2018 Semester 2
Advanced Topics in Geometry and Topology (MAST90124)
--Vector bundles over Riemann surfaces.
Instructor Yaping Yang
Office: Peter Hall building, Office 165
Office Hours: Mondays 10-11:30pm, Thursdays 2-3:30 pm.
Peter Hall-G03 (Evan Williams Theatre)
Monday 15:15-16:15. Thursday 11:00-12:00 Friday 11:00-12:00
Prerequisites(from the handbook):
Students must have completed one of the following subjects:
1. [Hitchin]: Integrable Systems, Twistors, Loop Groups, and Riemann Surfaces, N. J. Hitchin, G. B. Segal, and R. S. Ward.
(We will focus on the lecture: Riemann Surfaces and integrable Systems by Hitchin, Page11--52.)
2.[Gunning 1966]: Lectures on Riemann surfaces, by R.C. Gunning, Princeton University Press 1966. (1966)
3.[Gunning 1967]: Lectures on vector bundles over Riemann surfaces, by R.C. Gunning, Princeton University Press 1967. (1967)
4.[Forster]: Lectures on Riemann Surfaces, by Otto Forster, Graduate Texts in Mathematics.
1. [Lee]: Introduction to smooth manifolds, by John Lee.
Up to 50 pages of written assignments over the course of the semester, worth a total of 50% of the mark. A 3-hour written examination during the examination period worth 50% of the mark.
Assignment 1: is due on Aug 20, 2018.
Assignment 2: is due on Sep 7, 2018 (extended to Sep 14, 2018)
Assignment 3: due on Oct 12, 2018.
!!! Final Review !!!
Pre Exam Consultation: Monday Oct. 29, 2018: 10am-1pm.
Tentative Schedule and lecture notes:
Topics: Smooth manifolds, Riemann surfaces, holomorphic maps, examples.
23 July 2018:
Lecture notes 1, Read Page 1-4 of [Hitchin], Slides of Soibelman on "Moduli spaces of Higgs bundles in mathematics andphysics"
26 July 2018:
Lecture notes 2, Read Page 11-12 of [Hitchin], and read Section 1 of [Forster]
27 July 2018:
Lecture notes 3, Read Page 13-14 of [Hitchin], and read Section 2 of [Forster]
Line bundles, the canonical bundle
30 July 2018:
Lecture notes 4, Read Page 14-17 of [Hitchin].
2 August 2018
Lecture notes 4.5. Read Page 219 of [Forster].
3 August 2018
See Lecture notes 4. Read Page 14-17 of [Hitchin].
Sheaves, sheaf cohomology
6 August 2018
Read [Lee] P41-52, P57-58 on tangent bundles, and P65-P70 on cotangent bundles.
Read [Hitchin] P17-P18.
9 August 2018
10 August 2018
Read [Hitchin] P19-21
Properties of sheaf cohomology, the Picard group.
13 August 2018
16 August 2018
17 August 2018
Serre Duality, De Rham cohomology, the De Rham theorem.
20 August 2018
23 August 2018
24 August 2018
Vector bundles, rank and degree, sections, construction of new vector bundles, Chern classes, the Riemann–Roch theorem.
Classification of vector bundles on P^1
27 August 2018
30 August 2018
31 August 2018
Direct image of line bundles, the Riemann Hurwitz genus formula.
Hyperelliptic Riemann surfaces.
3 September 2018
6 September 2018
7 September 2018
The theta divisor, the Torelli theorem.
The Jacobian, Matrix polynomials and Lax pairs
10 September 2018
13 September 2018
14 September 2018
The Jacobian, Matrix polynomials and Lax pairs
Symplectic manifolds, Hamiltonian vector fields.
17 September 2018
20 September 2018
21 September 2018
Monday 24 September to Sunday 30 September:
Non Teaching Period & UA Common Vacation Week
Completely integrable hamiltonian systems
1 October 2018
4 October 2018
5 October 2018
(Semi)-stable vector bundles.
The Hitchin system: the moduli space of semi-stable rank r and degree d Higgs bundles
8 October 2018
11 October 2018
12 October 2018
The Hitchin fibration
15 October 2018
18 October 2018
19 October 2018
Try to read Hitchin's paper.
2 November 2018, Friday.
HOW TO LEARN abstract MATHEMATICS (by Ivan Mirković, from http://people.math.umass.edu/~mirkovic/B.UndergraduateCourses/370/1.html)
- The following is what I see as the BASIC approach towards learning mathematics at the conceptual level. The procedure is
- (0) You start by hearing (or reading) of a new idea, new procedure, new trick.
- (1) To make sense of it you check what it means in sufficiently many examples. You discuss it with teachers and friends.
- (2) After you see enough examples you get to the point where you think that you more or less get it. Now you attempt the last (and critical) step:
- (3) retell this idea or procedure, theorem, proof or whatever it is, to yourself in YOUR OWN words.
- More on step (3).
- Trying to memorize someone else's formulation, is a beginning but it is far from what you really need.
- You should get to the stage where you can tell it as a story, as if you are teaching someone else.
- When you can do this, and your story makes sense to you, you are done. You own it now.
- However, if at some point you find a piece that does not make sense, then you have to return to one of the earlier steps (1--3) above. Repeat this process as many times as necessary.