Suchergebnis: Katalogdaten im Herbstsemester 2020

Mathematik Bachelor Information
Kernfächer
Kernfächer aus Bereichen der reinen Mathematik
NummerTitelTypECTSUmfangDozierende
401-3531-00LDifferential Geometry I
Höchstens eines der drei Bachelor-Kernfächer
401-3461-00L Funktionalanalysis I / Functional Analysis I
401-3531-00L Differentialgeometrie I / Differential Geometry I
401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory
ist im Master-Studiengang Mathematik anrechenbar. Die Kategoriezuordnung können Sie in diesem Fall nicht selber in myStudies vornehmen, sondern Sie müssen sich dazu nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link) wenden.
W10 KP4V + 1UW. Merry
KurzbeschreibungThis 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.
Lernziel
LiteraturThere 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.
401-3461-00LFunctional Analysis I Belegung eingeschränkt - Details anzeigen
Höchstens eines der drei Bachelor-Kernfächer
401-3461-00L Funktionalanalysis I / Functional Analysis I
401-3531-00L Differentialgeometrie I / Differential Geometry I
401-3601-00L Wahrscheinlichkeitstheorie / Probability Theory
ist im Master-Studiengang Mathematik anrechenbar. Die Kategoriezuordnung können Sie in diesem Fall nicht selber in myStudies vornehmen, sondern Sie müssen sich dazu nach dem Verfügen des Prüfungsresultates an das Studiensekretariat (Link) wenden.
W10 KP4V + 1UA. Carlotto
KurzbeschreibungBaire 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.
LernzielAcquire 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.
LiteraturRecommended references include the following:

Michael Struwe: "Funktionalanalysis I" (Skript available at Link)

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.
Voraussetzungen / BesonderesSolid 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).
401-3001-61LAlgebraic Topology I Information W8 KP4GP. Biran
KurzbeschreibungThis 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.
Lernziel
Literatur1) 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:
Link

See also:
Link


3) E. Spanier, "Algebraic topology", Springer-Verlag
Voraussetzungen / BesonderesYou 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 KP4V + 1UP. Yang
KurzbeschreibungThis course is an introduction to Algebraic Geometry (algebraic varieties).
LernzielLearning Algebraic Geometry.
LiteraturPrimary 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, Link
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry Link

"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, Link

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.
Voraussetzungen / BesonderesLinear Algebra
401-3132-00LCommutative Algebra Information
Findet dieses Semester nicht statt.
401-3132-00L Commutative Algebra is not offered in the Autumn Semester 2020. However, a core course 401-3145-70L Algebraic Geometry I is offered instead.
W10 KP4V + 1Ukeine Angaben
KurzbeschreibungThis course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry.
LernzielWe 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
LiteraturPrimary 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)
Voraussetzungen / BesonderesPrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
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