Course Page: Algebraic Curves and Riemann Surfaces

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 and recordings

Please note that the recordings will be deleted after a week due to limited space constraints.

  • 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: Recording, Notes.
  • Lecture 16: November 9: Recording, Notes.
  • Lecture 17 November 15, Recording, Notes
  • Lecture 18: November 16, Recording, 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.
  • 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.

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:

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