Introduction to Manifolds
August- November 2017


Click here for the home-works.
First lecture on August 14th. First assignment due on August 24th.
Lectures: Monday 10:30 am, Thursday 11:50am
Classroom: Monday: 804, Thursday: 803
Instructor: Priyavrat Deshpande.
Contact: Office: 403
phone: 962
email: pdeshpande AT cmi DOT ac DOT in
Office Hours:
TBA
Texts: An Introduction to Manifolds by Loring Tu (2nd ed.)

Useful reading material:
  • Topology and Geometry by Glen Bredon, Springer-Verlag, GTM 139, 1997.
  • Differential Topology by V. Guillemin and A. Pollack.
  • Differential Forms and Applications by M. do Carmo
Prerequisites:   First course in topology and/or permission from the faculty advisor.
Grading:
  • Homework 30%
  • Mid semester exam 35%
  • Final exam 35%
Web: http://www.cmi.ac.in/~pdeshpande/manif17.html

Course description

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. The aim of this course is to get aquainted with the basic theory and lots of examples of manifolds. Towards the end of the course we will learn to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology.


Homework 

The homework will be assigned roughly every 2 weeks. It is your duty to submit the solutions on time. Copying and/or plagiarism will not be tolerated. Here are a few writing guidelines you might want to follow.
  1. Feel free to work together, but you should submit your own work.
  2. Your questions/comments/suggestions are most welcome. I will also be fairly generous with the hints. However, do not expect any kind of help, including extensions, on the day a homework is due.
  3. Please turn in a neat stapled stack of papers. Refrain from using blank / printing paper. Use ruled paper.
  4. Your final finished version should be as polished as you can make it. This probably means that you cannot submit sketchy solutions or sloppily written first versions. Please expect to do a fair amount of rewriting. Do not hand in work with parts crossed out; either use a pencil and erase or rewrite.
  5. Please write complete sentences that form paragraphs and so forth. It might be a good idea to use short simple sentences; avoid long complicated sentences.
  6. Do use commonly accepted notation (e.g., for functions, sets, etc.) and never invent new notation when there is already some available.
  7. Make sure you provide a statement clearly indicating precisely what it is that you are about to prove. You can, if you want, label your statement as a Theorem or Claim or whatever. Write the word Proof, and then give your proof.
  8. Throughout your writing, constantly tell the reader (i.e. me, :-)) exactly what it is that you are about to demonstrate. Be sure to indicate the end of your proof.
  9. Your answers should combine “logic”, your hypotheses, and possibly other mathematical facts (e.g., theorems that we have proved in class) into an argument that establishes the asserted conclusion.
  10. Please be extra careful about the order in which you use your quantifiers.
  11. In your arguments you can make free use of anything that we have proven in class, and of course all logic rules and basic axioms and definitions. Anything else that you use should be proved.
The homework problems.
  1. Assignment 1 pdf. Due date: 24/08/2017.
  2. Assignment 2 pdf. Due date: 04/09/2017.
  3. Assignment 3 pdf. Due date: 07/09/2017.
  4. Assignment 4 pdf. Due date: 21/09/2017.

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