An InternationalMath Circle
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Programme
Programme

The course of study

The programme runs on two tracks, side by side: rigorous pre-college mathematics, from the middle-school years to the end of high school; and the courses in pure mathematics that universities teach at the upper and honours levels. Every course is taught by the director. Each subject is laid out as a sequence of definitions and problems, in the manner of the Moscow mathematical schools, and the pupils, guided throughout the seminar, eventually prove its theorems themselves. The two tracks are one subject: school, olympiad and research mathematics differ only in the ingenuity they ask for and the scale of their difficulty.

Allow a difficulty to persist in your head for a long time, until you begin to feel yourself the natural host and inhabitant of its ideas.

Yaashaa Golovanov, from a letter to a family, October 2026
Seminars
Online, in groups of three to nine, formed by time zone and by mathematical preparation rather than by school grade.
Hours
In grades VII to IX, twenty-four hours of seminars a month; for senior pupils, forty to fifty, and for those resident in India more still. Written work fills the time between.
Duration
Pupils are admitted for no less than two years, or three for pupils resident in India, and every part of the programme is compulsory.
Contests
Pupils sit at least fifteen international contests a year, set in the United States, Canada, the United Kingdom, Australia and Iran, as exercises in thought, not as ends in themselves.
Plate II
B C A D E F G H K L
Euclid, Elements I.47, in the lettering of Heath’s translation. The square on BC is equal to the squares on BA and AC: the parallelogram BL is equal to the square GB, and CL to HC.

The course at a glance

Two tracks, side by side, over four years. Each step opens its place in the sections below.

Year I
Year II
Year III
Year IV
Pure mathematics
Year IA Transition to Pure Mathematicsproof, sets, functions and cardinality Year IIAnalysis and linear algebraat the upper and honours level
Year IIIDirected readingssixteen, by invitation Year IIIResearch, phase Iexpository, written as a thesis
Year IVResearch, phase IIoriginal work, the pupil’s own
The olympiad
Year ICore topics of the early olympiadgeometry, number theory, combinatorics, algebra Year IIAdvanced training in the olympiadsat least fifteen international contests a year Years III and IVPreparation for the International Mathematical Olympiadwhere a pupil shows the promise of reaching it, with the research lightened
I

Pre-college mathematics

Olympiad books are written around problems. The theory needed to solve those problems is what the lectures supply, and it forms their core.

  1. A Transition to Pure Mathematics

    Six to nine months. The logic and technique of proof (induction, direct proof, contrapositive and contradiction), then equivalence relations, partitions, functions, infinite sets and cardinality, each established by proof.

    Doud and Nielsen, A Transition to Advanced Mathematics

  2. The Structure of Arithmetic

    Some six months. The natural numbers, the integers and the rational numbers, constructed from first principles, with every law of arithmetic established by proof: a step that most school curricula skip and many university courses assume.

  3. Euclidean Geometry of Triangles and Circles

    A sequence of six courses: the topics central to the olympiad, and the central ideas of projective and hyperbolic geometry, in the language of groups of symmetries.

  4. Elementary Number Theory

    Burton, Elementary Number Theory · Andreescu and Andrica, Number Theory: Structures, Examples, and Problems

  5. A Walk Through Combinatorics

    Bóna, A Walk Through Combinatorics

  6. Topics in Algebra and Analysis

    The director’s own lecture notes

II

Undergraduate pure mathematics

Taught at the upper and honours level of research universities. Since 2025, at least half of every pupil’s course of study has been undergraduate pure mathematics, alongside the preparation for competitions.

  1. An Apprenticeship in Analysis on the Real Line

    Tao, Analysis I · Garling, A Course in Mathematical Analysis · Zorich, Mathematical Analysis I · Shilov, Elementary Real and Complex Analysis · Hardy, A Course of Pure Mathematics · Courant, Introduction to Calculus and Analysis · Marsden, Elementary Classical Analysis

  2. Abstract Linear Algebra

    Shilov, Linear Algebra · Friedberg, Insel and Spence, Linear Algebra · Golan, The Linear Algebra a Beginning Graduate Student Ought to Know · Kostrikin and Manin, Linear Algebra and Geometry

  3. Introduction to Modern Abstract Algebra: Groups, Rings and Fields

    Dummit and Foote, Abstract Algebra · Knapp, Basic Algebra · Elman, Lectures on Abstract Algebra

III

Year by year

Four years. The first two are the same for every pupil; the last two follow the prospects each pupil shows.

Year I

  • A Transition to Pure Mathematics
  • Introduction to Core Topics in Early Olympiad Preparations

Year II

  • Introduction to Mathematical Analysis and Linear Algebra
  • Advanced Training in the Mathematical Olympiads

Year III

  • Directed readings
  • Research, Phase I: expository work

Year IV

  • Research, Phase II: original work
Years III and IV

Shaped to the prospects each pupil shows, along one of three paths.

Expository research

Built on the directed readings, and written up as a thesis.

Original scientific work

The second phase of research, in work that is the pupil’s own.

The International Mathematical Olympiad

Where a pupil shows the promise of reaching the IMO, the load of research is lightened and the pupil is prepared for it.

IV

Directed reading

By invitation.

Sixteen readings, each built on a single textbook, taken by senior pupils beside the core curriculum. They lead to the expository thesis.

  1. 1Analysis on Metric SpacesTao; Garling
  2. 2Multidimensional Real AnalysisDuistermaat and Kolk
  3. 3Stokes’s Theorem and Whitney ManifoldsKnapp
  4. 4Manifolds and Differential FormsSjamaar
  5. 5Differential Geometry of Plane CurvesAlencar, Santos and Silva Neto
  6. 6Geometries: Euclidean, Projective, Spherical and HyperbolicSossinsky
  7. 7Fields and Galois TheoryKnapp, Basic Algebra
  8. 8Introduction to TopologyVassiliev
  9. 9Lectures on SurfacesKatok and Climenhaga
  10. 10Matrix GroupsTapp
  11. 11Introductory Real AnalysisKolmogorov and Fomin
  12. 12Complex AnalysisSilverman
  13. 13Ordinary Differential EquationsArnold
  14. 14Foundations of MechanicsAbraham and Marsden
  15. 15Topics in Differential GeometryMichor
  16. 16Topology, in the Polish traditionFrom the texts of the Polish school
V

The thesis

Senior pupils write expository theses on what they have read, often of more than a hundred and fifty pages. At the close of the programme each thesis is examined orally, in some twenty to twenty-five meetings of an hour, by the director and by mathematicians from outside the circle.

The AMMOC Theses
Tisya Chandrashekhar Rawat2022 Proofs of Some Classical Theorems Using Group Theory
Sarthak Dattatray Dhobale2024 Distance and Transformation: A Study of Topological and Algebraic Structures
Prasanna Mahesh Pawar2025 Symmetry and Structure: From Groups to the Fundamental Theorem of Galois Theory
VI

Courseware

Open to every reader.

The circle publishes its lecture notes here, for young mathematicians everywhere, whether or not they study in it. The first five are in preparation.

Lecture notes of the circle
  1. 1Linear Algebra: A Research-Oriented SeminarIn preparation
  2. 2Analysis IIn preparation
  3. 3Multidimensional Real AnalysisIn preparation
  4. 4POETICThe prequel to MOTIFIn preparation
  5. 5MOTIFA sequel to POETICIn preparation

Enquiries about the notes may be addressed to the director, golovanov@ammoc.org.