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Research

Research

Research at AMMOC begins in the directed reading of senior pupils and is completed in an expository thesis, examined orally by the director and by mathematicians from outside the circle. Beside it runs a steady practice of problem-solving for the mathematical journals.

Plate III
23
The sphere within its cylinder, which Archimedes proved to hold two-thirds of its volume and of its surface. He asked that the figure be set on his tomb, and by it Cicero found the tomb, near Syracuse, in 75 BC.
I

Theses

Expository theses written by senior pupils under the direction of Yaashaa Golovanov, and kept as a numbered series.

AMMOC Theses
No. 1 Proofs of Some Classical Theorems Using Group TheoryTisya Chandrashekhar Rawat · 2022Classical theorems of number theory, proved by the methods of group theory.Read the thesis (PDF, 14 pages)Cite asT. C. Rawat, Proofs of Some Classical Theorems Using Group Theory, AMMOC Theses 1, AMMOC — An International Math Circle, 2022.
No. 2 Distance and Transformation: A Study of Topological and Algebraic StructuresSarthak Dattatray Dhobale · 2024From metric spaces and compactness, through groups, their actions and Sylow’s theorems, to linear maps, rings and fields, with his counterexample to an exercise in T. W. Körner’s A Companion to Analysis; one hundred and sixty-nine pages.Read the thesis (PDF, 169 pages)Cite asS. D. Dhobale, Distance and Transformation: A Study of Topological and Algebraic Structures, AMMOC Theses 2, AMMOC — An International Math Circle, 2024.
No. 3 Symmetry and Structure: From Groups to the Fundamental Theorem of Galois TheoryPrasanna Mahesh Pawar · 2025An expository thesis in graduate algebra, of one hundred and eighty-three pages.Examined by Pradeep Das and Sridhar Tamilvanan.Read the thesis (PDF, 183 pages)Cite asP. M. Pawar, Symmetry and Structure: From Groups to the Fundamental Theorem of Galois Theory, AMMOC Theses 3, AMMOC — An International Math Circle, 2025.
In progress

Theses now being written, under working titles. Each joins the series when it has been examined.

  • Pricing Uncertainty: Analysis, Probability and the Black–Scholes FormulaVibhu Konduru
  • Structures and Dimensions: Abstract Algebra and Real Analysis in Several VariablesSergei Makarevich
  • The Real Line and Beyond: Analysis, Linear Algebra and Abstract AlgebraShaurya Patil
  • Towards Differential Forms: Linear Algebra and Analysis in Several VariablesAaditya Sahu
II

Directed reading

Sixteen readings, each on a single textbook, from analysis on metric spaces to the foundations of mechanics, prepare the ground for the thesis.

The sixteen readings

One reading written up in full: Nihad Hashimov’s Mathematical Analysis Studies (PDF, 41 pages).

Under way

Readings also run through the year for groups of pupils, always more than one at a time.

Topology Without Tears, by Sidney A. Morris: Noah Kotto, Sritha Uppaluru, Parth Sakharam Andhare and Yuri Mikhyalov.

Calculus, by Michael Spivak: Sritha Uppaluru, Yuri Mikhyalov, Alicia Zhong, Parth Sakharam Andhare, Noah Kotto and Dhruv Dubey.

Completed

Combinatorics, from A Walk Through Combinatorics, by Miklós Bóna, Principles and Techniques in Combinatorics, by Chen Chuan-Chong and Koh Khee-Meng, and Combinatorics: A Problem-Based Approach, by Pavle Mladenović, among other texts: Juan Silvera, Imana Vellathottam and Shaurya Patil, February 2026.

Its solutions written up in full: Directed Reading in Introductory Combinatorics (PDF, 51 pages).

III

Problem-solving in the journals

Pupils send solutions to the problems posed in Mathematical Reflections, the journal founded by Titu Andreescu, and in Crux Mathematicorum, published by the Canadian Mathematical Society.

Journal Issues Correct solutions
Mathematical Reflections2024, issues 3 to 686
Mathematical Reflections2025, issues 1 to 6197
Mathematical Reflections2026, issue 174
Crux MathematicorumVolume 52 (2026), issues 1 to 423

Crux Mathematicorum has also published original work by Shaurya Patil (MA344), and in September 2026 it printed Parth Sakharam Andhare’s solution to MA356 exactly as he wrote it, choosing it from fifteen submissions.

The problems and their solvers, issue by issue, are listed under Solutions in Mathematical Reflections.

IV

From the seminars

Records of seminars at which the pupils’ work met current research.

Research Seminar in Linear Algebra · September 2026

What does a basis cost?

A note by Yaashaa Golovanov · AMMOC Seminar Notes, No. 1

This week, in our Research Seminars in Linear Algebra for Middle and High-schoolers, my pupils proved that every vector space has a basis. Given Zorn’s Lemma it takes three lines.

Then we asked the better question. What does that theorem cost?

Over the Zermelo–Fraenkel axioms, five statements turn out to be the same statement:

  1. (1)The Axiom of Choice
  2. (2)Zermelo’s Well-Ordering Theorem
  3. (3)The Hausdorff Maximal Principle
  4. (4)Zorn’s Lemma
  5. (5)Every vector space over every field has a basis.

Courses use (4) ⇒ (5) and stop there. The direction that carries the content is (5) ⇒ (1), proved by Andreas Blass in 1984: the bare existence of bases hands back the full Axiom of Choice.

And Choice is not decided by the axioms at all. Gödel showed in 1938 that it cannot be refuted. Cohen showed in 1963, by inventing forcing, that it cannot be proved either. There are models of set theory in which the real numbers, viewed as a vector space over the rationals, have no basis whatsoever.

Blass’s proof carried one hypothesis he could not remove: the Axiom of Foundation. That question stood open for over forty years.

On 17 September 2026 a preprint appeared that closes it. Fernandes, Mezabarba, and Rodrigues give a foundation-free proof—and in fact they need only vector spaces over fields of characteristic zero (arXiv:2609.20140). It is a preprint and not yet refereed, so read it as news rather than as settled literature.

Here is what my pupils took from it. The sentence on our board is one a careful fourteen-year-old can prove. Its exact axiomatic strength was being sharpened in a paper posted the week before we covered it.

They were delighted. So was I.

There is no school mathematics and then real mathematics.
There is one subject, and if you teach it honestly, the frontier is never far away.

Participating pupils

Saarang Agarwal, Aaditya Sahu, Shaurya Patil, Vibhu Konduru, Sritha Uppaluru, Yuri Mikhyalov, Nikolai N. Mnev, Parth Sakharam Andhare, Noah Kotto and Ashwika Nukala.

References

A. Blass, ‘Existence of bases implies the axiom of choice’, in Axiomatic Set Theory, Contemporary Mathematics 31, American Mathematical Society, 1984, pp. 31–33.

G. Fernandes, R. M. Mezabarba and V. de O. Rodrigues, ‘Existence of bases implies the axiom of choice, a foundation-free proof’, preprint, 2026, arXiv:2609.20140.

V

The archive

The first of the circle’s booksIn preparation, as the prequel to MOTIF. POETIC
Research in Problem-SolvingSolutions by the circle’s pupils to problems posed in the mathematical journals. RPS