covers numerical linear algebra, quadrature, interpolation and approximation methods as well as their error analysis and implementation.
ObjectiveKnowledge of the fundamental numerical methods as well as
`numerical literacy': application of numerical methods for the solution
of application problems, mathematical foundations of numerical
methods, and basic mathematical methods of the analysis of
stability, consistency and convergence of numerical methods,
MATLAB implementation.
ContentRounding errors, solution of linear systems of equations, nonlinear equations,
interpolation (polynomial as well as trigonometric), least squares problems,
extrapolation, numerical quadrature, elementary optimization methods.
Lecture notesLecture Notes and reading list will be available.
LiteratureLecture Notes (german or english) will be made available to students of ETH BSc MATH.

Quarteroni, Sacco and Saleri, Numerische Mathematik 1 + 2, Springer Verlag 2002 (in German).

There is an English version of this text, containing both German volumes, from the same publisher. If you feel more comfortable with English, you can follow this text as well. Content and Indexing are identical in the German and the English text.
Prerequisites / NoticeAdmission Requirements:
Completed course
Linear Algebra I, Analysis I in ETH BSc MATH
Parallel enrolment in
Linear Algebra II, Analysis II in ETH BSc MATH

Weekly homework assignments involving MATLAB programming
are an integral part of the course.
Turn-in of solutions will be graded.
402-1782-00LPhysics II
Accompanying the lecture course "Physics II", among GESS Science in Perspective is offered: 851-0147-01L Philosophical Reflections on Physics II
O7 credits4V + 2UR. Wallny
AbstractIntroduction to theory of waves, electricity and magnetism. This is the continuation of Physics I which introduced the fundamentals of mechanics.
Objectivebasic knowledge of mechanics and electricity and magnetism as well as the capability to solve physics problems related to these subjects.
Compulsory Courses
Examination Block II
NumberTitleTypeECTSHoursLecturers
401-2284-00LMeasure and Integration Information O6 credits3V + 2UF. Da Lio
AbstractIntroduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces
ObjectiveBasic acquaintance with the abstract theory of measure and integration
ContentIntroduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces
Lecture notesNew lecture notes in English will be made available during the course
Literature1. L. Evans and R.F. Gariepy " Measure theory and fine properties of functions"
2. Walter Rudin "Real and complex analysis"
3. R. Bartle The elements of Integration and Lebesgue Measure
4. The notes by Prof. Michael Struwe Springsemester 2013, https://people.math.ethz.ch/~struwe/Skripten/AnalysisIII-FS2013-12-9-13.pdf.
5. The notes by Prof. UrsLang Springsemester 2019. https://people.math.ethz.ch/~lang/mi.pdf
6. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis: http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf
.
401-2004-00LAlgebra II Information O5 credits2V + 2UR. Pink
AbstractThe main topics are field extensions and Galois theory.
ObjectiveIntroduction to fundamentals of field extensions, Galois theory, and related topics.
ContentThe main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals.
LiteratureJoseph J. Rotman, "Advanced Modern Algebra" third edition, part 1,
Graduate Studies in Mathematics,Volume 165
American Mathematical Society

Galois Theory is the topic treated in Chapter A5.
401-2554-00LTopology Information Restricted registration - show details O6 credits3V + 2UA. Carlotto
AbstractTopics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.
ObjectiveAn introduction to topology i.e. the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.
LiteratureWe will follow these, freely available, standard references by Allen Hatcher:

i) http://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf

(for the part on General Topology)

ii) http://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf

(for the part on basic Algebraic Topology).

Additional references include:

"Topology" by James Munkres (Pearson Modern Classics for Advanced Mathematics Series)

"Counterexamples in Topology" by Lynn Arthur Steen, J. Arthur Seebach Jr. (Springer)

"Algebraic Topology" by Edwin Spanier (Springer).
Prerequisites / NoticeThe content of the first-year courses in the Bachelor program in Mathematics. In particular, each student is expected to be familiar with notion of continuity for functions from/to Euclidean spaces, and with the content of the corresponding basic theorems (Bolzano, Weierstrass etc..). In addition, some degree of scientific maturity in writing rigorous proofs (and following them when presented in class) is absolutely essential.
401-2654-00LNumerical Analysis II Information O6 credits3V + 2UH. Ammari
AbstractThe central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation.
ObjectiveThe course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments.
ContentChapter 1. Some basics
1.1. What is a differential equation?
1.2. Some methods of resolution
1.3. Important examples of ODEs
Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case
2.1. Banach fixed point theorem
2.2. Gronwall’s lemma
2.3. Cauchy-Lipschitz theorem
2.4. Stability
2.5. Regularity
Chapter 3. Linear systems
3.1. Exponential of a matrix
3.2. Linear systems with constant coefficients
3.3. Linear system with non-constant real coefficients
3.4. Second order linear equations
3.5. Linearization and stability for autonomous systems
3.6 Periodic Linear Systems
Chapter 4. Numerical solution of ordinary differential equations
4.1. Introduction
4.2. The general explicit one-step method
4.3. Example of linear systems
4.4. Runge-Kutta methods
4.5. Multi-step methods
4.6. Stiff equations and systems
4.7. Perturbation theories for differential equations
Chapter 5. Geometrical numerical integration methods for differential equation
5.1. Introduction
5.2. Structure preserving methods for Hamiltonian systems
5.3. Runge-Kutta methods
5.4. Long-time behaviour of numerical solutions
Chapter 6. Finite difference methods
6.1. Introduction
6.2. Numerical algorithms for the heat equation
6.3. Numerical algorithms for the wave equation
6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension
Chapter 7. Stochastic differential equations
7.1. Introduction
7.2. Langevin equation
7.3. Ornstein-Uhlenbeck equation
7.4. Existence and uniqueness of solutions in dimension one
7.5. Numerical solution of stochastic differential equations
Lecture notesLecture notes including supplements will be provided electronically.

Please find the lecture homepage here:

https://www.sam.math.ethz.ch/~grsam/SS20/NAII/

All assignments and some previous exam problems will be available for download on lecture homepage.
LiteratureNote: Extra reading is not considered important for understanding the
course subjects.

Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994.

Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996.

Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002.

L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009.

Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993.

Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972.

Walter: Ordinary differential equations, Springer-Verlag, New York, 1998.
Prerequisites / NoticeHomework problems involve Python implementation of numerical algorithms.
401-2604-00LProbability and Statistics Information O7 credits4V + 2UM. Schweizer
Abstract- Discrete probability spaces
- Continuous models
- Limit theorems
- Introduction to statistics
ObjectiveThe goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. This includes a mathematically rigorous treatment as well as intuition and getting acquainted with the ideas behind the definitions. The course does not use measure theory systematically, but does point out where this is required and what the connections are.
Content- Discrete probability spaces: Basic concepts, Laplace models, random walks, conditional probabilities, independence
- Continuous models: general probability spaces, random variables and their distributions, expectation, multivariate random variables
- Limit theorems: weak and strong law of large numbers, central limit theorem
- Introduction to statistics: What is statistics?, point estimators, statistical tests, confidence intervals
Lecture notesThere will be lecture notes (in German) that are continuously updated during the semester.
LiteratureA. DasGupta, Fundamentals of Probability: A First Course, Springer (2010)
J. A. Rice, Mathematical Statistics and Data Analysis, Duxbury Press, second edition (1995)
Core Courses
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
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.
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.
401-3372-00LDynamical 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.
» Core Courses: Pure Mathematics (Mathematics Master)
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"
» Core Courses: Applied Mathematics and Further Appl.-Oriented Fields (Mathematics Master)
Core Courses: Further Application-Oriented Fields
402-0204-00L Electrodynamics is eligible in the Bachelor's programme in Mathematics as an applied core course, but only if 402-0224-00L Theoretical Physics (offered for the last time in FS 2016) isn't recognised for credits (neither in the Bachelor's nor in the Master's programme).
For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
402-0204-00LElectrodynamicsW7 credits4V + 2UR. Renner
AbstractDerivation and discussion of Maxwell's equations, from the static limit to the full dynamical case. Wave equation, waveguides, cavities. Generation of electromagnetic radiation, scattering and diffraction of light. Structure of Maxwell's equations, relativity theory and covariance, Lagrangian formulation. Dynamics of relativistic particles in the presence of fields and radiation properties.
ObjectiveDevelop a physical understanding for static and dynamic phenomena related to (moving) charged objects and understand the structure of the classical field theory of electrodynamics (transverse versus longitudinal physics, invariances (Lorentz-, gauge-)). Appreciate the interrelation between electric, magnetic, and optical phenomena and the influence of media. Understand a set of classic electrodynamical phenomena and develop the ability to solve simple problems independently. Apply previously learned mathematical concepts (vector analysis, complete systems of functions, Green's functions, co- and contravariant coordinates, etc.). Prepare for quantum mechanics (eigenvalue problems, wave guides and cavities).
ContentClassical field theory of electrodynamics: Derivation and discussion of Maxwell equations, starting from the static limit (electrostatics, magnetostatics, boundary value problems) in the vacuum and in media and subsequent generalization to the full dynamical case (Faraday's law, Ampere/Maxwell law; potentials and gauge invariance). Wave equation and solutions in full space, half-space (Snell's law), waveguides, cavities, generation of electromagnetic radiation, scattering and diffraction of light (optics). Application to various specific examples. Discussion of the structure of Maxwell's equations, Lorentz invariance, relativity theory and covariance, Lagrangian formulation. Dynamics of relativistic particles in the presence of fields and their radiation properties (synchrotron).
LiteratureJ.D. Jackson, Classical Electrodynamics
W.K.H Panovsky and M. Phillis, Classical electricity and magnetism
L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of continuus media
A. Sommerfeld, Electrodynamics / Optics (Lectures on Theoretical Physics)
M. Born and E. Wolf, Principles of optics
R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures of Physics, Vol II
W. Nolting, Elektrodynamik (Grundkurs Theoretische Physik 3)
Electives
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
401-3056-00LFinite 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.
401-3556-20LTopics 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.
401-3574-61LIntroduction 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
No offering in this semester yet
Selection: Numerical Analysis
No offering in this semester yet
Selection: Probability Theory, Statistics
NumberTitleTypeECTSHoursLecturers
401-6102-00LMultivariate Statistics
Does not take place this semester.
W4 credits2Gnot available
AbstractMultivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications.
ObjectiveAfter the course, you should be able to:
- describe the various methods and the concepts and theory behind them
- identify adequate methods for a given statistical problem
- use the statistical software "R" to efficiently apply these methods
- interpret the output of these methods
ContentVisualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Lecture notesNone
LiteratureThe course will be based on class notes and books that are available electronically via the ETH library.
Prerequisites / NoticeTarget audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background.

Prerequisite: A basic course in probability and statistics.

Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units.
401-4626-00LAdvanced Statistical Modelling: Mixed ModelsW4 credits2VM. Mächler
AbstractMixed Models = (*| generalized| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms.

In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.
Objective- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models.

- Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis:
"fit -> interpret & diagnose -> modify the fit -> interpret & ...."

- Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.
ContentThe lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online.

Inference (significance of factors, confidence intervals) will focus on the more realistic *un*balanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.
Lecture notesWe will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples.

These lecture notes and all R scripts are made available from
https://github.com/mmaechler/MEMo
Literature(see web page and lecture notes)
Prerequisites / Notice- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..).

Typically this means at least two classes of (math based) statistics, say
1. Intro to probability and statistics
2. (Applied) regression including Matrix-Vector notation Y = X b + E

- Basic (1 semester) "Matrix calculus" / linear algebra is also assumed.

- If familiarity with [R](https://www.r-project.org/) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00LEmpirical Process Theory and ApplicationsW4 credits2VS. van de Geer
AbstractEmpirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc.
Objective
ContentIn this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
Selection: Financial and Insurance Mathematics
NumberTitleTypeECTSHoursLecturers
401-3888-00LIntroduction to Mathematical Finance Information
A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.
W10 credits4V + 1UC. Czichowsky
AbstractThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.
ObjectiveThis is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and also study convex duality in utility maximization.
ContentThis course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
Lecture notesThe course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.
LiteratureLiterature:

Michael U. Dothan, "Prices in Financial Markets", Oxford University Press

Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter

Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press

Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer
Prerequisites / NoticeA related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In other words, it is also not possible to earn credit points with one for the Bachelor and with the other for the Master degree.

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
401-3629-00LQuantitative Risk Management Information W4 credits2V + 1UP. Cheridito
AbstractThis 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, dependence structures and operational risk.
ObjectiveThe goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.
Content1. 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 notesCourse material is available on https://people.math.ethz.ch/~patrickc/qrm
LiteratureQuantitative 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 / NoticeThe course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-3923-00LSelected Topics in Life Insurance MathematicsW4 credits2VM. Koller
AbstractStochastic Models for Life insurance
1) Markov chains
2) Stochastic Processes for demography and interest rates
3) Cash flow streams and reserves
4) Mathematical Reserves and Thiele's differential equation
5) Theorem of Hattendorff
6) Unit linked policies
Objective
401-3917-00LStochastic Loss Reserving MethodsW4 credits2VR. Dahms
AbstractLoss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.
ObjectiveOur goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.
ContentWe will present the following stochastic claims reserving methods/models:
- Stochastic Chain-Ladder Method
- Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods
- Distributional Models
- Linear Stochastic Reserving Models, with and without inflation
- Bootstrap Methods
- Claims Development Result (solvency view)
- Coupling of portfolios
LiteratureM. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination periods.

This course will be held in English and counts towards the diploma "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-3956-00LEconomic Theory of Financial Markets
Does not take place this semester.
W4 credits2VM. V. Wüthrich
AbstractThis lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
ObjectiveThis lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.
ContentWe treat the following topics:
- Fundamental concepts in economics
- Portfolio theory
- Mean variance analysis, capital asset pricing model
- Arbitrage pricing theory
- Cash flow theory
- Valuation principles
- Stochastic discounting, deflator techniques
- Interest rate modeling
- Utility theory
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3936-00LData Analytics for Non-Life Insurance PricingW4 credits2VC. M. Buser, M. V. Wüthrich
AbstractWe study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.
ObjectiveThe student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.
ContentWe present the following chapters:
- generalized linear models (GLMs)
- generalized additive models (GAMs)
- neural networks
- credibility theory
- classification and regression trees (CARTs)
- bagging, random forests and boosting
Lecture notesThe lecture notes are available from:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2870308
Prerequisites / NoticeThis course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch

Good knowledge in probability theory, stochastic processes and statistics is assumed.
401-4920-00LMarket-Consistent Actuarial ValuationW4 credits2VM. V. Wüthrich, H. Furrer
AbstractIntroduction to market-consistent actuarial valuation.
Topics: Stochastic discounting, full balance sheet approach, valuation portfolio in life and non-life insurance, technical and financial risks, risk management for insurance companies.
ObjectiveGoal is to give the basic mathematical tools for describing insurance products within a financial market and economic environment and provide the basics of solvency considerations.
ContentIn this lecture we give a full balance sheet approach to the task of actuarial valuation of an insurance company. Therefore we introduce a multidimensional valuation portfolio (VaPo) on the liability side of the balance sheet. The basis of this multidimensional VaPo is a set of financial instruments. This approach makes the liability side of the balance sheet directly comparable to its asset side.

The lecture is based on four sections:
1) Stochastic discounting
2) Construction of a multidimensional Valuation Portfolio for life insurance products (with guarantees)
3) Construction of a multidimensional Valuation Portfolio for a run-off portfolio of a non-life insurance company
4) Measuring financial risks in a full balance sheet approach (ALM risks)
LiteratureMarket-Consistent Actuarial Valuation, 3rd edition.
Wüthrich, M.V.
EAA Series, Springer 2016.
ISBN: 978-3-319-46635-4

Wüthrich, M.V., Merz, M.
Claims run-off uncertainty: the full picture.
SSRN Manuscript ID 2524352 (2015).

England, P.D, Verrall, R.J., Wüthrich, M.V.
On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins.
Insurance: Mathematics and Economics 85 (2019), 74-88.

Wüthrich, M.V., Embrechts, P., Tsanakas, A.
Risk margin for a non-life insurance run-off.
Statistics & Risk Modeling 28 (2011), no. 4, 299--317.

Financial Modeling, Actuarial Valuation and Solvency in Insurance.
Wüthrich, M.V., Merz, M.
Springer Finance 2013.
ISBN: 978-3-642-31391-2

Cheridito, P., Ery, J., Wüthrich, M.V.
Assessing asset-liability risk with neural networks.
Risks 8/1 (2020), article 16.
Prerequisites / NoticeThe exams ONLY take place during the official ETH examination period.

This course will be held in English and counts towards the diploma of "Aktuar SAV".
For the latter, see details under www.actuaries.ch.

Knowledge in probability theory, stochastic processes and statistics is assumed.
Selection: Mathematical Physics, Theoretical Physics
In the Bachelor's programme in Mathematics 402-0204-00L Electrodynamics is eligible as an elective course, but only if 402-0224-00L Theoretical Physics isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3814-00LQuantum Mechanics for Mathematicians
NOTICE: The class scheduled for 5 March 2020 has been cancelled.
W4 credits2VJ. Wisniewska
AbstractIntroduction to quantum mechanics aimed at mathematics students
ObjectiveThe course begins with the fundamentals of classical mechanics and its mathematical description i.e. Hamiltonian dynamics. We will introduce the notion of states and observables in the classical setting and further on its counter parts in the quantum setting. We then will discuss quantisation and the mathematical formulation of quantum mechanics. Further on we will study the Heisenberg’s uncertainty relations and quantum entanglement. The course then goes on to study the dynamics of quantum systems described by the Schrödinger’s equation.
Content1. Hamiltonian mechanics and fundamentals of symplectic geometry
2. Classical observables and Poisson bracket
3. Basic principles of quantum mechanics and quantisation
4. Heisenberg’s uncertainty relations
5. Quantum entanglement and EPR paradox
6. Schrödinger’s equation
LiteratureTakhtajan, Leon A.
Quantum mechanics for mathematicians.
Graduate Studies in Mathematics, 95. American Mathematical Society, Providence, RI, 2008. xvi+387 pp. ISBN: 978-0-8218-4630-8
Prerequisites / NoticePrerequisites:
Essential: Differential Geometry 1
Recommended: basic Functional Analysis and Algebraic Topology
401-2334-00LMethods of Mathematical Physics II Information Restricted registration - show details W6 credits3V + 2UG. Felder
AbstractGroup theory: groups, representation of groups, unitary and orthogonal groups, Lorentz group. Lie theory: Lie algebras and Lie groups. Representation theory: representation theory of finite groups, representations of Lie algebras and Lie groups, physical applications (eigenvalue problems with symmetry).
Objective
402-0206-00LQuantum Mechanics IIW10 credits3V + 2UG. Blatter
AbstractMany-body quantum physics rests on symmetry considerations that lead to two kinds of particles, fermions and bosons. Formal techniques include Hartree-Fock theory and second-quantization techniques, as well as quantum statistics with ensembles. Few- and many-body systems include atoms, molecules, the Fermi sea, elastic chains, radiation and its interaction with matter, and ideal quantum gases.
ObjectiveBasic command of few- and many-particle physics for fermions and bosons, including second quantisation and quantum statistical techniques. Understanding of elementary many-body systems such as atoms, molecules, the Fermi sea, electromagnetic radiation and its interaction with matter, ideal quantum gases and relativistic theories.
ContentThe description of indistinguishable particles leads us to (exchange-) symmetrized wave functions for fermions and bosons. We discuss simple few-body problems (Helium atoms, hydrogen molecule) und proceed with a systematic description of fermionic many body problems (Hartree-Fock approximation, screening, correlations with applications on atomes and the Fermi sea). The second quantisation formalism allows for the compact description of the Fermi gas, of elastic strings (phonons), and the radiation field (photons). We study the interaction of radiation and matter and the associated phenomena of radiative decay, light scattering, and the Lamb shift. Quantum statistical description of ideal Bose and Fermi gases at finite temperatures concludes the program. If time permits, we will touch upon of relativistic one particle physics, the Klein-Gordon equation for spin-0 bosons and the Dirac equation describing spin-1/2 fermions.
Lecture notesQuanten Mechanik I und II in German.
LiteratureG. Baym, Lectures on Quantum Mechanics (Benjamin, Menlo Park, California, 1969)
L.I. Schiff, Quantum Mechanics (Mc-Graw-Hill, New York, 1955)
A. Messiah, Quantum Mechanics I & II (North-Holland, Amsterdam, 1976)
E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1998)
C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics I & II (John Wiley, New York, 1977)
P.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (Mc Graw-Hill, New York, 1965)
A.L. Fetter and J.D. Walecka, Theoretical Mechanics of Particles and Continua (Mc Graw-Hill, New York, 1980)
J.J. Sakurai, Modern Quantum Mechanics (Addison Wesley, Reading, 1994)
J.J. Sakurai, Advanced Quantum mechanics (Addison Wesley)
F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley, New York, 1993)
Prerequisites / NoticeBasic knowledge of single-particle Quantum Mechanics
Selection: Mathematical Optimization, Discrete Mathematics
No offering in this semester yet
Auswahl: Theoretical Computer Science
In the Bachelor's programme in Mathematics 401-3052-05L Graph Theory is eligible as an elective course, but only if 401-3052-10L Graph Theory isn't recognised for credits (neither in the Bachelor's nor in the Master's programme). For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
252-0408-00LCryptographic Protocols Information W6 credits2V + 2U + 1AM. Hirt, U. Maurer
AbstractThe course presents a selection of hot research topics in cryptography. The choice of topics varies and may include provable security, interactive proofs, zero-knowledge protocols, secret sharing, secure multi-party computation, e-voting, etc.
ObjectiveIndroduction to a very active research area with many gems and paradoxical
results. Spark interest in fundamental problems.
ContentThe course presents a selection of hot research topics in cryptography. The choice of topics varies and may include provable security, interactive proofs, zero-knowledge protocols, secret sharing, secure multi-party computation, e-voting, etc.
Lecture notesthe lecture notes are in German, but they are not required as the entire
course material is documented also in other course material (in english).
Prerequisites / NoticeA basic understanding of fundamental cryptographic concepts
(as taught for example in the course Information Security or
in the course Cryptography Foundations) is useful, but not required.
263-4660-00LApplied Cryptography Information Restricted registration - show details
Number of participants limited to 150.
W8 credits3V + 2U + 2PK. Paterson
AbstractThis course will introduce the basic primitives of cryptography, using rigorous syntax and game-based security definitions. The course will show how these primitives can be combined to build cryptographic protocols and systems.
ObjectiveThe goal of the course is to put students' understanding of cryptography on sound foundations, to enable them to start to build well-designed cryptographic systems, and to expose them to some of the pitfalls that arise when doing so.
ContentBasic symmetric primitives (block ciphers, modes, hash functions); generic composition; AEAD; basic secure channels; basic public key primitives (encryption,signature, DH key exchange); ECC; randomness; applications.
LiteratureTextbook: Boneh and Shoup, “A Graduate Course in Applied Cryptography”, https://crypto.stanford.edu/~dabo/cryptobook/BonehShoup_0_4.pdf.
Prerequisites / NoticeIdeally, students will have taken the D-INFK Bachelors course “Information Security" or an equivalent course at Bachelors level.
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-4944-20LMathematics of Data ScienceW8 credits4GA. Bandeira
AbstractMostly self-contained, but fast-paced, introductory masters level course on various theoretical aspects of algorithms that aim to extract information from data.
ObjectiveIntroduction to various mathematical aspects of Data Science.
ContentThese topics lie in overlaps of (Applied) Mathematics with: Computer Science, Electrical Engineering, Statistics, and/or Operations Research. Each lecture will feature a couple of Mathematical Open Problem(s) related to Data Science. The main mathematical tools used will be Probability and Linear Algebra, and a basic familiarity with these subjects is required. There will also be some (although knowledge of these tools is not assumed) Graph Theory, Representation Theory, Applied Harmonic Analysis, among others. The topics treated will include Dimension reduction, Manifold learning, Sparse recovery, Random Matrices, Approximation Algorithms, Community detection in graphs, and several others.
Lecture noteshttps://people.math.ethz.ch/~abandeira/TenLecturesFortyTwoProblems.pdf
Prerequisites / NoticeThe main mathematical tools used will be Probability, Linear Algebra (and real analysis), and a working knowledge of these subjects is required. In addition
to these prerequisites, this class requires a certain degree of mathematical maturity--including abstract thinking and the ability to understand and write proofs.


We encourage students who are interested in mathematical data science to take both this course and ``227-0434-10L Mathematics of Information'' taught by Prof. H. Bölcskei. The two courses are designed to be
complementary.
A. Bandeira and H. Bölcskei
252-0220-00LIntroduction to Machine Learning Information Restricted registration - show details
Limited number of participants. Preference is given to students in programmes in which the course is being offered. All other students will be waitlisted. Please do not contact Prof. Krause for any questions in this regard. If necessary, please contact studiensekretariat@inf.ethz.ch
W8 credits4V + 2U + 1AA. Krause
AbstractThe course introduces the foundations of learning and making predictions based on data.
ObjectiveThe course will introduce the foundations of learning and making predictions from data. We will study basic concepts such as trading goodness of fit and model complexitiy. We will discuss important machine learning algorithms used in practice, and provide hands-on experience in a course project.
Content- Linear regression (overfitting, cross-validation/bootstrap, model selection, regularization, [stochastic] gradient descent)
- Linear classification: Logistic regression (feature selection, sparsity, multi-class)
- Kernels and the kernel trick (Properties of kernels; applications to linear and logistic regression); k-nearest neighbor
- Neural networks (backpropagation, regularization, convolutional neural networks)
- Unsupervised learning (k-means, PCA, neural network autoencoders)
- The statistical perspective (regularization as prior; loss as likelihood; learning as MAP inference)
- Statistical decision theory (decision making based on statistical models and utility functions)
- Discriminative vs. generative modeling (benefits and challenges in modeling joint vy. conditional distributions)
- Bayes' classifiers (Naive Bayes, Gaussian Bayes; MLE)
- Bayesian approaches to unsupervised learning (Gaussian mixtures, EM)
LiteratureTextbook: Kevin Murphy, Machine Learning: A Probabilistic Perspective, MIT Press
Prerequisites / NoticeDesigned to provide a basis for following courses:
- Advanced Machine Learning
- Deep Learning
- Probabilistic Artificial Intelligence
- Seminar "Advanced Topics in Machine Learning"
263-5300-00LGuarantees for Machine Learning Information Restricted registration - show details W5 credits2V + 2AF. Yang
AbstractThis course teaches classical and recent methods in statistics and optimization commonly used to prove theoretical guarantees for machine learning algorithms. The knowledge is then applied in project work that focuses on understanding phenomena in modern machine learning.
ObjectiveThis course is aimed at advanced master and doctorate students who want to understand and/or conduct independent research on theory for modern machine learning. For this purpose, students will learn common mathematical techniques from statistical learning theory. In independent project work, they then apply their knowledge and go through the process of critically questioning recently published work, finding relevant research questions and learning how to effectively present research ideas to a professional audience.
ContentThis course teaches some classical and recent methods in statistical learning theory aimed at proving theoretical guarantees for machine learning algorithms, including topics in

- concentration bounds, uniform convergence
- high-dimensional statistics (e.g. Lasso)
- prediction error bounds for non-parametric statistics (e.g. in kernel spaces)
- minimax lower bounds
- regularization via optimization

The project work focuses on active theoretical ML research that aims to understand modern phenomena in machine learning, including but not limited to

- how overparameterization could help generalization ( interpolating models, linearized NN )
- how overparameterization could help optimization ( non-convex optimization, loss landscape )
- complexity measures and approximation theoretic properties of randomly initialized and
trained NN
- generalization of robust learning ( adversarial robustness, standard and robust error tradeoff )
- prediction with calibrated confidence ( conformal prediction, calibration )
Prerequisites / NoticeIt’s absolutely necessary for students to have a strong mathematical background (basic real analysis, probability theory, linear algebra) and good knowledge of core concepts in machine learning taught in courses such as “Introduction to Machine Learning”, “Regression”/ “Statistical Modelling”. It's also helpful to have heard an optimization course or approximation theoretic course. In addition to these prerequisites, this class requires a certain degree of mathematical maturity—including abstract thinking and the ability to understand and write proofs.
401-3502-20LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W2 credits4ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3503-20LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W3 credits6ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
401-3504-20LReading Course Restricted registration - show details
To start an individual reading course, contact an authorised supervisor
Link
and register your reading course in myStudies.
W4 credits9ASupervisors
AbstractFor this Reading Course proactive students make an individual agreement with a lecturer to acquire knowledge through independent literature study.
Objective
Core Courses and Electives (Mathematics Master)
» Electives (Mathematics Master)
» Core Courses (Mathematics Master)
Further Courses Suitable for the Second Year
NumberTitleTypeECTSHoursLecturers
401-2334-00LMethods of Mathematical Physics II Information Restricted registration - show details W6 credits3V + 2UG. Felder
AbstractGroup theory: groups, representation of groups, unitary and orthogonal groups, Lorentz group. Lie theory: Lie algebras and Lie groups. Representation theory: representation theory of finite groups, representations of Lie algebras and Lie groups, physical applications (eigenvalue problems with symmetry).
Objective
401-2200-13LRepresentation Theory of Finite Groups Restricted registration - show details
Primarily for students Bachelor Mathematics 4th semester (or 6th semester if not fully booked).
Number of participants limited to 12.
W4 credits2SR. Pink
Abstract-Grundlegende Begriffe aus der Darstellungstheorie
-Zerlegung in irreduzible Darstellungen
-Charaktertheorie
-Berechnung von Charaktertabellen
-Anwendungen zur Gruppentheorie, insbesondere Satz von Burnside
ObjectiveMethoden und Resultate der Darstellungstheorie.
Vortragstechnik.
LiteratureRepresentations and Characters of Groups, Gordon James & Martin Liebeck, Cambridge Verlag.
Prerequisites / NoticeDas Seminar richtet sich primär an Studierende im 4. Semester, die die Vorlesung Algebra I bei mir besucht haben.
Am Donnerstag den 6. Januar um 15:00 im Raum HG G43 findet eine Vorbesprechung statt, an der Sie unbedingt teilnehmen sollten.
Seminars
This semester, many seminars have a waiting list with special selection procedure. If no other criteria apply, a definitive registration will be granted first of all to students who haven't got another seminar registration. Here is the best procedure for dealing with two waiting lists: first choose your preferred seminar and afterwards choose an alternative seminar.
NumberTitleTypeECTSHoursLecturers
401-2200-13LRepresentation Theory of Finite Groups Restricted registration - show details
Primarily for students Bachelor Mathematics 4th semester (or 6th semester if not fully booked).
Number of participants limited to 12.
W4 credits2SR. Pink
Abstract-Grundlegende Begriffe aus der Darstellungstheorie
-Zerlegung in irreduzible Darstellungen
-Charaktertheorie
-Berechnung von Charaktertabellen
-Anwendungen zur Gruppentheorie, insbesondere Satz von Burnside
ObjectiveMethoden und Resultate der Darstellungstheorie.
Vortragstechnik.
LiteratureRepresentations and Characters of Groups, Gordon James & Martin Liebeck, Cambridge Verlag.
Prerequisites / NoticeDas Seminar richtet sich primär an Studierende im 4. Semester, die die Vorlesung Algebra I bei mir besucht haben.
Am Donnerstag den 6. Januar um 15:00 im Raum HG G43 findet eine Vorbesprechung statt, an der Sie unbedingt teilnehmen sollten.
401-3110-20LQuadratic Forms, Markov Numbers and Diophantine Approximation Information Restricted registration - show details
Number of participants limited to 22.
W4 credits2SP. Bengoechea Duro
AbstractIn 1880 Andrei A. Markov discovered beautiful connections between minima of binary real quadratic forms, badly approximable numbers by rationals, and a certain Diophantine equation which describes an affine cubic surface, now and days called Markov surface. We will use Markov's theory as a unifying thread to talk about quadratic forms, Diophantine approximation and hyperbolic geometry.
Objective
ContentContinued fractions; representation of real numbers by rationals; Hurwitz's theorem; Lagrange spectrum; badly approximable numbers; binary quadratic forms; Markov numbers; Markov tree; geometric interpretation of Markov numbers; the still open Unicity Conjecture
401-3180-20LIntroduction to Homotopy Theory and Model Category Structure Restricted registration - show details
Number of participants limited to 24.
W4 credits2SJ. Ducoulombier
AbstractIntroductory seminar about category theory and techniques in Algebraic topology such as model category structure and "homotopy" limits and colimits.
ObjectiveIt is well known that topological spaces are endowed with a homotopy theory classifying objects up to continuous deformations. Model categories provide a natural setting for homotopy theory and it has been used in some parts of algebraic K-theory, algebraic geometry and algebraic topology, where homotopy-theoretic approaches led to deep results.


The goal of this seminar is to get an introduction to model structures through examples (topological spaces, simplicial sets, chains complexes...). To get further into this theory, we develop the notion of derived functors with applications to homotopy limits and colimits.
Literature"Homotopy theories and model categories" Dwyer and Spalinski

"A primer on homotopy colimits" Dugger

"Model categories" Hovey
Prerequisites / NoticeThe students are expected to be familiar with topological spaces
and fundamental groups.

This seminar takes the form of a working group, where interactions are encouraged. Participants are expected to attend the seminar, give a presentation and write a report. Topic will be assigned during the first meeting.
401-3200-69LA Survey of Geometric Group Theory Information Restricted registration - show details
Number of participants limited to 24.
W4 credits2SM. Cordes
AbstractIn this class we will cover some of the tools, techniques, and groups central to the study of geometric group theory. After introducing the basic concepts (groups and metric spaces), we will branch out and sample different topics in geometric group theory based on the interest of the participants.
ObjectiveTo learn and understand a wide range of tools and groups central to the field of geometric group theory.
ContentPossible topics include: properties of free groups and groups acting on trees, large scale geometric invariants (Dehn functions, hyperbolicity, ends of groups, asymptotic dimension, growth of groups), and examples of notable and interesting groups (Coxeter groups, right-angled Artin groups, lamplighter groups, Thompson's group, mapping class groups, and braid groups).
LiteratureThe topics will be chosen from "Office Hours with a Geometric Group Theorist" edited by Matt Clay and Dan Margalit.
Prerequisites / NoticeOne should be familiar with the basics of groups, metric spaces, and topology (should be familiar with the fundamental group).
401-3030-19LThe Axiom of Choice Restricted registration - show details
Number of participants limited to 44.
If the number of registrations exceeds 30, then the seminar will be duplicated.
W4 credits2SL. Halbeisen
AbstractEs werden verschiedene Aspekte des Auswahlaxioms untersucht. Einerseits werden Konsequenzen des Auswahlaxioms behandelt, andererseits werden auch Modelle der Mengenlehre konstruiert, in denen das Auswahlaxiom nicht gilt.
Objective
ContentEs werden verschiedene Aspekte des Auswahlaxioms untersucht.

Im Seminar A, von 13-15 Uhr, werden äquivalente Formen und Konsequenzen des Auswahlaxioms behandelt

Im Seminar B, von 15-17 Uhr, werden Modelle der Mengenlehre konstruiert, in denen nur abgeschwächte Formen des Auswahlaxioms gelten, nicht aber das volle Auswahlaxiom.
401-3200-16LPower Sums of Coxeter Exponents (With Some Insight Into the Evolution of an Article) Restricted registration - show details
Number of participants limited to 12.
W4 credits2SR. Suter
AbstractIn addition to its mathematical content, this seminar shall provide an insight into what is usually hidden away from the reader of an article.
ObjectiveThe gradual development from an initial wish to make progress on a certain topic towards a published article is usually hidden away in the final text. The idea of this seminar is to have a look at the two author paper "Power sums of Coxeter exponents" (Advances in Mathematics 231 (2012), 1291-1307), that arose entirely by means of email correspondence, and to make accessible some excerpts from this correspondence in order to gain some insight into how the article evolved. This might be instructive in particular with regard to the students' own research ambitions.
LiteratureJ. Burns, R. Suter: Power sums of Coxeter exponents, Adv. Math. 231 (2012), 1291-1307. www.sciencedirect.com/science/article/pii/S0001870812002411
Prerequisites / NoticeNo prior knowledge of Coxeter exponents is required because some relevant stuff about Coxeter groups and root systems shall be explained in an early seminar talk.
401-3600-20LStudent Seminar in Probability Theory Restricted registration - show details
Limited number of participants.
Registration to the seminar will only be effective once confirmed by email from the organizers.
W4 credits2SA.‑S. Sznitman, J. Bertoin, V. Tassion, W. Werner
Abstract
Objective
401-3620-20LStudent Seminar in Statistics: Inference in Non-Classical Regression Models Restricted registration - show details
Number of participants limited to 24.
Mainly for students from the Mathematics Bachelor and Master Programmes who, in addition to the introductory course unit 401-2604-00L Probability and Statistics, have heard at least one core or elective course in statistics. Also offered in the Master Programmes Statistics resp. Data Science.
W4 credits2SF. Balabdaoui
AbstractReview of some non-standard regression models and the statistical properties of estimation methods in such models.
ObjectiveThe main goal is the students get to discover some less known regression models which either generalize the well-known linear model (for example monotone regression) or violate some of the most fundamental assumptions (as in shuffled or unlinked regression models).
ContentLinear regression is one of the most used models for prediction and hence one of the most understood in statistical literature. However, linearity might too simplistic to capture the actual relationship between some response and given covariates. Also, there are many real data problems where linearity is plausible but the actual pairing between the observed covariates and responses is completely lost or at partially. In this seminar, we review some of the non-classical regression models and the statistical properties of the estimation methods considered by well-known statisticians and machine learners. This will encompass:
1. Monotone regression
2. Single index model
3. Unlinked regression
4. Partially unlinked regression
Lecture notesNo script is necessary for this seminar
LiteratureIn the following is the material that will read and studied by each pair of students (all the items listed below are available through the ETH electronic library or arXiv):

1. Chapter 2 from the book "Nonparametric estimation under shape constraints" by P. Groeneboom and G. Jongbloed, 2014, Cambridge University Press

2. "Nonparametric shape-restricted regression" by A. Guntuoyina and B. Sen, 2018, Statistical Science, Volume 33, 568-594

3. "Asymptotic distributions for two estimators of the single index model" by Y. Xia, 2006, Econometric Theory, Volume 22, 1112-1137

4. "Least squares estimation in the monotone single index model" by F. Balabdaoui, C. Durot and H. K. Jankowski, Journal of Bernoulli, 2019, Volume 4B, 3276-3310

5. "Least angle regression" by B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, 2004, Annals of Statsitics, Volume 32, 407-499.

6. "Sharp thresholds for high dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso)" by M. Wainwright, 2009, IEEE transactions in Information Theory, Volume 55, 1-19

7."Denoising linear models with permuted data" by A. Pananjady, M. Wainwright and T. A. Courtade and , 2017, IEEE International Symposium on Information Theory, 446-450.

8. "Linear regression with shuffled data: statistical and computation limits of permutation recovery" by A. Pananjady, M. Wainwright and T. A. Courtade , 2018, IEEE transactions in Information Theory, Volume 64, 3286-3300

9. "Linear regression without correspondence" by D. Hsu, K. Shi and X. Sun, 2017, NIPS

10. "A pseudo-likelihood approach to linear regression with partially shuffled data" by M. Slawski, G. Diao, E. Ben-David, 2019, arXiv.

11. "Uncoupled isotonic regression via minimum Wasserstein deconvolution" by P. Rigollet and J. Weed, 2019, Information and Inference, Volume 00, 1-27
401-3900-16LAdvanced Topics in Discrete Optimization Information Restricted registration - show details
Number of participants limited to 12.
W4 credits2SC. Angelidakis, A. A. Kurpisz, R. Zenklusen
AbstractIn this seminar we will discuss selected topics in discrete optimization. The main focus is on mostly recent research papers in the field of Combinatorial Optimization.
ObjectiveThe goal of the seminar is twofold. First, we aim at improving students' presentation and communication skills. In particular, students are to present a research paper to their peers and the instructors in a clear and understandable way. Second, students learn a selection of recent cutting-edge approaches in the field of Combinatorial Optimization by attending the other students' talks. A very active participation in the seminar helps students to build up the necessary skills for parsing and digesting advanced technical texts on a significantly higher complexity level than usual textbooks.

A key goal is that students prepare their presentations in a concise and accessible way to make sure that other participants get a clear idea of the presented results and techniques.

Students intending to do a project in optimization are strongly encouraged to participate.
ContentThe selected topics will cover various classical and modern results in Combinatorial Optimization.

Contrary to prior years, a very significant component of the seminar will be interactive discussions where active participation of the students is required.
LiteratureThe learning material will be in the form of scientific papers.
Prerequisites / NoticeRequirements: We expect students to have a thorough understanding of topics covered in the course "Mathematical Optimization".
401-3940-20LStudent Seminar in Mathematics and Data: Optimization of Random Functions Restricted registration - show details
Number of participants limited to 12.
W4 credits2SA. Bandeira
AbstractMore information at course webpage:
https://people.math.ethz.ch/~abandeira/Spring2020.StudentSeminar.html
Objective
401-3530-20LStokes Phenomenon and Isomonodromy Equations Restricted registration - show details
Does not take place this semester.
Number of participants limited to 12.
The seminar does not take place in the Spring Semester 2020.
W4 credits2SG. Felder
AbstractOrdinary differential equations with irregular singularities, Stokes phenomenon, isomonodromy deformations and appications.
Objective
ContentThis seminar is about the study of ordinary differential equations with poles and its application in mathematical physics.
When a system of differential equations has an irregular singularity, such as a pole of order two or higher, a solution may fail to have a well-defined asymptotic expansion at the singular locus. Instead, there is a collection of angular sectors surrounding the singular locus, in each of which an asymptotic expansion is defined. The existence of such sectorial asymptotic expansions is what is called the “Stokes phenomenon”.

The Stokes phenomenon has found remarkable applications in different areas of mathematics and physics, such as in cohomological field theory, the study of stability conditions, noncommutative Hodge theory, cluster algebras, quantum groups and so on. In particular, the Stokes phenomenon is the essential ingredient in an irregular version of the Riemann-Hilbert correspondence, where the moduli space of differential equations with irregular singularities is described in terms of its associated generalized monodromy data (Stokes matrices). Moreover, the crucial role of the Stokes phenomenon in the study of representation theory and integrable systems is only beginning to emerge. The first 9 talks will include a general introduction to Stokes phenomenon and isomonodromy deformation. The last 3 talks of the seminar will focus on its applications.
LiteratureWerner Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Chapter 1-9, Springer.

P. Boalch, Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. Math. 146 (2001), 479–506.

P. Boalch, G-bundles, isomonodromy and quantum Weyl groups, Int. Math. Res. Not. (2002), no. 22, 1129–1166.

B. Dubrovin, Geometry of 2d topological field theory, Lecture 1-3, https://arxiv.org/pdf/hep-th/9407018.pdf.

B. Dubrovin, Painleve transcendents in two-dimensional topological field theory, The Painleve property, Springer, New York, 1999, pp. 287–412.
252-4102-00LSeminar on Randomized Algorithms and Probabilistic Methods Restricted registration - show details
The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar.

Number of participants limited to 24.
W2 credits2SA. Steger
AbstractThe aim of the seminar is to study papers which bring the students to the forefront of today's research topics. This semester we will study selected papers of the conference Symposium on Discrete Algorithms (SODA18).
ObjectiveRead papers from the forefront of today's research; learn how to give a scientific talk.
Prerequisites / NoticeThe seminar is open for both students from mathematics and students from computer science. As prerequisite we require that you passed the course Randomized Algorithms and Probabilistic Methods (or equivalent, if you come from abroad).
263-4203-00LGeometry: Combinatorics and Algorithms Information
The deadline for deregistering expires at the end of the second week of the semester. Students who are still registered after that date, but do not attend the seminar, will officially fail the seminar.
W2 credits2SB. Gärtner, M. Hoffmann, E. Welzl, M. Wettstein
AbstractThis seminar complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent.
ObjectiveEach student is expected to read, understand, and elaborate on a selected research paper. To this end, (s)he should give a 45-min. presentation about the paper. The process includes

* getting an overview of the related literature;
* understanding and working out the background/motivation:
why and where are the questions addressed relevant?
* understanding the contents of the paper in all details;
* selecting parts suitable for the presentation;
* presenting the selected parts in such a way that an audience
with some basic background in geometry and graph theory can easily understand and appreciate it.
ContentThis seminar is held once a year and complements the course Geometry: Combinatorics & Algorithms. Students of the seminar will present original research papers, some classic and some of them very recent. The seminar is a good preparation for a master, diploma, or semester thesis in the area.
Prerequisites / NoticePrerequisite: Successful participation in the course "Geometry: Combinatorics & Algorithms" (takes place every HS) is required.
» Seminars (Mathematics Master)
Minor Courses
no minor course offered in this semester
Bachelor's Thesis
NumberTitleTypeECTSHoursLecturers
401-2000-00LScientific Works in Mathematics
Target audience:
Third year Bachelor students;
Master students who cannot document to have received an adequate training in working scientifically.
O0 creditsÖ. Imamoglu, E. Kowalski
AbstractIntroduction to scientific writing for students with focus on publication standards and ethical issues, especially in the case of citations (references to works of others.)
ObjectiveLearn the basic standards of scientific works in mathematics.
Content- Types of mathematical works
- Publication standards in pure and applied mathematics
- Data handling
- Ethical issues
- Citation guidelines
Lecture notesMoodle of the Mathematics Library: https://moodle-app2.let.ethz.ch/course/view.php?id=519
Prerequisites / NoticeDirective Link
401-2000-01LLunch Sessions – Thesis Basics for Mathematics Students
Details and registration for the optional MathBib training course: https://www.math.ethz.ch/mathbib-schulungen
Z0 creditsSpeakers
AbstractOptional course "Recherchieren in der Mathematik" (held in German) by the Mathematics Library.
Objective
401-3990-10LBachelor's Thesis Restricted registration - show details
Successful participation in the course unit 401-2000-00L Scientific Works in Mathematics is required.
For more information, see www.math.ethz.ch/intranet/students/study-administration/theses.html
O8 credits11DSupervisors
AbstractThe purpose of the BSc thesis is to deepen knowledge in a certain subject chosen by the student. In their BSc thesis, students should demonstrate their ability to carry out independent work in mathematics and to organize results in a written report.
Objective
GESS Science in Perspective
Science in Perspective
» see Science in Perspective: Type A: Enhancement of Reflection Capability
» Recommended Science in Perspective (Type B) for D-MATH
Language Courses
» see Science in Perspective: Language Courses ETH/UZH
Additional Courses
NumberTitleTypeECTSHoursLecturers
401-5000-00LZurich Colloquium in Mathematics Information E-0 creditsR. Abgrall, P. L. Bühlmann, M. Iacobelli, A. Iozzi, S. Mishra, R. Pandharipande, further lecturers
AbstractThe lectures try to give an overview of "what is going on" in important areas of contemporary mathematics, to a wider non-specialised audience of mathematicians.
Objective
401-5990-00LZurich Graduate Colloquium Information E-0 creditsA. Iozzi, University lecturers
AbstractThe Graduate Colloquium is an informal seminar aimed at graduate students and postdocs whose purpose is to provide a forum for communicating one's interests and thoughts in mathematics.
Objective
402-0101-00LThe Zurich Physics Colloquium Information
Cancelled until further notice.
E-0 credits1KS. Huber, A. Refregier, University lecturers
AbstractResearch colloquium
ObjectiveThe goal of this event is to bring you closer to current day research in all fields of physics. In each semester we have a set of distinguished speakers covering the full range of topics in physics. As a participating student should learn how to follow a research talk. In particular, you should be able to extract key points from a colloquium where you don't necessarily understand every detail that is presented.
402-0800-00LThe Zurich Theoretical Physics Colloquium Information E-0 credits1KO. Zilberberg, University lecturers
AbstractResearch colloquium
Objective
Prerequisites / NoticeTalks in German are also possible.
251-0100-00LComputer Science Colloquium Information E-0 credits2KLecturers
AbstractInvited talks on the entire spectrum of Computer Science. External guests are welcome. A detailed program is published at the beginning of every semester.
Objective
ContentEingeladene Vorträge aus dem gesamten Bereich der Informatik, zu denen auch Auswärtige kostenlos eingeladen sind. Zu Semesterbeginn erscheint jeweils ein ausführliches Programm.