Teaching
Mathematics is not a thing to suffer but a thing to enjoy.
~ Masaki Yoshida in the preface of the book Hypergeometric Functions, My Love!

Seminar Course on Branched Riemann Surfaces: Jan-April 2023.

This is a zero credit elective course for undergraduate students at IISER Pune, every weekly lecture will be of 1.5 hrs with a break of 5 minutes in between. 

In the slides here is the planned overview of the course and the following is the detailed plan up to mid-semester:

Lecture 1: Without requiring any prerequisite, we will start by recalling five postulates of Euclid and study some interesting alternatives of the fifth postulate e.g. Clairaut's and Playfair's axioms. In the second part of the lecture we will study some theorems of neutral geometry and non-Euclidean geometry (with emphasis on hyperbolic geometry) developed by Saccheri and Legendre. Lecture Notes part-1: and part-2.

Lecture 2:  In this lecture, continuing from the previous lecture, we will study arguments of Saccheri and Legendre (for their wrong proofs of the 5th postulates) with special emphasis on the "defect" of the triangle. we will discuss the relation between angle defect and concept of area in the hyperbolic geometry. Next we introduce a model of the hyperbolic plane (namely the upper-half plane model), and their elements (lines, circles, triangles, tessellations, angles, boundary etc). Notes part-1 and part-2.

Lecture 3: In this lecture, after recalling a few things from the previous lecture, we will study elementary geometric transformations such as rotation, translation, reflection, glide reflection with an emphasis on inversion in a circle. Notes part-1 and part-2

Lecture 4:  In this lecture we extend our study of inversions in circle from the previous lecture. And then we discuss elementary hyperbolic transformations. Next we give representation of Euclidean and Hyperbolic transformations in terms of complex numbers. And towards the end we discuss Mobius transformations. Notes.

Lecture 5 : In this lecture we focus on group of Mobius transformations on Riemann sphere CP^1 and its important subset namely upper half plane.
We will see which Mobius transformations corresponds to which transformations. After discussing its fixed points, we will define Fuchsian groups,
as a discrete and equivalently as properly discontinuous group. In the end we will note a few important but basic facts which often causes confusion to beginners. Notes 

Lecture 6: Fundamental domain/region is an essential concept in the study of action of Fuchsian group on hyperbolic plane. In this lecture we will focus on this and an special Fundamental domain known as Dirichlet’s domain and related concepts. Notes are here

Lecture-7: In this last lecture to define and discuss the concept of “Limit set” of a Fuchsian group, we revisit the notion of properly discontinuous action. We further discuss a couple of important examples of Fuchsian groups e.g. congruence subgroups of modular group, triangle group with emphasize on Hecke triangle group. Notes are here.

Disclaimer: Notes posted here are sketchy and imprecise, maybe sometimes with error (conceptual and typographical both), please use them with care and reach out me in case you have confusion and corrections to share :)

Tentative plan of upcoming lectures: Brief review of topological aspects of complex analysis, Idea and the definition of a Riemann surface, some examples.
Problems in classical function theory, Riemann surface of multi-valued functions, monodromy of elliptic functions, modular group and elliptic curve, Weirstrass p-function, modular function j.
Functions on the Riemann surface. Holomorphic mapping, differential, meromorphic function, Riemann existence theorem, function fields, and examples.
Branched Riemann Surface. Review of unramified covering theory, de gree, ramified coverings, Riemann-Hurwitz formula, three-point branched covers of P^1(\C),

Old Teaching/Tutorship.

• Introduction to Hyperbolic Geometry, a 2-hour lecture for under-graduate students, 20th Sep 2023.

• Twelve lectures (of 1:30 hrs each) on branched Riemann surface at IISER Pune from Jan-April 2023. 

• Differential geometry course, Virtual AFS-II at Shiv Nadar University, 29th November 2021 to 8th January 2022.

• Complex Analysis course, Virtual AFS program, SGGS Nanded, May 2020 .

• 15 lectures (of one hour each) on advanced calculus (inverse and implicit function theorem)
  to M.Sc. Ist year, Abeda Inamdar College, Pune from July-Sep 2019.

• Differential Geometry course, Annual Foundation School, Part 2, BP, Pune, June-2019.

• Complex Analysis course, Annual Foundation School, Part 1, BP Pune, December-2018.

• Complex Analysis course, Annual Foundation School, Part 1, IIT Delhi, December-2017.

• Differential Topology course, Annual Foundation School, Part 2, CMS Almora, May-2014.

• A remedial course in Calculus - I to first year students in CEBS, Kalina Campus, Mumbai 2013.