Here is a list of topics I plan to cover throughout the semester.

Midterm Review

There will be a midterm review session on Tuesday, 2017-10-10, 5-6pm in C204 Wells Hall. Here is a Practice Midterm, its solution, and here are some comments about what you should know for the midterm.


The midterm will take place on Wednesday, 2017-10-18, 9.10-10am in A334 Wells Hall.

Final Exam

The Final Exam will take place on Thursday, 2017-12-14 2017 7.45am-9.45am in A334 Wells Hall. Here is a list of topics you might want to review in preparation for the Final Exam

Qualifying Exam

Here is a copy of the Final Exam for you to review in preparation for the qualifying exam. Here are some remarks on the Qualifying Exam.


All homework sets are due at the beginning of the class on the date indicated below. You are encouraged to work in groups to find the solutions, but you need to write them up yourself.

Homework # Due
Homework 10 2017-12-06
Homework 9 2017-11-20
Homework 8 2017-11-10
Homework 7 2017-11-03
Homework 6 2017-10-27
Homework 5 2017-10-13
Homework 4 2017-10-06
Homework 3 2017-09-29
Homework 2 2017-09-22
Homework 1 (PDF,LaTeX source,Hint) 2017-09-11

Class Schedule

Date Topics Reading
2017-12-4 Variation of curvature, gauge invariance of curvature, Yang--Mills functional
2017-12-1 Gauge transformations, covariant exterior derivatives, and curvature
2017-11-29 Covariant Derivatives on associated bundles
2017-11-27 Covariant Derivatives
2017-11-22 Hodge Theory
2017-11-20 De Rham cohomology of connected sums pp. 440-466
2017-11-17 Computation of the de Rham cohomology of CPn pp. 440-466
2017-11-15 Poincaré duality pp. 440-466
2017-11-13 Computation of the de Rham cohomology of Sn pp. 440-466
2017-11-10 The Mayer–Vietoris Theorem pp. 440-466
2017-11-8 Homotopy invariance of de Rham cohomology pp. 440-466
2017-11-6 De Rham cohomology pp. 440-466
2017-11-3 Stokes' Theorem pp. 411-439
2017-11-1 Stokes' Theorem pp. 411-439
2017-10-30 Integration on Manifolds pp. 400-411
2017-10-27 Integration on Manifolds pp. 400-411
2017-10-25 Lie derivatives of differential forms, Cartan's magic fomula pp. 372-373, notes
2017-10-23 Exterior Derivative pp. 362-373
2017-10-20 Differential Forms (cont.) pp. 349-361
2017-10-16 Differential Forms pp. 349-361
2017-10-13 Vector bundle constructions pp. 249-268
2017-10-11 Multilinear algebra (cont.) pp. 304-316
2017-10-09 Multilinear algebra pp. 304-316
2017-10-06 Frobenius's Theorem pp. 496-501
2017-10-04 Flows and the Lie derivative pp. 209-236
2017-10-02 Vector fields and integral curves pp. 205-209
2017-09-29 Partitions of unity and the Whitney embedding theorem pp. 40-47 and 131-136
2017-09-27 Submanifolds (cont.) pp. 98-123 and 657-662
2017-09-25 Submanifolds (cont.) pp. 98-123 and 657-662
2017-09-22 Submanifolds pp. 98-123
2017-09-20 Vector fields pp. 174-189
2017-09-18 The tangent bundle pp. 50-75
2017-09-15 Review session with the TA
2017-09-13 first student seminar
2017-09-11 Tangent vectors and tangent spaces:
the geometer's and the algebraist's approach
pp. 50-75, my notes
2017-09-08 Tangent vectors and tangent spaces:
the physicist's approach
pp. 50-73
2017-09-04 Smooth maps between smooth manifolds pp. 32-40
2017-09-01 What is a smooth manifold? pp. 10-29
2017-08-30 What is a topological manifold? pp. 1-10 and Appendix A

Student Seminar

Here is a list of possible topics for the student seminar. If you have your own ideas for topics, please, don't hesitate to propose them.

The student seminar takes place on Thursday 2-3pm or Friday 3-4pm in C117. For details, please, see the schedule below.

You can sign up for the student seminar by simply emailing me at, talking to me after class, or coming to my office.

Date Topics Speakers Room
2017-12-8 3-4pm Linking numbers Aaron Roach C117 WH
2017-12-7 2-3pm Hopf–Rinow Theorem Dimitris Vardakis C117 WH
2017-12-1 3-4pm Geodesics in Riemannian Geometry Zack Bezemek C117 WH
2017-11-30 2-3pm Lie groups actions and de Rham cohomology Jon Miles C117 WH
2017-11-17 3-4pm Symplectic geometry and Hamiltonian flows Mike Annunziata and Keshav Sutrave C117 WH
2017-11-16 2-3pm Complex geometry Christopher Grow C117 WH
2017-11-3 3-4pm The connected sum of manifolds David Storey C117 WH
2017-11-2 2-3pm Morse theory Alex Ginsberg C117 WH
2017-10-26 2-3pm Brouwer's Fixed-Point Theorem Mark Roach C117 WH
2017-10-5 2-3pm Lie groups, Lie algebras Franciska Nekaien, Rachel Domagalski C117 WH
2017-09-29 3-4pm classification of 1–manifolds Joshua Schroeder C117 WH
2017-09-22 3-3.30pm The Grassmann manifold Stan Tan C117 WH
2017-09-21 2.10-3.40pm point-less topology Brian Pinsky C117 WH
2017-09-13 9-10am Manifolds with boundary, Quotient Manifolds Alex Wilson, Joe Melby A334 WH

If you cannot make it to the student seminar or simply don't want to give a presentation in the student seminar, you can alternatively do a little project which will count for 10% of the grade.


The final course grade will be a weighted average and computed as follows:
Final Grade = 20% Midterm + 25% Final Exam + 45% Homework + 10% Student Seminar.