MTH 612 (Fall 2018)
This is a graduate level course of Topology for Fall 2018. You may find the course material here.
We will follow our IISER, Pune Ph. D. syllabus (please find the attachment).
Evaluation:
1. Mid-Sem-30%
2. End-Sem-30%.
3. Class tests-10%+10%
4. Short tests in the tutorials-20% (Four of them ).
Class timing:
Monday-Thursday-Friday 2-3 pm.
Announcements:
1. There will be class on (08/08/2018), 3.10-4. 10pm (Kindly note an unusual time)
2. There will be a short test on 13/08/2018 at 2 pm.
3. There will be classes on (24/10/2018) and (31/10/2018) , 3.10-4. 10pm (Kindly note an unusual time)
4.Short test III will be conducted on either 11/10/2018 (3-4 pm) or 12/10/2018 (2-3 pm). You can take any of the test.
5. No assignments. Students are expected to solve exercises of ``Atiyah Macdonald" without looking at the solution manuals.
There will be 3 tests before mid-sem as part of the continuous evaluation and 3 tests after mid-sem.
Details of Lectures:-
Week 0
1. (02/ 08/2018):-Introduction to the course, quotient topology, different topological spaces.
2. (06/08/2018):- Introduction to fundamental groups.
Week 1
3. (06/08/2018):-Calculations of Fundamental groups. Assignment 1 uploaded.
4. (08/08/2018), 3.10-4. 10pm (Kindly note an unusual time), Statement and some applications of Seifert-Van-Kampen's theorem.
No class on 09/08/2018 and 10/08/2018
Week 2
13/08/2018: Short test 1:
5. 16/08/2018 Proof of the Seifert-Van-Kampen's theorem.
6. 17/08/2018):- Covering spaces. Assignment 2 uploaded.
Week 3
7. (20/08/2018):- Path and homotopy lifting theorem, general lifting criteria.
8. (23/08/2018)- Deck transformations, group actions.
9. 24/08/2018)- Groups actions, regular/normal coverings, universal covering spaces.
Week 4
10. 27/08/2018- Construction of Universal covering space for semi-locally simply connected spaces.
11. (30/08/2018):- Class test 1.
12. (31/08/2018) : Classification theorem for covering spaces. Assignment 3 uploaded.
Week 5
13. (03/09/2018):-CW complexes,
14. (06/09/2018):- fundamental groups of CW complexes, Covering
spaces of CW complexes.
15. (07/09/2018):-Higher homotopy groups, Commutativity of higher homotopy groups, Higher homotopy groups of covers, (Serre) fibrations, Assignment 4 uploaded.
Week 6
14. (10/09/2018):-Revision, tutorial.
15. 13/09/2018 Short test II at 2 pm,
16. (14/09/2018): Maths symposium.
Week 7
16. 17/09/2018: Smooth manifolds.
17.20/09/2018: Smooth maps between manifolds.
18. 21/09/2018: Holiday, Muharam.
Week 8
Mid-sem Week. No lecture.
Week 9.
04/10/2018: Lie goups, Bump functions
05/10/2018: Partition of unity. Extension Lemma, Implicit and inverse function theorem for Euclidean space and Manifolds,
Week 10.
08/10/2018: Tangent vectors for \R^n.Tangent space for arbitrary manifolds, Push-forwards.
11/10/2018 Tangent bundles. 3-4 (Short test III; Question paper I)
12/10/2018: Short test III (Question paper II)
Week 11.
Festival break.
Week 12.
22/10/2018: Introduction to Vector bundles.
24/10/2018: Implicit function theorem Manifolds revisted, Immersions, Submanifolds, submersions. (3.10-4.10 pm)
25/10/2018:. -Sections, cotangent bundles. Assignment 5 uploaded.
26/10/2018: Differential Forms.
Week 13.
29/10/2018: Wedge products, Exterior Derivatives.
31/10/2018: Vector fields.
01/11/2018: Class test II.
02/11/2018: Orientation, Riemannian metric, Riemann volume forms.
Week 14.
05/11/2018: Integration on Manifolds using differential forms. Assignment 6 and 7 uploaded,
08/11/2018:Definition and basic properties of De Rham cohomology groups. Mayer Vietoris and homotopy properties of De Rham cohomology group.
09/11/2018: Stoke's theorem
Week 15.
12/11/2018): Short test IV.
15/11/2018: Level sets, statement of Sard's theorem,
16/11/2018: Bundle maps, tensor products, Whitney sums .
Week 16.
19/11/2018:Tutorial
20/11/2018 onwards: End semester examination.