# 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

Read [Hitchin] P19-21

Week 4

Properties of sheaf cohomology, the Picard group.

13 August 2018

16 August 2018

17 August 2018

Week 5

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

20 August 2018

23 August 2018

24 August 2018

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

30 August 2018

31 August 2018

Week 7

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

Hyperelliptic Riemann surfaces.

3 September 2018

6 September 2018

7 September 2018

Week 8

The theta divisor, the Torelli theorem.

The Jacobian, Matrix polynomials and Lax pairs

10 September 2018

13 September 2018

14 September 2018

Week 9

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

Week 10

Coadjoint orbits

Completely integrable hamiltonian systems

Calogero-Moser system.

1 October 2018

4 October 2018

5 October 2018

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

11 October 2018

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 Mirkovi__ć,__ 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.