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Differential Topology (ΓΕ4): Διαφορά μεταξύ των αναθεωρήσεων

Differential Topology (ΓΕ4): Διαφορά μεταξύ των αναθεωρήσεων

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(Νέα σελίδα με 'Graduate Courses Outlines - [https://math.uoi.gr Department of Mathematics] === General === {| class="wikitable" |- ! School | School of Science |- ! Academic Unit | Department of Mathematics |- ! Level of Studies | Graduate |- ! Course Code | ΧΧΧ |- ! Semester | 000 |- ! Course Title | ΧΧΧ |- ! Independent Teaching Activities | Lectures (Weekly Teaching Hours: 3, Credits: 7.5) |- ! Course Type | General Background |- ! Prerequisite Courses | - |- ! L...')
 
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(11 ενδιάμεσες αναθεωρήσεις από τον ίδιο χρήστη δεν εμφανίζεται)
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[[Graduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
* [[Διαφορική Τοπολογία (ΓΕ4)|Ελληνική Έκδοση]]
{{Course-Graduate-Top-EN}}
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=== General ===
=== General ===
Γραμμή 15: Γραμμή 17:
|-
|-
! Course Code
! Course Code
| ΧΧΧ
| ΓΕ4
|-
|-
! Semester
! Semester
| 000
| 2
|-
|-
! Course Title
! Course Title
| ΧΧΧ
| Differential Topology
|-
|-
! Independent Teaching Activities
! Independent Teaching Activities
Γραμμή 27: Γραμμή 29:
|-
|-
! Course Type
! Course Type
| General Background
| Special Background
|-
|-
! Prerequisite Courses
! Prerequisite Courses
Γραμμή 34: Γραμμή 36:
! Language of Instruction and Examinations
! Language of Instruction and Examinations
|
|
ΧΧΧ
Greek
|-
|-
! Is the Course Offered to Erasmus Students
! Is the Course Offered to Erasmus Students
| Yes
| Yes (in Greek)
|-
|-
! Course Website (URL)
! Course Website (URL)
Γραμμή 49: Γραμμή 51:
! Learning outcomes
! Learning outcomes
|
|
ΧΧΧ
In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds.
|-
|-
! General Competences
! General Competences
|
|
ΧΧΧ
* Work autonomously
* Work in teams
* Develop critical thinking skills
|}
|}


=== Syllabus ===
=== Syllabus ===


ΧΧΧ
* 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 ===
=== Teaching and Learning Methods - Evaluation ===
Γραμμή 66: Γραμμή 80:
! Delivery
! Delivery
|
|
ΧΧΧ
Face-to-face
|-
|-
! Use of Information and Communications Technology
! Use of Information and Communications Technology
|
| -
ΧΧΧ
|-
|-
! Teaching Methods
! Teaching Methods
Γραμμή 81: Γραμμή 94:
| 39
| 39
|-
|-
| ΧΧΧ
| Autonomous Study
| 000
| 78
|-
|-
| ΧΧΧ
| Solution of Exercises-Homeworks
| 000
| 70.5
|-
|-
| Course total  
| Course total  
Γραμμή 93: Γραμμή 106:
! Student Performance Evaluation
! Student Performance Evaluation
|
|
ΧΧΧ
Weakly HomeWorks, presentations in the blackboard of HomeWorks, written final examination.
|}
|}


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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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