BR Maths Corner-1

[Amber G.]

Just realized, Nicușor Daniel Dan , Romania's new president won IMO gold medals in two years and perfect scores. Probably a first.

Sharing:

Nicuşor Dan, Romania’s president and a two-time International Mathematics Olympiad gold medallist.

Nicușor Dan, is a fascinating figure — a former math prodigy who scored perfect marks twice at the International Math Olympiad. He’s known not just for his sharp analytical mind, but also for being down-to-earth and thoughtful. (Heard him a few years ago - and liked it - seems to combine deep mathematical thinking with a practical sense of human nature, which is rare)

His election reminds me of another leader (only I can recall) with a strong math background — Lee Hsien Loong of Singapore, who was Senior Wrangler at Cambridge.

More: Nicușor Dan, the maths prodigy who beat an ultranationalist for Romanian presidency

[Vayutuvan]

https://ramanujanexplained.org/

Came across the above interesting website founded by Gaurav Bhatnagar

First Lecture

Abstract
We begin our study of Ramanujan’s identities. The Rogers-Ramanujan identities appear Chapter 16 of Volume 3 of Berndt’s Ramanujan’s notebooks. One of the key results is the q-binomial theorem. We will begin with a discovery approach to the binomial theorem and then give Ramanujan’s own proof of the q-binomial theorem. Ramanujan developed hypergeometric series in an earlier chapter, so chances are that he was motivated to find a more general series with an additional parameter.

[Amber G.]

Big congrats to Eshan Chattopadhyay and David Zuckerman for winning the 2025 Gödel Prize! Their groundbreaking research on explicit two-source extractors is a game-changer in theoretical computer science. Eshan, an Indian-origin professor at Cornell University, and David, a professor at UT Austin, have made significant contributions to pseudorandomness and computational complexity. Their work has opened up new avenues in randomness extraction and Ramsey graphs. Cheers to their achievement! #GodelPrize #ComputerScience #Mathematics

Link:The 2025 Gödel Prize has been awarded to Eshan Chattopadhyay and David Zuckerman.

[Vayutuvan]

https://www.youtube.com/watch?v=e4kOh7qlsM4

~ 29 min video on Riemann Hypothesis. Totally self-contained (including a quick 5 min introduction to complex number plane) with lovely animations.

What is the Riemann Hypothesis REALLY about?

HexagonVideos
674,508 views Dec 12, 2022

Solve one equation and earn a million dollars! We will explorer the secrets behind the Riemann Hypothesis - the most famous open problem in mathematics - and what it would tell us about prime numbers.

I should have mentioned one additional property, namely zeros are mirrored along the line 1/2, even though non of them are found and maybe even non of them even exist. This way, every zero not on the line would give a harmonic with Re(s) greater than 1/2, thereby breaking the estimates for the prime counting function.

[Amber G.]

The 66ᵗʰ International Mathematical Olympiad (IMO) 2025:

-Taking place July 10–20, 2025, with official events beginning July 10 and contest days on July 15–16
- Hosted in Sunshine Coast, Queensland, Australia, at the Sunshine Coast Convention Centre.
At present the official IMO site confirms the country registrations but doesn’t list exact participant numbers. India and US both are taking part, but their team member names are not 'officially' announced yet on the the website(s) - both International and local/national official sites. (Wish they kept it more up-to-date). India's team was selected around end of May (after HBCSE Training Camp). I wished the list should have been public by now.

(Good luck and best wishes to Archit Manas, Able, Kanav, Adhitya, Aarav and Adish. - Picture Credit - SM of a friend)


Both teams are particularly strong. (in 2024, Team USA scored 192 points, securing 1st place)

[Amber G.]

It is not officially posted (yet):

India secured 3 Gold Medals, 2 Silver Medals and 1 Bronze Medal at the International Mathematical Olympiad (IMO) 2025 held at Sunshine Coast, Australia. The team achieved the highest ever total score of 193 since India's first participation in IMO in 1989.

1. Adhitya Mangudy (GOLD)
2. Kanav Talwar (GOLD)
3. Aarav Gupta (GOLD)
4. Abel George Mathew (SILVER)
5. Aadish Jain (SILVER)
6. Archit Manas (BRONZE)

Congratulations!

[Amber G.]

^^^
Looks like two people missed the gold by just 2 points and Archit lost silver by 2 points...

Results are not *official* yet.. but China, United States, South Korea, Japan, and Poland are on top 5 . India is (most likely) at 7th. (Standing may change).. I think Iran will be just after India.

The official site still does not have them results at the time of this posting.. )

[Vayutuvan]

The above page is not loading. Only main page https://www.imo-official.org/ is loading. Maybe too much load on the website.

[Amber G.]

^^^ As said, results were/are not official - though I am sure of medal cut-off points and Indian team scores (The team achieved the highest ever total score since it starting taking part).

-- just added --
But now as I post this, the results are official. It should be on the imo site. now.

- Tid bits - 5 people scored perfect scores! (Q-6 was relatively hard - most missed it.
- USA with 5 gold and 1 silver was in second place.

[Vayutuvan]

- USA with 5 gold and 1 silver was in second place.

Does that mean China in the first place got 6 golds? Wow.

[Prem Kumar]

I think China has been getting 6 golds for a while now, with a couple of them getting a perfect score each year

China is at #2 as well - we will know when we look at the US IMO team picture

[Vayutuvan]

I think China has been getting 6 golds for a while now, with a couple of them getting a perfect score each year
China is at #2 as well - we will know when we look at the US IMO team picture

[Amber G.]

Kudos to Prof. Shanta Laishram and the team!


(All scores, and medals, I put here yesterday are confirmed officially)

Indian students shine again at the International Mathematical Olympiad!

[Amber G.]

Okay I don't know how much serious interest is here in advanced math/science but for those into who love math/science — bright high schoolers or anyone who knows one in India — this is worth a look!

Only a few days left to register for IOQM — the first step toward representing India at the International Math Olympiad. Challenge yourself with the best.

[Vayutuvan]

https://www.youtube.com/watch?v=Cqlpr0W0-34

I love this teacher's excitement and joy. Very infectious.

[Vayutuvan]

[Vayutuvan]

This is same as the original Euler's solution

Then there is a physical/geometric derivation with beautiful animation.

[A_Gupta]

@Vayutuvan, any suggested reading? (Re: viewtopic.php?p=2654756#p2654756 )

[Amber G.]

This is relatively easy problem

One day, Arya decided to make up a number sequence just for fun.
She started with:
𝑎₂ = 2
𝑎₃ = 3
Then she came up with two simple rules to create the rest of the numbers:

To get a number in an even position (like 𝑎₄ 𝑎₆ 𝑎₈ etc), she adds 2 to the numbers at position k and k+1:
𝑎₂ₖ₊₂=2+𝑎ₖ+𝑎ₖ₊₁

To get a number in an odd position (like 𝑎₃, 𝑎₅. 𝑎₇ etc), she takes 2 times the number at position k, and adds 2:
𝑎₂ₖ₊₁ = 2 + 2𝑎ₖ

Now Arya wonders:
For which positive numbers n’s (𝑎ₙ/n) whole number (an integer)?
Can you help her figure it out?

[Vayutuvan]

@Vayutuvan, any suggested reading? (Re: viewtopic.php?p=2654756#p2654756 )

I will try to organize my thoughts. It would be help if I know two things.

1. Why do you ask how to generate problems
2. Your domain. My guess is that you are very good at engineering physics. I read on BRF that you took courses with Richard Feynman. Are you a physicist? You also have done course in ML at Stanford (online certificate). I heard it is a very tough course.

It is not necessary though.

[Amber G.]

Historic Breakthrough in Mathematics: Geometric Langlands Conjecture Proven

One of the most important breakthroughs in modern mathematics has just been achieved — and it’s quietly making waves. The long-elusive geometric Langlands conjecture, a central pillar of the so-called Langlands programme (often dubbed the "grand unified theory of mathematics"), has finally been proven and published. This proof — nearly 1,000 pages long and years in the making — was developed by a team led by Dennis Gaitsgory and Sam Raskin, and it's already drawing major attention in the math world and beyond.

As a physicist, what makes this especially fascinating is the unexpected bridge to quantum field theory. The same deep symmetry (S-duality) that shows up in the equations of electromagnetism and gauge theory also appears in the Langlands correspondence. Edward Witten and others have shown that what once seemed like abstract, esoteric math may actually reflect hidden structures in the fabric of physical law. In other words, this is not just beautiful mathematics — it might be a shadow of deeper symmetries in nature itself.

If there's interest — and people want to dive deeper or understand what this really means — let me know.
-Amber G.

[Prem Kumar]

Yes, please share. I have vaguely heard of the Langlands Programme. Would love to learn more, dumbed down, of course

[A_Gupta]

I will try to organize my thoughts. It would be help if I know two things.

1. Why do you ask how to generate problems
2. Your domain. My guess is that you are very good at engineering physics. I read on BRF that you took courses with Richard Feynman. Are you a physicist? You also have done course in ML at Stanford (online certificate). I heard it is a very tough course.

It is not necessary though.

I will unwind this once only, in the hope of getting gyaan from you and other members of this board.

1. These days, I dabble in stuff like https://projecteuler.net.

2. I want to see if generating problems that are tractable is a faster way to learning than solving problems. I believe that there are fundamental results in mathematics and physics that one must understand, not just "know of", for the good of one's soul; and my pace of learning is inadequate for the number of years likely left to me. Am seeking ways to speed it up.

3. A larger goal from which I have been distracted is to understand Paul Cohen's forcing and the independence of the Axiom of Choice and the Continuum hypothesis from conventional set theory. On the way Gödel's incompleteness theorems in their full technicality I have understood, but not yet with the grasp of fundamentals learned in my younger days.

4. As a graduate student, I and another worked for about six months with Feynman, but that research problem did not get anywhere.

5. Unless the Stanford credential is via Coursera, I must plead not guilty. I did do a lot of ML study, but it has decayed because where I am we are consumers of Ai models, not creators. (Incidentally, I once got a call from a debt collector, and after he satisfied himself that I was not the person he was looking for, he told me he had located 57 or so people with my name, so that might be source of confusion.)

6. My mathematical limitations you may guess from that I could never get very far or very interested in superstring theory, though I was resident at one of the centers of the first superstring revolution, and then a young aspiring physicist.

7. My work for many many years does not require mathematics or physics of any significance; what I have is what I am able to spend some time on.

[Amber G.]

Yes, please share. I have vaguely heard of the Langlands Programme. Would love to learn more, dumbed down, of course

Hope this is useful.


Historic Breakthrough in Mathematics: Geometric Langlands Conjecture Proven:

One of the most profound achievements in modern mathematics unfolded quietly in 2023–2024: a team of nine mathematicians, led by Dennis Gaitsgory and Sam Raskin, proved the geometric Langlands conjecture, a central part of the broader Langlands programme — often described as a “grand unified theory of mathematics.”

This monumental proof, detailed in five papers spanning ~1,000 pages, has been hailed as a triumph not just for what it solves, but for the new mathematical directions it opens. Gaitsgory won the $3M Breakthrough Prize, and Raskin received a New Horizons Prize for this work.


What is the Langlands Programme?:

Proposed in the 1960s by Robert Langlands, the programme aims to connect distant areas of mathematics — especially number theory and harmonic analysis. It inspired key milestones like Andrew Wiles’ proof of Fermat’s Last Theorem.

The geometric Langlands conjecture, a variant formulated in the 1980s, focuses on Riemann surfaces (shapes like spheres or doughnuts) and explores a correspondence between:

Fundamental group representations (loops on a surface, captured as matrices)

Sheaves (geometric objects assigning vector spaces to points)

Why It is important:

Unification: The proof suggests deep connections across fields, including physics.

Foundation for more: It boosts work on local versions of the Langlands conjecture and the more mysterious arithmetic version.

New techniques: It refines prior work from Drinfeld, Beilinson, and others using advanced algebraic geometry and representation theory.

Link to Physics:

Surprisingly, the geometric Langlands programme resonates with quantum field theory, especially through S-duality, a symmetry in electromagnetism and gauge theory. Work by Edward Witten and Anton Kapustin showed parallels between this duality and Langlands symmetry, revealing potential connections between abstract math and the standard model of particle physics.

Impact:

Peter Scholze and Laurent Fargues built a "wormhole" connecting local arithmetic Langlands conjectures to the global geometric version.

Physicist-mathematicians like Minhyong Kim are working to formalize analogies between number theory and quantum physics.


The proof of the geometric Langlands conjecture is a landmark in mathematics — not as a final destination, but as a launchpad. It opens up profound possibilities across math and physics, inching closer to a long-dreamed unity of ideas.

One important link from Nature (For further reading):

The breakthrough proof bringing mathematics closer to a grand unified theory

--
For many people like me, the beauty of this deep mathematics–physics connection evokes the spirit of Freeman Dyson and Mehta — whose work linked Riemann’s zeta function , number theory and work of Ramanujan - to the energy levels of complex atomic nuclei, revealing how abstract patterns in math can mirror the hidden order in physical systems.

[A_Gupta]

This likely won’t help, but read up and down the thread and one wont feel loneliness in one’s ignorance
:

https://golem.ph.utexas.edu/category/20 ... ml#c034370

[Vayutuvan]

This likely won’t help, but read up and down the thread and one wont feel loneliness in one’s ignorance
:

https://golem.ph.utexas.edu/category/20 ... ml#c034370

Nice find.

By the way wikipedia says but I am quoting from the linked thread as well

> "it has something to do with representation theory, algebraic geometry and number theory"

It is possible to understand (speaking for myself, I do understand at a 30K feet level) representative theory, algebraic geometry, and some NT (just a few theorems here and there to get apply the concepts designing and analyzing algorithms and data structures for numerical analysis (my job is R&D and D and D), the first two are lot more abstract. I don;t use them on a day to day basis and hence even if I read the definitions, understand them, and read a few proofs theorems, I usually don't retain any of that information beyond a week or two.

Representation theory is probably easier than algebraic geometry and Analytic NT just because I think I have reasonable amount of mastery over Linear Algebra.

Probably understanding what this latest advance means is a up hill task given that I want to learn more about Quantum Computing, Logic, and Foundations of Mathematics (three areas which are my side interest).

[Vayutuvan]

From the above, I don't understand this.

> The diagonal matrices T in U(n) are a “maximal torus”.

A diagonal matrix is associated with an ellipsoid in normal linear algebra which has genus 0 topologically speaking. Torus is a genus 1, i.e. it has one handle. How can an object with genus 0 be "maximal/minimal/<anything>" of an object with a genus 1? No transformations of vector spaces can make one into the other (unless there is projection in to lower dimensions). K-torus can be projected into k-1 sphere.

[Amber G.]

At the request of BRF admin, I hosted this thread and I've been curating this space for over 17 years, sharing hundreds of posts on important — and often fun — topics in mathematics. It's always been a pleasure to explore deep ideas together and make them accessible to a wider audience.

In that spirit, I recently shared the Nature article on the historic proof of the geometric Langlands conjecture — a breakthrough that marks a major milestone in the grand vision of the Langlands programme. At the request of some here, I’ve also tried to dive a bit deeper and explain, in layperson’s terms, what this means and why it's exciting — not only for mathematicians but also for physicists and anyone who appreciates the beauty of deep structure. The thing to note that even many professional mathematicians avoid Langlands because it’s so deep and technical.

As you’ve seen in the past, sometimes these threads get hijacked or sidetracked — by trolling, off-topic debates (including pseudoscience like astrology), or jargon-laden comments that sound technical but don't really hold up. I usually ignore such attempts, but I do want to gently emphasize:

Let’s not let impressive-sounding jargon derail meaningful discussion.

Just as one example: referring to diagonal matrices as being genus 0 surfaces, and questioning their role as a "torus," mixes up very different concepts from linear algebra, topology, and Lie group theory. In the context of U(n), the maximal torus refers to a group-theoretic structure (like (𝑆₁)ⁿ )
— not a genus-1 surface. . It’s unrelated to the geometric concept of genus or ellipsoids. Mixing those concepts leads to confusion, especially in discussions of something as deep as Langlands.
There is confusion here and it's important not to conflate unrelated ideas.

Let’s keep this a place for real learning and exploration — and as always, if people want to delve deeper or clarify any part of this topic, I’m happy to help.

[A_Gupta]

Via the previous link:

"That the possible number fields of degree n are restricted in nature by the irreducible infinite dimensional representations of GL(n) was the visionary conjecture of R. P. Langlands. "

This is decipherable with the help of AI.

This sentence is a poetic summary of one of the boldest ideas in modern mathematics: the Langlands program. Let’s unpack it step by step.

What it's saying

Number fields of degree n: These are extensions of the rational numbers ℚ that have dimension n as vector spaces over ℚ. Think of them as more complex number systems built from ℚ.

GL(n): This is the general linear group of degree n, which consists of all invertible n × n matrices. It plays a central role in representation theory.

Irreducible infinite-dimensional representations: These are ways GL(n) can act on infinite-dimensional vector spaces in a way that can’t be broken down into simpler actions. They’re like the “atoms” of symmetry in this context.

Restricted in nature: This means that not all number fields are possible—only those that correspond to certain representations.

Langlands’ visionary conjecture

Langlands proposed that there’s a deep, hidden correspondence between:

Galois representations (which encode the symmetries of number fields), and

Automorphic representations of GL(n) (which come from analysis and geometry).

In essence, he suggested that the structure of number fields—which seems purely algebraic—is governed by the representation theory of GL(n), which is analytic and geometric. This was revolutionary because it connected disparate areas of mathematics: number theory, harmonic analysis, and algebraic geometry.

Why it matters

Langlands’ conjecture implies that understanding the representations of GL(n) could classify all number fields of degree n, or at least tell us which ones are “allowed” by the universe of mathematics. It’s like saying the symmetries of a certain kind of space determine which kinds of numbers can exist.

Added - I don't know whether the Ai is misleading or not but my intuition says it is correct. The punchline I've put in bold.

[Vayutuvan]

Let’s keep this a place for real learning and exploration — and as always, if people want to delve deeper or clarify any part of this topic, I’m happy to help.

Thanks.

I will wait for somebody who can explain to me at my level. I know my limitations and specified clearly what I know. As is the case with everybody else, what I know is much much much smaller than what I don't know.

[Vayutuvan]

This is decipherable with the help of AI.

Langlands’ conjecture implies that understanding the representations of GL(n) could classify all number fields of degree n, or at least tell us which ones are “allowed” by the universe of mathematics. It’s like saying the symmetries of a certain kind of space determine which kinds of numbers can exist.

Added - I don't know whether the Ai is misleading or not but my intuition says it is correct. The punchline I've put in bold.

My dumb question, if you can answer or ask the AI.

Are those numbers known? I will use some jargon (which is unavoidable). Are they constructible (I am striking out as it is too narrow) computable. If they are definable, that is also somewhat OK.

I would have no opinion on those numbers which cannot exist.

I realized it is a dumb question.

[Vayutuvan]

...
1. These days, I dabble in stuff like https://projecteuler.net.
2. I want to see if generating problems that are tractable is a faster way to learning than solving problems. ...
3. ... On the way Gödel's incompleteness theorems in their full technicality I have understood, ...
...
7. My work for many many years does not require mathematics or physics of any significance; what I have is what I am able to spend some time on.

(I edited your post down to the points on which I have an opinion/comment/seeking info)

1. Those problems seem to be quite hard. What motivates people to attempt these problems and spend their valuable time? (not rhetorical question)

2. I think that is a good strategy and even the preferred strategy. Formulating a question correctly is half of the battle. Some questions can be simply stated but resisted proofs in the positive/negative. Two famous examples from the top of my head are FLT and FCT. Six color theorem is easy to prove. Five color is a little bit harder but FCT took much longer.
...
7. Any references? My understanding is from

"Godel's Proof Revised Edition
by Ernest Nagel (Author), James R. Newman (Author), Douglas R. Hofstadter (Editor, Foreword)".
It is a favorite of a couple of logician friends of mine.

Godel's papers themselves (originally German, several English translations exist) are supposedly very dense.
I have this book called "The Annotated Godel: A Reader's Guide to his classical paper on logic and incompleteness" by Hal Prince. I scanned it before and now before writing this post. My impression is that I would get incrementally more understanding than Nagel's book.

But understanding Godel Numbering is the first step which would also help in understanding Matiysevich's book on how he gave the final proof (in the negative) of Hilbert's 10th problem. The theorem is know as Matiyasevich–Robinson–Davis–Putnam (MRDP) theorem. His book is a pleasure to read.

Some random ramblings FWIW.

[Vayutuvan]

https://archive.org/details/gdelsproof00nage/mode/2up

1958 edition of Godel's Proof is free on archive.org I don't know how better the revised edition is though since I haven't read this book. In anycase, there may not be any substantial additions as the length is about 130 pages in either of the editions.