u/Necessary-Wolf-193

I often ask mathematicians what their favorite shape is; mine is the cone on a trefoil knot, a rather peculiar object.

Why is this my favorite shape? Because it appears naturally in algebraic geometry -- and in fact, the observation that it appeared in algebraic geometry led Milnor, Grothendieck, Goresky, MacPherson, and many more to develop some of the most widely used tools in modern algebraic geometry and singularity theory! If you'd like to know how this trefoil knot appears in algebraic geometry, read this week's hidden-phenomena blogpost!

https://hidden-phenomena.com/articles/gelfand

How does one define a shape? The usual definitions are based on starting with the set of all points of a shape. However, the great Russian mathematician Gelfand found another approach to defining shapes. This approach is more natural from the point of view of certain abstract shapes which arise in physics (namely, phase spaces of physical systems), and leads naturally to the modern formulation of quantum mechanics; however, perhaps surprisingly, this idea of Gelfand also leads to modern algebraic geometry!

My friend and I wrote a blog post describing Gelfand's observation at https://hidden-phenomena.com/articles/gelfand , and using Gelfand's idea to understand better the Chinese remainder theorem of elementary number theory. It's a really cool fact of reality that geometry connects so intimately to algebra, and we hope you'll enjoy the article!

A symplectic structure is some at first strange sounding extra structure you can put on a manifold. By a miracle, a lot of the shapes arising in geometric representation theory have this extra structure, and this structure can be exploited to prove very useful things.

In this blogpost, my friend and I motivate the definition of symplectic structures from mechanics, and say a little at the end about where they appear in pure math (by the way, the two of us are mathematicians, and we only learned physics to better appreciate the symplectic structures which were showing up in our work!).

In mathematics, oftentimes choosing a good coordinate system can drastically simplify a problem, as long as you pay a small upfront cost in translating the problem to these new coordinates.

Along these lines, Hamilton discovered a surprising way of re-formulating Lagrangian mechanics, which has had great success. In this article, we explain Hamilton's change of coordinates, and use it to discover the Fradkin tensor: a very mysterious symmetry of the 2-d harmonic oscillator!

Have you ever wanted to do calculus on the surface of a sphere, or other exotic shapes? Have you ever wondered what 'dx' really means?

My friend and I wrote a blog post at https://hidden-phenomena.com/articles/diff-forms covering the basic ideas of doing calculus on exotic shapes, with lots of fun animated widgets.

https://preview.redd.it/ro383u7yfpnh1.png?width=1182&format=png&auto=webp&s=af4d2488b591304e7b92dbf4eff83c824c22649f

It has always been somewhat strange to me how popular category theory and infinity-category theory are on the 'mathematical internet', despite how few working mathematicians actually need them.

However, over the past decade, infinity categories have grown in importance in more classical mathematics research -- especially in my own field of arithmetic geometry!

My friend and I wrote a blog post on infinity groupoids -- these are to infinity categories as sets are to ordinary categories. The goal of the blog post was to show, in as elementary a way as possible, what uses infinity groupoids have, to try and give readers a taste of why they've become so helpful in modern mathematics.

https://hidden-phenomena.com/articles/anima

As a sneak peak: groupoids are, in some situations, a more convenient object than sets for handling group actions!

The mathematician Emmy Noether made a fundamental, and very beautiful, discovery: continuous symmetries in the laws of physics give rise to conservation laws in physics! In this way, conservation of energy, conservation of momentum, and conservation of angular momentum all come from symmetries of the laws of physics: energy is conserved because the laws of physics are independent of time; momentum is conserved because the laws of physics are translation invariant; and angular momentum is conserved because the laws of physics are rotation invariant.

At the end of the article, we also say a little about Lie groups and Lie algebras, because secretly they are the mechanism by which mathematicians formalize continuous symmetries; for conservation of energy and conservation of momentum, it's easy to get by without them, but to really understand conservation of angular momentum, it is very helpful to think about the Lie algebra of the group SO(3).

When someone first learns classical mechanics, they're given Newtons equations of motion. But there is another -- at first highly counter-intuitive -- approach to classical mechanics, discovered by Lagrange: you can reformulate classical mechanics as about solving optimization problems! This is very strange (at least for me) to ponder, because Newtonian mechanics explains things in "cause and effect" terms which fit better into my head. The sci-fi writer Ted Chiang wrote a short story (and now feature film!) Arrival which tried to imagine humans meeting an alien species who thought in terms of Lagrangian mechanics instead of Newtonian mechanics.

But one thing I had always been upset by with Lagrangian mechanics was how most treatments start by telling you what the Lagrangian is, instead of telling you how Lagrange might have invented it (even Feynman's lectures on physics start by telling you what the Lagrangian is). My friend and I wrote a derivation of the Lagrangian, and of the Euler-Lagrange equation (a fantastic result of the calculus of variations!) at https://hidden-phenomena.com/articles/lagrangian ; we hope this can be a fun introduction!

One of the most important theorems in my area of research is the Riemann--Hilbert correspondence. Roughly, it tells you that you can convert differential equations into certain geometric objects, and that this conversion process loses no information. In particular, one can convert questions about differential equations into geometric questions, and conversely one can convert certain geometric questions into problems about differential equations.

In https://hidden-phenomena.com/articles/monodromy , my friend and I wrote a blog post showing this example in a very simple case. The differential equation in question is very simple: f'(x) = f(x)/2x, and the geometric object is related to the Möbius strip!

Pascal's triangle, with entries colored according to parity

If you take Pascal's triangle, and color each entry according to whether it is even or odd, you get a funny pattern, which resembles Sierpinski's triangle. To understand this, it's helpful to know Lucas' theorem, which tells you when an entry in Pascal's triangle will be even or odd. If you've never seen it, you might enjoy the article https://hidden-phenomena.com/articles/lucas that we just wrote about it!

Lucas' theorem is a great result, which even tells you about how to compute (n choose k) modulo p. It is a wonderful piece of elementary number theory, and suitable as a fun but challenging exercise for the end of an elementary number theory course. Recently, one of us had to invoke Lucas' theorem in a modern math research paper https://arxiv.org/abs/2604.20054 about some relatively fancy arithmetic geometry! We thought this was a good example of how small results from introductory courses can be helpful in your research career in completely unexpected ways! The article itself isn't about the paper (which isn't very elementary), but Lucas' theorem is still helpful, and will hopefully come in handy.

In high school, many are taught the quadratic formula. But how do solve a cubic equation, like x^3 + 6x^2 + 9x + 3 = 0? There is a formula for such equations, but it's rarely taught since it's a lot more complicated than the quadratic formula:

The solution to ax^3 + bx^2 + cx + d = 0

Despite its apparent complication, we will explain at https://hidden-phenomena.com/articles/cubic how you could have come up with this formula!

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Last week, we posted a blog post here about how to solve quartic equations, assuming you knew how to solve cubic equations. If you read today's post on how to solve cubics, then by putting the two posts together, you can solve any cubic or quartic equation!

In school, people are often taught the quadratic formula, but almost never told that there is a formula to solve cubic and quartic equations (like x^3 + 4x + 2 = 0 or x^4 - 5x^3 + 6x^2 - 7x + 8 = 0).

This is for a good reason: the cubic and quartic formulas are considerably more complicated than the quadratic one! It took people a long, long time to discover them.

However, the French mathematician Galois had a wonderful idea that allows people to re-discover the cubic and quartic formulas much more efficiently: one could use symmetries of numbers to derive the formula. When people discuss Galois theory, they usually use it talk about a negative result: Galois theory proves there is no formula to solve quintics. But this positive result uses the same basic ideas, and has the benefit of giving you a cubic formula at the end!

At https://hidden-phenomena.com/articles/quartic , my friend and I wrote a blog post to explain how to use the symmetries of numbers, a la Galois, to derive the quartic formula. Next week, we'll explain how to solve the cubic formula.

This order might seem funny to you, but actually it follows history: the Italian mathematician Ferrari discovered how to solve quartics in terms of cubics, and then later his teacher Cardano found (by asking Tartaglia...) how to solve cubics! So, Ferrari knew how to solve quartics in terms of cubics before he knew how to solve cubics.

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For experts, here I will say a little about the modern Galois theory way of describing this solution, but if you don't know Galois theory, please read the blog post https://hidden-phenomena.com/articles/quartic instead, as it is entirely elementary!

Anyway, here it goes. There is a surjective group homomorphism S_4 -> S_3 (coming from the fact that 4 = 2+2 in three ways), with kernel Z/2 \oplus Z/2. In particular, Z/2 + Z/2 is a normal subgroup of S_4, and so Galois theory tells us that if L/k is any S_4-extension, say with k characteristic 0, then there is an intermediate field F so that F/k is Galois with Galois group S_3, and L/F is Galois with Galois group Z/2 + Z/2. In particular, L is obtained by adjoining two square roots to F; the cubic formula will tell us how to build F from k with cube roots and square roots, so that L can be built out of k from cube roots and square roots, and hence we can find a quartic formula using cube roots and square roots!

The factorial of an integer can, perhaps surprisingly, be evaluated even at non-integer entries. For example, you might even see these factorials of non-integers appearing in formulas for the volumes of high dimensional balls (usually, these factorials appear in the guise of the 'gamma function').

The extension of the factorials to non-integers is usually done with a certain integral formula, but Euler's originally derivation actually used some simple combinatorial identities, which he realized allowed him to write down a formula for x! which only involved factorials of integers and certain standard arithmetic operations. This let Euler define x! in general, as a certain limit.

At https://hidden-phenomena.com/articles/gamma , you can see this derivation in full -- it's quite cool!