• Religion of Rigor & Proofs—Transition to Pure Mathematics. In this course we cover [in six to nine months]
  • • Logics and Techniques of Proofs: Induction, Direct Proofs, Proof by Contrapositive, and Proofs by Contradiction using the textbook ‘How to Prove It’ by Velleman.
    • Equivalence relations, partitions, functions, infinite sets & cardinalities using Mathematical Proofs by Garry Chartrand.
    • Mathematical Thinking: Problem-Solving and Proofs by Douglas B. West and John D’Angelo, both at the UIUC. 
• Euclidean Geometry of Triangles & Circles. It is a sequence of six courses in geometry at AMMOC, and we cover topics central to the Olympiad as well as central ideas of projective & hyperbolic geometries in the languages of groups of symmetries.  
• Elementary Number Theory by David M. Burton and its application to problems of mathematical contests and olympiads, which are very well documented by Titu Andreescu in his definitive text, Number Theory: Structures, Examples and Problems. 
• A Wall Through Combinatorics, by Miklós Bóna.
• Topics in Algebra & Analysis using course notes of the Director, Yaashaa Golovanov.

Note that many of the books written on olympiad mathematics are entirely “problem-oriented,” and therefore, rigorous and detailed theoretical supplements needed to solve these problems are what constitute the core of Golovanov’s lecture for mentees at AMMOC.

• Apprenticeship in Analysis on the real line from textbooks
  • “Analysis I” by Terence Tao and
  • “Analysis I” by D.G.H Garling
  • “Analysis I” by Vladimir Zorich
• Abstract Linear Algebra from the textbooks
  • “Linear Algebra,” by Georgi Shilov 
  • “Linear Algebra,” Friedberg, Spence, and Incel. 
  • “Linear Algebra Every Graduate Ought to Know,” Jonathan Golan
  • “Linear Algebra & Geometry” by Yu. I Manin
• Introduction to Modern Abstract Algebra: Groups, Rings, and Fields from the textbooks
  • “Abstract Algebra,” Dummit & Foote, 
  • “Basic Algebra,’ Anthony W. Knapp (Stonybrook)
  • “Lectures on Abstract Algebra,’ Richard Elman (UCLA).

The most advanced protégé, Prasanna Mahesh Pawar (mentee of AMMOC since 2021), did a systematic study of Groups & Vector spaces, Rings & Modules, Fields & Galois theory. His senior thesis, spanning 180 pages can be accessed here.

• Analysis on metric spaces using the textbooks written by Terence Tao & D.J.H. Garling. 
• Multidimensional Real Analysis, using the textbook written by J.J. Dieustermaat. This is an abstract treatment of differentiation, the inverse & implicit function theorem, manifolds, and tangent space.
• Stokes’ Theorem and Whitney Manifolds using the textbook written by Anthony W. Knapp. A soft copy is freely available at his website. 
• Manifolds and Differential Forms, using the notes of Reyer Sajamar (Cornell). A soft copy is freely available at the website of Prof. Sjamaar. 
• Differential Geometries of Plane Curves using the book with the same title written by Hilario Alencar.
• Basic Study of Different Kinds of Geometries: Euclidean, Projective, Spherical, and Hyperbolic [Dd]—written by A.B. Sossinsky.
• Fields and Galois Theory using the textbook ‘Basic Algebra,’ written by Anthony W. Knapp. A soft copy is freely available at his website. 
• Introduction to Topology using the textbook with the same title written by V.A. Vassiliev.
• Study of Surfaces Almost everything you need to know’ by using the textbook written by Anatole Katok.
• Study of Matrix Group [Dh]—using the textbook with the same title by Kristopher Tapp.

The most advanced protégé, Sarthak Dattatray Dhobale (a freshman at Princeton), has done many of these directed reading projects. He received a full ride to study pure math at Princeton, Class of 2029.