MU:MU25006 Global Analysis - Course Information

MU25006 Global Analysis

The course is not taught in Summer 2025

Extent and Intensity

2/2/0. 6 credit(s). Type of Completion: zk (examination).

Teacher(s)

RNDr. Petr Vojčák, Ph.D. (lecturer)
RNDr. Jiřina Jahnová, Ph.D. (seminar tutor)

Guaranteed by

RNDr. Petr Vojčák, Ph.D.
Mathematical Institute in Opava

Prerequisites (in Czech)

TYP_STUDIA(BN)

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Geometry and Global Analysis (programme MU, NMgr-M)

Course objectives

The classical mathematical analysis deals with functions, vector fields, differential equations, etc., in Euclidean space R^n. The global analysis strives for an application of similar methods on more general sets, so-called smooth manifolds. The main goal of this course is a study of maps between manifolds. Primarily we will use such concepts which are invariant with respect to changes of coordinates.

Syllabus

• Differentiable maps between manifolds, algebra of smooth functions on manifolds and its derivations.
  Maps of constant rank, immersion, submersion, embeddings.
  Critical points and the Sard theorem; the Whitney theorems.
  Tensor bundles on manifolds, tangent and cotangent bundles, local and global sections, bundle maps.
  Vector distributions, the Frobenius theorem.
  Differential forms, de Rham cohomology, the Poincare lemma, the Poincare duality.
  

Literature

required literature
• J. M. Lee. Introduction to Smooth Manifolds. Springer-Verlag, New York, 2002. info
• D. W. Kahn. Introduction to Global Analysis. Academic Press, 1980. ISBN 0-12-394050-8. info

recommended literature
• L. Krump, V. Souček, J. A. Těšínský. Matematická analýza na varietách. Praha, Karolinum, 1998. info
• D. Krupka. Úvod do analýzy na varietách. SPN, Praha, 1986. info
• F. Warner. Foundations of differentiable manifolds and Lie groups. Springer-Verlag, N.Y.-Berlin, 1971. info
• R. Narasimhan. Analysis on real and complex manifolds. North-Holland Publishing Company, Amsterdam, 1968. info
• M. Spivak. Calculus on Manifolds. 1965. info

Language of instruction

Czech

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.

Teacher's information

To obtain the course credits it is necessary to actively participate in the seminar and solve homework problems. The additional potential requirements are set by the tutor. The final exam consists of a written and an oral part. In the written part, it is necessary to solve two assigned problems and potentially be able to explain some details of the solutions. The oral part comprises two theoretical questions.

Activity	Difficulty [h]
Cvičení	26
Přednáška	26
Summary	52

• Enrolment Statistics (Summer 2025, recent)
  
• Permalink: https://is.slu.cz/course/sumu/summer2025/MU25006