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Differential Topology (ΓΕ4): Διαφορά μεταξύ των αναθεωρήσεων

Differential Topology (ΓΕ4): Διαφορά μεταξύ των αναθεωρήσεων

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[[Graduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Διαφορική Τοπολογία (ΓΕ4)|Ελληνική Έκδοση]]
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=== General ===
=== General ===
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! Prerequisite Courses
! Prerequisite Courses
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Topology, Differential Geometry (ΓΕ2)
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! Language of Instruction and Examinations
! Language of Instruction and Examinations
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! Learning outcomes
! Learning outcomes
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In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds. Emphasis is give to Morse theory.
In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds.
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! General Competences
! General Competences
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=== Syllabus ===
=== Syllabus ===


* Homology and cohomology.
* Manifolds.
* Betti numbers.
* Immersions, embeddings and submersions.
* Attaching and gluing manifolds.
* Milnor’s proof of the fundamental theorem of algebra.
* Morse functions.
* Sard’s theorem and Morse functions.
* Sard’s Theorem.
* Partition of unity and Whitney’s embedding theorem.
* Passing through a critical value.
* Homotopy and isotopy.
* Regular interval theorem.
* Brouwer’s degree.
* CW decomposition of manifolds.
* Whitney’s approximation theorem.
* Morse inequalities.
* Differential forms and integration.
* Total curvature and Gauss maps.
* Hopf's invariant.
* Hopf's degree theorem.


=== Teaching and Learning Methods - Evaluation ===
=== Teaching and Learning Methods - Evaluation ===
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Τελευταία αναθεώρηση της 16:28, 15 Ιουνίου 2023

General

School School of Science
Academic Unit Department of Mathematics
Level of Studies Graduate
Course Code ΓΕ4
Semester 2
Course Title Differential Topology
Independent Teaching Activities Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type Special Background
Prerequisite Courses -
Language of Instruction and Examinations

Greek

Is the Course Offered to Erasmus Students Yes (in Greek)
Course Website (URL) See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds.

General Competences
  • Work autonomously
  • Work in teams
  • Develop critical thinking skills

Syllabus

  • Manifolds.
  • Immersions, embeddings and submersions.
  • Milnor’s proof of the fundamental theorem of algebra.
  • Sard’s theorem and Morse functions.
  • Partition of unity and Whitney’s embedding theorem.
  • Homotopy and isotopy.
  • Brouwer’s degree.
  • Whitney’s approximation theorem.
  • Differential forms and integration.
  • Hopf's invariant.
  • Hopf's degree theorem.

Teaching and Learning Methods - Evaluation

Delivery

Face-to-face

Use of Information and Communications Technology -
Teaching Methods
Activity Semester Workload
Lectures 39
Autonomous Study 78
Solution of Exercises-Homeworks 70.5
Course total 187.5
Student Performance Evaluation

Weakly HomeWorks, presentations in the blackboard of HomeWorks, written final examination.

Attached Bibliography

  • T. Bröcker, K. Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
  • V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
  • J. Milnor, Morse Theory, Annals of Mathematical Studies, 51. Princeton University Press, Princeton, N.J. 1963.
  • J. Milnor, Topology from a differentiable viewpoint, The University Press of Virginia, Charlottesville, Va. 1965.
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