MAT367 Differential Geometry
Course information
Code: MAT367S
Instructor: Boris Khesin
E-mail: khesin (at) math.toronto.edu
Class schedule: Thu 4-5 in GB 120, Fri 12-2 in MP 134.
TA office hours: Mon 4:30-5:30 in HU1027.
Instructor office hours: Fri 2-3:30 in BA6228 (tentative)
Note: There will be no tutorials for the course
Schedule changes: TBA
Teaching assistant: Ilia Kirillov
Term Exams (bring your student ID): on Thursdays Feb. 27 (50min) and Mar. 26 (to be announced), the usual class time, in the usual class room GB 120
Test 1
Mock Test Solutions to the Mock Test
Solutions to Test2
Final Exam: Monday April 13, 2020, 9am-12pm, at BA 2195.
Final
Marking Scheme: 20% Homework (best 5/6), 20% Test1, 20% Test2, 40% Final.
Assignments
Assignments will be sent online, via Quercus, every two weeks. You will be asked to submit the solutions electronically, via Quercus.
Problem Set 2 (due Thu Feb. 6)
Problem Set 3 (due Thu Feb. 20)
Problem Set 4 (due Thu Mar. 5)
Problem Set 5 (due Thu Mar. 26)
Problem Set 6 (due Thu Apr. 9)
Lecture notes and suggested references
Lecture notes:
- Introduction to Differential Geometry, book in progress by E. Meinrenken and G. Gross; its newer version with exercises solved is on Quercus.
- Lecture (March 19) on Cartan Calculus: Exterior differential, cohomology groups, Lie derivatives, pull-backs, and contractions of differential forms
Exercises for Lecture March 19
- Lecture (March 20) on integration of differential forms and the Stokes theorem
- Lecture (March 27) on the winding number, degree of a map, linking number, and volume forms
- Bonus* (not required): Lecture (April 2-3) on homology groups, Euler characteristics, Hopf invariant, index of a zero of a vector field, the Poincare-Hopf theorem
- Bonus* (not required): Fun problems in Geometry
In addition, the following texts are recommended:
- Calculus on manifolds, notes by S. Yakoveno, 87pp.
- An introduction to differentiable manifolds and Riemannian geometry, by W.M.Boothby, Academic Press.
- Morse theory, by J.Milnor, ISBN 0691080089,
Additional sources:
- The Hopf Fibration: interactive visualization
- A note on the Fundamental Theorem of Algebra (lecture in detail)
- Huygens and Barrow, Newton and Hooke, by V.Arnold
Overview of topics we intend to cover
- Manifolds, definitions and examples
- Smooth maps and their properties
- Submanifolds
- Vector fields and their flows
- Lie brackets
- Frobenius’ theorem
- Differential forms
- The exterior derivative
- Cartan calculus
- Integration and Stokes’ theorem for general manifolds
- Linking and winding numbers