The blog is full of my random thoughts.
The category of random thoughts contributes to the fifth largest categories among tagged entries, after "diary", "notes", "maths" and "works". Most of them were in Chinese though.
And today I come up with something different: three math related random thoughts in a single entry.
This is new in the sense that all "math diary" in the past are more like "math but too casual" that were still focused on a single topic. This is also new because it's 3 math-related entries in a row...
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I recently bought another sixth form "textbook" from an antique store. Titled "Polynomials and equations" and claimed "part of the chapters applicable for HKALE or equivalent examinations". This is nothing special: polynomials is an essential part in the curriculum with questions commonly on binomial sums and Vieta applications.
I bought the book at the price of half a cup of coffee, made myself a proper cup of coffee and started diving in...
- Introduction...starts with the notation $R[x]$ and $C[x]$.
- Factorization...gcd and lcm, then the Euclidean algorithm, unique factorization and FToA.
- Solving equations...then start talking about the meaning of discriminant in these solutions together with the insolvability of quintic equations.
- Integral solutions...some number theory, then Eisenstein without warning.
- Derivatives...Taylor and approximating continuous functions, IVT, Rolle.
- Root bounding and separation...approximates, Strum and Fourier(surprise!)...
Oh yeah, the "new math" style old math textbook.
The context was a bit different though. The book I bought was published in 1992, and indeed part of the chapters fitted the HKALE pure maths syllabus and exam depth adequately. The pure math exam was hard more because HKALE itself was designed to be an ultimate challenge to filter true elite to enter university.
Should it be a true "new math" style book, it has to be much earlier like in the 1970s or 1980s, it would also be covering topics straight at university level abstractness. I have another true "new math" era "sixth form" textbook with the title "group theory" that followed M.Artin's line all the way up to similarity. Can you imagine average sixth form students doing that?
Nonetheless the bridging between high school and university has always been an intriguing topic for discussion especially regions adopting the sixth form/matriculation system. Students at that level are matured enough to take another step above the cert level but not quite ready for university. What would you teach them, and what would you examine?
It always feel weird to me that geometry has been playing the vital role in most A-level or even high school level syllabuses across the globe. Coordinate geometry on conics, vector geometry, 3D vector calculus, parametric equations, geometry on complex plane -- do we really need them? Are they really useful other than serving the purpose of filtering students who can't handle messiness?
These geometry topics are almost never used in universities. You will use vector but they are not for solving Euclidean geometry problems. You will see them in linear algebra or calculus II with completely different intents. Conics are almost never used except for optics which is extremely niche and still look simpler than questions you will encounter in A-level exams. Complex geometry is beautiful but you will only study complex analysis in undergrad, not complex geometry.
There are so many misc topics you can fill into the roster instead. What about functions and relations? Injective and surjective functions? Equivalence relations? Group theory from matrices? Inequalities, epsilon-N and epsilon-delta?
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Youtube forwarded me a video on quintic insolvability without Galois on the same day. Frankly this is what motivated me to write something, not the sixth form textbook.
The video was about Arnold's proof, who managed to open the field of topological Galois theory from this result. The concept of "multivalued pullback" is somewhat unexpected yet so beautiful to understand.
When I first read M.Artin's algebra on branched coverings, it was too abstract for me. I simply jotted down and didn't understand a word of it. It wasn't even in the exam so I never came back to the topic until I took algebraic geometry course much, much later. Even so, it was never my focus.
On the other hand, the idea of travelling from a branch to another continuously is something that is much common. Winding is everywhere in geometry, topology and analysis. The moment they showed the loop (the projections of the infinite branched sheets onto complex plane) it's like a spark in my brain and everything started to come together.
The rest is all about bridging between loop commutators and roots permutation, but they looks so complicated even as presented in those modified sources. Here is the logic I would rather present instead:
- Given any solution formula for a given degree of polynomial equation, it should be invariant upon transversal along any loop based at solution point.
- In particular, it must be invariant to the commutators of loops, the commutators of commutators,... and so on because they are all loops.
- The key result is that suppose the minimal (least number of nested roots) formulae that are invariant to two loops are both $k$ and that the two loops do not commute, then the minimal formulae that can be invariant to the two loops and its commutator requires at least $k+1$ nested roots (!!!).
- Thus, the existence of the formula relies on the requirement that commutator of commutators eventually becomes trivial.
- Quadratic case: the commutators of $S_2$ is trivial, hence there is just 1 square root.
- Cubic case: the commutator group chain is $S_3, A_3, \left\{ e\right\}$, hence two nested roots.
- Quartic case: the commutator group chain is $S_4, A_4, V, \left\{ e \right\}$ so it works too.
- Quintic case? The commutator group is stuck at $A_5$ because it's damn simple!
In the first half the proof may look completely new and creative, but once you saw the subgroup chains it reduces to "oh it's the damn old $A_5$ again...".
The video avoided saying $A_5$ being simple directly. Instead it said commutator (sub)group generates the whole permutation group which means any finite nested root doesn't work.
Abel's proof (for him proving the result before Galois) also avoided the use of $A_5$. He used the language of algebraic independence to show that no further field extension is possible after the quadratic extension (equivalently from $S_5$ to $A_5$), just without all modern tools.
Both approaches avoided the use of $A_5$ although they are simply equivalent in the algebraic sense. The commutator group is in fact the minimal normal subgroup such that the quotient is Abelian. Since $A_5$ is simple, the only normal subgroup is the trivial group or itself, but the quotient upon trivial group is not Abelian, so the commutator group got stuck at $A_5$. Abel's approach is merely the elementary way of showing that there is no transitive field extension (by prime characteristic each step) to $S_5$ because $A_5$ is simple.
That makes me wondering what counts as "Galois"? The use of Galois theory, or even the fact that $A_5$ is simple? Even in modern algebraic sense, we could have avoided the use of Galois theory simply by using the language of field extension. It could have been much easier than Abel's proof too. Does that count as non-Galois proof?
I have no idea.
But one thing for sure: I would be extremely impressed if someone proved irresolvability without using any algebraic structure related to $A_5$.
Anyway, it's a good math video almost at 3B1B level, a hidden treasure that is definitely worth a look.
Some other great references:
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Speaking of Galois theory I've got one more thing to say: the math LLM benchmarks.
Forget about testing against AMC or AIME. Forget about testing against IMO or Putnam problems. We are now straight into the most advanced problems we can scramble out of our garage.
Previously I have been looking into the HLE (humanity's last exam) benchmarks as they aim to gather top questions from all subjects where math plays a big part of it. However, the mix between math and other subjects means that it is very hard to evaluate how good at math LLMs had became. There are also questions that are pure travia or multiple choice questions that aren't foul proof and hardly represent any meaningful results.
That is, until my friend told me the existence of FrontierMath.
I really like the fact that cover a wide range of breadth while keeping everything hard and foul proof. Top tier questions are only answered using frontier research results and are absolutely into the most niche bit of mathematics at the frontier. I really wonder how did the LLMs managed even to answer any single one of them...we can learn much more from what they managed to answer instead of what they can't for now.
Can I give any prediction on their progress of solving these questions? Not really. I still think they will be stopped by these questions for a long time but I might as well give an opposite answer by the end of 2025...
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