Search result: Catalogue data in Autumn Semester 2019

Mathematics Bachelor Information
Core Courses
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3531-00LDifferential Geometry I 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.
W10 credits4V + 1UU. Lang
AbstractIntroduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
Objective
Lecture notesPartial lecture notes are available from https://people.math.ethz.ch/~lang/
LiteratureDifferential geometry in R^n:
- Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces
- Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten
- Christian Bär: Elementare Differentialgeometrie
Differential topology:
- Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds
- Victor Guillemin & Alan Pollack: Differential Topology
- Morris W. Hirsch: Differential Topology
401-3461-00LFunctional Analysis I 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.
W10 credits4V + 1UM. Struwe
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.
LiteratureWe will be using the Lecture Notes on

"Funktionalanalysis I" by Michael Struwe.

Other useful, and recommended references include the following books:

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

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

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.

Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.

Dirk Werner, "Funktionalanalysis". Springer-Lehrbuch, 8. Auflage. Springer, 2018
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 measure theory, Lebesgue integration and L^p spaces).
401-3371-00LDynamical Systems IW10 credits4V + 1UW. Merry
AbstractThis course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics.
ObjectiveMastery of the basic methods and principal themes of some aspects of dynamical systems.
ContentTopics covered include:

1. Topological dynamics
(transitivity, attractors, chaos, structural stability)

2. Ergodic theory
(Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures)

3. Low-dimensional dynamics
(Poincare rotation number, dynamical systems on [0,1])
LiteratureThe most relevant textbook for this course is

Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.

I will also produce full lecture notes, available from my website

https://www.merry.io/teaching/
Prerequisites / NoticeThe material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory.
401-3001-61LAlgebraic Topology I Information W8 credits4GA. Sisto
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) 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

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

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.
401-3114-69LIntroduction to Algebraic Number Theory Information W8 credits3V + 1UÖ. Imamoglu
AbstractThis is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory.
Objective
ContentThis is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory.
401-3132-00LCommutative Algebra Information W10 credits4V + 1UE. Kowalski
AbstractThis course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry.
ObjectiveWe shall cover approximately the material from
--- most of the textbook by Atiyah-MacDonald, or
--- the first half of the textbook by Bosch.
Topics include:
* Basics about rings, ideals and modules
* Localization
* Primary decomposition
* Integral dependence and valuations
* Noetherian rings
* Completions
* Basic dimension theory
LiteraturePrimary Reference:
1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)
Secondary Reference:
2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013)
Tertiary References:
3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)
5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)
Prerequisites / NoticePrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
» Core Courses: Pure Mathematics (Mathematics Master)
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