Week 1: August 28 - September 1
Monday
Reading:
Section 1: Introduction and Overview.
Wednesday Differential vector calculus
(
slides).
Reading:
Appendix A. Vector Calculus in Euclidean Spaces.
Friday Topology
(
slides).
Reading:
Appendix B. Topology.
Reading questions.
Some
breaking
news.
Turn in: HW 1 (
Overleaf
solution template).
Video of the
week
What is a manifold?
Week 2: September 4 - 8
Monday
No class: Labor Day.
Wednesday Definition of a manifold
(
slides).
Reading:
Sections 1 and 2: Introduction, and Definition of a Manifold
Model for the projective
plane (crosscap)
Friday Differentiable Maps.
Reading:
Section 3: Differentiable Maps
(
slides).
Turn in: HW 2 (
Overleaf
solution template).
Video of the
week
A Möbius strip hidden in the projective plane.
Week 3: September 11 - 15
Monday Tangent space
(
slides).
Reading: Sections 4.1, 4.2, and 4.3. (Three versions of tangent space, their
equivalence and standard bases w.r.t. a chart.)
Reading questions.
Wednesday Tangent space
(
slides).
Quiz topics.
Reading: Sections 4.1, 4.2, and 4.3.
Friday The tangent mapping (differential)
(
slides).
Reading: Section 4.4,
The differential of a mapping of manifolds.
Turn in: HW3 (
Overleaf
solution template).
Video of the
week
What is a moduli space?
Week 4: September 18 - 22
Monday Tensors
(
slides).
Reading: Linear algebra sections 5.1–3.1.
Wednesday Tensors and duality
(
slides).
Quiz topics.
Reading: Section 5.4.
Friday Tensors and duality. (See Wednesday's slides.)
Reading: Review linear algebra (Section 5).
Turn in: HW4 (
Overleaf
solution template).
Video of the
week
What is the tensor product, anyway?
Week 5: September 25 - 29
Monday Vector bundles
(
slides).
Reading: Section 6.
Wednesday Tangent bundles
(
slides).
Quiz topics.
Reading: Section 7.
Friday Differential forms
(
slides).
Reading: Section 8.
Turn in: HW 5 (
Overleaf
solution template).
Week 6: October 2 - 6
Monday Exterior derivatives; orientations
(
slides).
Reading: Sections 8.2 and 9.
Wednesday Exterior derivatives; orientations
(
slides).
Quiz topics.
Reading: Sections 8.2 and 9.
Friday Integration of forms
(
slides).
Reading: Appendix C (Measure theory); Section 10.1..
Midterm exam distributed. The midterm covers the first four weeks of class.
Thus, tensors on a vector space will be covered, but not vector bundles.
Here is the
midterm (
Overleaf
solution template).
Turn in: HW 6 (
Overleaf
solution template).
Week 7: October 9 - 13
Monday Integration
(
slides).
Wednesday No class. No quiz.
Friday Manifolds with boundary.
No class. Click on the title for today to access the lecture video (or go to
our Moodle page).
Reading: Section 10.2.
Midterm due by 1:10 pm.
Week 8: October 23 - 27
Monday Stokes' theorem
(
slides).
Reading: Section 10.3.
Wednesday Stokes' theorem (continued from Monday).
Quiz topics.
Reading: Section 10.3.
Friday de Rham cohomology
(
slides).
Reading: Section 11.1.
Turn in: HW 7 (
Overleaf
solution template).
Week 9: October 30 - November 3
Monday Homotopy invariance of de Rham cohomology
(
slides).
Reading: Section 11.2.
Wednesday Homotopy invariance of de Rham cohomology; exact sequences
(
slides).
Quiz topics.
Reading: Section 11.2.
Friday Exact sequences
(
slides).
Reading: Appendix D (read D.1(exact sequences) carefully; at least skim D.2 (simplicial homology)).
Turn in: HW 8 (
Overleaf
solution template).
Week 10: November 6 - 10
Monday Mayer-Vietoris
(
slides).
Reading: Section 11.3.
Wednesday More topics in de Rham cohomology.
(
slides).
Quiz topics.
Reading: Section 11.
Friday Toric varieties: Introduction
(
slides).
Reading: Section 13.1.
Turn in: HW 9 (
Overleaf
solution template).
Week 11: November 13 - 17
Monday Toric varieties: Introduction.
(
slides, same as
last Friday's).
Reading: Section 13.1.
Wednesday Toric varieties: polytopes, cohomology
(
slides).
Quiz topics.
Reading: Section 13.3.
Friday Toric varieties: cohomology; Hirzebruch surfaces
(
slides).
Reading: Section 13.3.
Turn in: HW 10 (
Overleaf
solution template).
Week 12: November 20 - 24
Monday Grassmannians
(
slides).
Reading: Section 14.1.
Wednesday Plücker embedding
(
slides).
Quiz: none this week.
Reading: Section 14.2.
Turn in: HW 11 (
Overleaf
solution template).
Friday Thanksgiving break.
Reading: None.
Week 13: November 27 - December 1
Monday Schubert varieties
(
slides).
Reading: Section 14.3.1 and 14.3 and
Biography
of Hermann Schubert.
Wednesday Schubert calculus.
Quiz: none this week.
Reading: Section 14.3. Also see problem 15 in
Hilbert's problems, and
the introduction to
Grassmannians, flag varieties, and Gelfand–Zetlin
polytopes.
Friday Schubert calculus
(
slides).
Reading: Section 14.3.
Turn in: HW 12 (
Overleaf
solution template).
Video of the
week
Hilbert's 15th Problem: Schubert Calculus.
Week 14: December 4 - 8
Monday Schubert calculus.
Reading: Section 14.3.
Group problems
(
solutions).
Wednesday Four lines.
(
slides).
Quiz: none this week.
Reading: Section 14.3.
Friday No class.
\(\LaTeX\) document preparation system resources:
- \(\LaTeX\)
(getting started in \(\LaTeX\))
- Some find Overleaf an easy
way to get started.
- Change .pdf to .tex in the URL of (nearly) any file accessed from
this page to get the \(\LaTeX\) source code.
- short
guide