Search result: Catalogue data in Autumn Semester 2020

Mathematics Master Information
Core Courses
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3225-00LIntroduction to Lie Groups Information
Self-service registration for this course unit in myStudies has been closed.
W8 credits4GA. Iozzi
AbstractTopological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.
ObjectiveThe goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.
LiteratureA. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)
A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
H. Samelson: "Notes on Lie algebras" (Springer, '90)
S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)
Prerequisites / NoticeTopology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.

Course webpage: https://metaphor.ethz.ch/x/2020/hs/401-3225-00L/
401-3001-61LAlgebraic Topology I Information W8 credits4GP. Biran
AbstractThis is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include:
singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms.
Objective
Literature1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.


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

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

See also:
http://www.math.cornell.edu/~hatcher/#anchor1772800


3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeYou should know the basics of point-set topology.

Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology").

Some knowledge of differential geometry and differential topology is useful but not strictly necessary.

Some (elementary) group theory and algebra will also be needed.
401-3145-70LAlgebraic Geometry I
Registration for this course unit has been closed.
W10 credits4V + 1UP. Yang
AbstractThis course is an introduction to Algebraic Geometry (algebraic varieties).
ObjectiveLearning Algebraic Geometry.
LiteraturePrimary reference:
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publ., 1969.

Secondary reference:
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* 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.
* D. Eisenbud: Commutative algebra. With a view towards algebraic geometry, GTM 150, Springer Verlag, 1995.
* H. Matsumura, Commutative ring theory, Cambridge University Press 1989.
* N. Bourbaki, Commutative Algebra.

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
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
Prerequisites / NoticeLinear Algebra
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
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NumberTitleTypeECTSHoursLecturers
401-3651-00LNumerical Analysis for Elliptic and Parabolic Partial Differential Equations Information
Course audience at ETH:
3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course
"Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.
W10 credits4V + 1UC. Schwab
AbstractThis course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
ObjectiveParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
ContentThe course will address the mathematical analysis of numerical solution methods
for linear and nonlinear elliptic and parabolic partial differential equations.
Functional analytic and algebraic (De Rham complex) tools will be provided.
Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed.

Particular attention will be placed on developing mathematical foundations
(Regularity, Approximation theory) for a-priori convergence rate analysis.
A-posteriori error analysis and mathematical proofs of adaptivity and optimality
will be covered.
Implementations for model problems in MATLAB and python will illustrate the
theory.

A selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
LiteratureSUPPLEMENTARY Literature (core material will be in lecture notes)


Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp.

A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer,
1st Ed. 2004, 2nd Ed. 2015.

R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013

Additional Literature:
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)

Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp.

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]

V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).
Prerequisites / NoticePractical exercises based on MATLAB

Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations
401-3621-00LFundamentals of Mathematical Statistics Information W10 credits4V + 1US. van de Geer
AbstractThe course covers the basics of inferential statistics.
Objective
401-3622-00LStatistical Modelling Information W8 credits4GP. L. Bühlmann, M. Mächler
AbstractIn regression, the dependency of a random response variable on other variables is examined. We consider the theory of linear regression with one or more covariates, high-dimensional linear models, nonlinear models and generalized linear models, robust methods, model choice and nonparametric models. Several numerical examples will illustrate the theory.
ObjectiveIntroduction into theory and practice of a broad and popular area of statistics, from a modern viewpoint.
ContentIn der Regression wird die Abhängigkeit einer beobachteten quantitativen Grösse von einer oder mehreren anderen (unter Berücksichtigung zufälliger Fehler) untersucht. Themen der Vorlesung sind: Einfache und multiple Regression, Theorie allgemeiner linearer Modelle, Hoch-dimensionale Modelle, Ausblick auf nichtlineare Modelle. Querverbindungen zur Varianzanalyse, Modellsuche, Residuenanalyse; Einblicke in Robuste Regression. Durchrechnung und Diskussion von Anwendungsbeispielen.
Lecture notesLecture notes
Prerequisites / NoticeThis is the course unit with former course title "Regression".
Credits cannot be recognised for both courses 401-3622-00L Statistical Modelling and 401-0649-00L Applied Statistical Regression in the Mathematics Bachelor and Master programmes (to be precise: one course in the Bachelor and the other course in the Master is also forbidden).
401-4889-00LMathematical Finance Information W11 credits4V + 2UJ. Teichmann
AbstractAdvanced course on mathematical finance:
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- additional topics
ObjectiveAdvanced course on mathematical finance, presupposing good knowledge in probability theory and stochastic calculus (for continuous processes)
ContentThis is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.

Topics include
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- and probably others
Lecture notesThe course is based on different parts from different books as well as on original research literature.

Lecture notes will not be available.
Literature(will be updated later)
Prerequisites / NoticePrerequisites are the standard courses
- Probability Theory (for which lecture notes are available)
- Brownian Motion and Stochastic Calculus (for which lecture notes are available)
Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts.

This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II.

For an overview of courses offered in the area of mathematical finance, see Link.
401-3901-00LMathematical OptimizationW11 credits4V + 2UR. Zenklusen
AbstractMathematical treatment of diverse optimization techniques.
ObjectiveThe goal of this course is to get a thorough understanding of various classical mathematical optimization techniques with an emphasis on polyhedral approaches. In particular, we want students to develop a good understanding of some important problem classes in the field, of structural mathematical results linked to these problems, and of solution approaches based on this structural understanding.
ContentKey topics include:
- Linear programming and polyhedra;
- Flows and cuts;
- Combinatorial optimization problems and techniques;
- Equivalence between optimization and separation;
- Brief introduction to Integer Programming.
Literature- Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018.
- Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes.
- Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
- Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.
Prerequisites / NoticeSolid background in linear algebra.
Bachelor Core Courses: Pure Mathematics
Further restrictions apply, but in particular:
401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme.
Analogously for:
401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II
401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II
401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry
For the category assignment take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
NumberTitleTypeECTSHoursLecturers
401-3461-00LFunctional Analysis I Restricted registration - show details
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
E-10 credits4V + 1UA. Carlotto
AbstractBaire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces.
ObjectiveAcquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps.
LiteratureRecommended references include the following:

Michael Struwe: "Funktionalanalysis I" (Skript available at https://people.math.ethz.ch/~struwe/Skripten/FA-I-2019.pdf)

Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011.

Peter D. Lax: "Functional analysis". Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.

Elias M. Stein and Rami Shakarchi: "Functional analysis" (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.

Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017.

Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
Prerequisites / NoticeSolid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with topology and measure theory, in part. Lebesgue integration and L^p spaces).
401-3531-00LDifferential Geometry I
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
E-10 credits4V + 1UW. Merry
AbstractThis will be an introductory course in differential geometry.

Topics covered include:

- Smooth manifolds, submanifolds, vector fields,
- Lie groups, homogeneous spaces,
- Vector bundles, tensor fields, differential forms,
- Integration on manifolds and the de Rham theorem,
- Principal bundles.
Objective
LiteratureThere are many excellent textbooks on differential geometry. A friendly and readable book that covers everything in Differential Geometry I is:

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

A more advanced (and far less friendly) series of books that covers everything in both Differential Geometry I and II is:

S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volumes I and II (1963, 1969) Wiley.
Bachelor Core Courses: Applied Mathematics ...
Further restrictions apply, but in particular:
401-3601-00L Probability Theory can only be recognised for the Master Programme if neither 401-3642-00L Brownian Motion and Stochastic Calculus nor 401-3602-00L Applied Stochastic Processes has been recognised for the Bachelor Programme.
402-0205-00L Quantum Mechanics I is eligible 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
401-3601-00LProbability Theory Information
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits.
E-10 credits4V + 1UA.‑S. Sznitman
AbstractBasics of probability theory and the theory of stochastic processes in discrete time
ObjectiveThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
ContentThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Lecture notesavailable in electronic form.
LiteratureR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
402-0205-00LQuantum Mechanics I Information W10 credits3V + 2UG. M. Graf
AbstractIntroduction to quantum theory: Wave mechanics, Schrödinger equation, angular momentum, central force problems, potential scattering, spin. General structure: Hilbert space, states, observables, equation of motion, density matrix, symmetries, Schrödinger and Heisenberg picture. Approximate methods:
perturbation theory, variational approach, quasi-classics.
ObjectiveIntroduction to single-particle quantum mechanics. Familiarity with basic ideas and concepts (quantisation, operator formalism, symmetries, angular momentum, perturbation theory) and generic examples and applications (bound states, tunneling, hydrogen atom, harmonic oscillator). Ability to solve simple problems.
ContentThe beginnings of quantum theory with Planck, Einstein and Bohr; wave and matrix mechanics; the formalism of quantum mechanics (states and observables, Hilbert spaces and operators), the measurement process, symmetries (translation, rotations), quantum mechanics in one dimension (bound states, scattering problems, tunnel effect, resonances) as well as in three (central force problems, potential scattering), perturbation theory, variational methods, angular momentum and spin; relationship of QM to classical physics; possibly composite systems and entanglement.
Lecture notesAuf Moodle, in deutscher Sprache
LiteratureG. Baym, Lectures on Quantum Mechanics
E. Merzbacher, Quantum Mechanics
L.I. Schiff, Quantum Mechanics
R. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals
J.J. Sakurai: Modern Quantum Mechanics
A. Messiah: Quantum Mechanics I
S. Weinberg: Lectures on Quantum Mechanics
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Algebra, Number Thy, Topology, Discrete Mathematics, Logic
NumberTitleTypeECTSHoursLecturers
401-3119-70Lp-Adic NumbersW4 credits2VP. Bengoechea Duro
AbstractThis course is an introduction to the p-adic numbers. We will see how the field of p-adic numbers Q_p is build. We will explore the (strange) topology and the arithmetic of Q_p, as well as some elementary analytic concepts such as functions, continuity, integrals, etc. We will explain an algebraic and an analytic reasons of interest for the existence of p-adic numbers.
Objective
Content- Absolute values on Q and Completions
- Topology and Arithmetic of Q_p, p-adic Integers
- Equations over p-adic numbers and Hensel's Lemma
- Local-global principle
- Hasse-Minkowski's Theorem on binary quadratic forms
- Elementary Analysis in Q_p
- the p-adic Riemann zeta function
Literature"p-adic Numbers. An Introduction", Fernando Q. Gouvea (Springer)
"p-adic Numbers, p-adic Analysis, and Zeta-Functions", Neal Koblitz (Springer)
"p-adic numbers and Diophantine equations", Yuri Bilu (online notes 2013)
Prerequisites / NoticeThe courses Topology, Measure and Integration, Algebra I/II are required prerequisites.
401-3059-00LCombinatorics II
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.
Selection: Geometry
NumberTitleTypeECTSHoursLecturers
401-3533-70LDifferential Geometry IIIW4 credits2VU. Lang
AbstractTopics in Riemannian geometry in the large: the structure of complete, non-compact Riemannian manifolds of non-negative sectional curvature, including Perelman's (1994) proof of the Cheeger-Gromoll soul conjecture; the Besson-Courtois-Gallot barycenter method (1996) and the proofs of the minimal entropy theorem and the Mostow rigidity theorem for rank one locally symmetric spaces.
Objective
401-4531-66LTopics in Rigidity Theory Information W6 credits3VM. Burger
AbstractThe aim of this course is to give detailed proofs of Margulis' normal subgroup theorem and his superrigidity theorem for lattices in higher rank Lie groups.
ObjectiveUnderstand the basic techniques of rigidity theory.
ContentThis course gives an introduction to rigidity theory, which is a set of techniques initially invented to understand the structure of a certain class of discrete subgroups of Lie groups, called lattices, and currently used in more general contexts of groups arising as isometries of non-positively curved geometries. A prominent example of a lattice in the Lie group SL(n, R) is the group SL(n, Z) of integer n x n matrices with determinant 1. Prominent questions concerning this group are:
- Describe all its proper quotients.
- Classify all its finite dimensional linear representations.
- More generally, can this group act by diffeomorphisms on "small" manifolds like the circle?
- Does its Cayley graph considered as a metric space at large scale contain enough information to recover the group structure?
In this course we will give detailed treatment for the answers to the first two questions; they are respectively Margulis' normal subgroup theorem and Margulis' superrigidity theorem. These results, valid for all lattices in simple Lie groups of rank at least 2 --like SL(n, R), with n at least 3-- lead to the arithmeticity theorem, which says that all lattices are obtained by an arithmetic construction.
Literature- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984.
- D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv
- Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage.
- M.Burger: "Rigidity and Arithmeticity", European School of Group Theory, 1996, handwritten notes, will be put online.
Prerequisites / NoticeFor this course some knowledge of elementary Lie theory would be good. We will however treat Lie groups by examples and avoid structure theory since this is not the point of the course nor of the techniques.
401-4141-70LCurves, Jacobians, and Modern Abel-Jacobi Theory Information W6 credits3VR. Pandharipande
Abstract
Objective
401-3057-00LFinite Geometries IIW4 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.
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4355-70LElliptic Regularity TheoryW8 credits4VM. Struwe
AbstractWe extend the theory developed in Functional Analysis II in various directions, including variants of the maximum principle, Harnack's inequality, L^p-theory, and systems. Certain limit cases will be discussed. Examples, including the harmonic map system, will illustrate the use of these methods.
Objective
LiteratureGiaquinta, Mariano: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

Gilbarg, David; Trudinger, Neil S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin, 2001.

Further references will be given in the lectures.
Selection: Further Realms
NumberTitleTypeECTSHoursLecturers
401-3502-70LReading 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
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