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.

Email: yaping.yang1@unimelb.edu.au

Class meeting

Location:

Peter Hall-G03 (Evan Williams Theatre)

Time:

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:

References:

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.

Background reading:

1. [Lee]: Introduction to smooth manifolds, by John Lee.

http://webmath2.unito.it/paginepersonali/sergio.console/lee.pdf

Assessment:

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:

Week 1

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]

Week 2

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].

Week 3

Sheaves, sheaf cohomology

6 August 2018

Lecture notes 5.5, Lecture notes 5.

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

Lecture notes 6

Read [Hitchin] P19-21

Week 4

Properties of sheaf cohomology, the Picard group.

13 August 2018

Lecture notes 6

16 August 2018

17 August 2018

Lecture notes 7

Week 5

Serre Duality, De Rham cohomology, the De Rham theorem.

20 August 2018

Lecture notes 8

23 August 2018

Lecture notes 9

24 August 2018

Lecture notes 9

Week 6

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

Lecture notes 10

30 August 2018

Lecture notes 10

31 August 2018

Lecture notes 11

Week 7

Direct image of line bundles, the Riemann Hurwitz genus formula.

Hyperelliptic Riemann surfaces.

3 September 2018

Lecture notes 12

6 September 2018

Lecture notes 12

7 September 2018

Lecture notes 13

Week 8

The theta divisor, the Torelli theorem.

The Jacobian, Matrix polynomials and Lax pairs

10 September 2018

Lecture notes 14

13 September 2018

14 September 2018

Lecture notes 15

Week 9

The Jacobian, Matrix polynomials and Lax pairs

Symplectic manifolds, Hamiltonian vector fields.

17 September 2018

Lecture notes 15

Lecture notes 16

20 September 2018

Lecture notes 17

21 September 2018

Lecture notes 17

Monday 24 September to Sunday 30 September:

Non Teaching Period & UA Common Vacation Week

Week 10

Coadjoint orbits

Completely integrable hamiltonian systems

Calogero-Moser system.

1 October 2018

4 October 2018

5 October 2018

Lecture notes 18.5

Week 11

(Semi)-stable vector bundles.

The Hitchin system: the moduli space of semi-stable rank r and degree d Higgs bundles

8 October 2018

Lecture notes 18

11 October 2018

Lecture notes 19

12 October 2018

Week 12

The Hitchin fibration

15 October 2018

18 October 2018

19 October 2018

Try to read Hitchin's paper.

Final:

2 November 2018, Friday.

8:30am-11:30am

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.