### Dates: 20th September 2021-21st December 2021

**Spatiotemporal Coordinates: **Monday and Tuesday 9:10- 10:25 am @Zoom.

Email me in case you would like to attend the course.

## Lecture notes

Lecture 1: September 20th**: ** **Notes**

Lecture 2: September 21st**: Notes**

Lecture 3: September 27th: **Notes****.**

Lecture 4: September 28th: **Notes.**

Errata: Quotient of a Hausdorff space is not Hausdorff: See here . And a locally Euclidean space also need not be Hausdorff. In the exercise sheet 1, we saw the counter example for both, it is the line with double origin.

Lecture 5: October 4th:** Notes. **

Lecture 6: October 5th:** Notes. **

Lecture 7: October 11th: **Notes.**

Lecture 8: October 12th:** Notes.**

Lecture 9: October 18th: **Notes.**

Lecture 10: October 19th: **Notes. **

Lecture 11: October 25th: Notes.

Lecture 12: October 26th: **Notes.**

Lecture 13: November 1: **Notes.**

Lecture 14: November 2: Notes.

Lecture 15: November 8: Notes.

Lecture 16: November 9: Notes.

Lecture 17 November 15: Notes

Lecture 18: November 16: Notes.

Lecture 19: November 22: Notes

Lecture 20: November 29: Notes

Lecture 21: November 30: Notes

Lecture 22: December 6: Notes

Lecture 23: December 7: Notes

Lecture 24: December 13: Notes

Lecture 25: December 14: Notes

Lecture 26: December 20: Notes

Lecture 27: December 21: Notes

### Exercise Sheets

If you have doubts or questions about the exercise sheet, ask me.

- Exercise sheet 1
- Deadline is 27th September 2021, till 23:59. Have fun with it.
**Plagiarism will be strictly dealt with, so don’t copy.**You are free to discuss with your friends about the exercises. Explain in our own words. Latex-ing is highly encouraged. If you are scanning,**please ensure**that the**hand writing is legible**(much much better if you latex the answers) and all the**pages are in order**, the**pages are numbered**and**correctly oriented**and**the answer sheet/ solution sheet contains your Name.**

- Deadline is 27th September 2021, till 23:59. Have fun with it.
- Exercise sheet 2
- Deadline is 4th October 2021, till 23:59.

- Exercise Sheet 3
- Deadline is 17th October 2021, till 8:59.

- Exercise sheet 4
- Deadline is 22 nd October 2021 till 8:59.

- Exercise sheet 5
- Deadline is 2nd November 2021 till 23:59.

- Exercise sheet 6
- Deadline is 23rd November 2021 till 23:59.

- Exercise sheet 7
- Exercise sheet 8

## FINAL EXAM

Final exam was a take home exam and quite long (8 pages of questions). The goal was to reprove the Abel-Ruffini’s theorem on unsolvability of a general quintic equation by radicals using the theory of Riemann surfaces and not Galois theory. The exam questions were mostly taken from the book “Abel’s Theorem in Problems and Solutions” based on the lectures of Professor V.I. Arnold by V.B. Alekseev.

The idea behind it was that the students would see monodromy properties in a different light than what was presented in the course. And this would be close to the historical reason of why and how Riemann surfaces were first constructed.

Final exam Question paper: PDF.

**Course Plan**

- The
**aim of the course**is to introduce the students to algebraic curves and discuss their geometric properties. - The
**plan**is to follow mostly Miranda’s s book from chap 1-7 (skipping the last section of chap 7). There will be a lot of examples and techniques used in proof would be complex analytic. We will prove Abel-Ruffini theorem, Riemann-Roch theorem and classify curves of lower genus. - What I plan to do differently from the book is to introduce also along with Riemann surfaces, algebraic curves over any fields. In this case one would require different techniques than complex analytic to handle things, but we gain two immediate advantages, first, we don’t have to restrict to smooth curves and second, we don’t have to restrict to complex numbers. The algebraic geometric techniques which replace the complex analytic ones will be revealed in the exercise sheets. (These techniques can also be learnt in the parallel algebraic geometry course at CMI). At the very end of the course we will use Riemann-Roch theorem to prove Weil conjectures for curves. This conjecture (now a theorem) shows how working over finite fields can give us results about complex geometry.
- The
**weekly exercise sheets**will consist of two parts, one about the complex analytic theory of Riemann surfaces and the second part where I will introduce the techniques in algebraic world used to prove the results discussed in the class. The algebraic part can be made optional, depending on the student’s level of interest. **Pre-requisites:**Complex analysis in one variable, abstract topology, abstract algebra (rings, modules, ideals).- (Optional Pre-requisites, which may render slight advantage) Some knowledge of manifolds and fundamental groups

**Grading policy :**

60% Home work

10% Quizzes

30% Final exam.

Exam and quizzes will be open book.

References:

**Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics Volume: 5; 1995; AMS.**- Sam Raskin, The Weil conjectures for curves, https://math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Raskin.pdf
- Mircea Mustata, Zeta functions in Algebraic Geometry, http://www.math.lsa.umich.edu/~mmustata/zeta_book.pdf
- Henryk Żołądek, The topological proof of Abel-Ruffini theorem, Topol. Methods Nonlinear Anal. 16(2): 253-265 (2000).
- Hannah Santa Cruz, A survey on the monodromy of algebraic functions, http://math.uchicago.edu/~may/REU2016/REUPapers/SantaCruz.pdf.
- W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf.
- Historical Fun reading: Pampu, Patrick, What is the genus?, History of Mathematics , Subseries, 2162, Springer, 2016.