Mir Publications

Tue, 10 Sep 2024 12:51:09 +0000

I loathe American textbooks from the bottom of my heart. I learned math from these tiny 100 page books that made the point directly and succinctly and trusted the reader’s intelligence to figure out the rest. Textbooks here seem to be written for people who are incapable of thinking for themselves.

To be great surprise I discovered this website containing MIR publication books:

https://archive.org/details/mir-titles

The first book listed (at least for me) is Irodov’s Physics, every engineering student’s nightmare and yet one of the best books to learn physics from. I hope younger students find these and I myself am looking forward to reading some of the old gems and finding some new ones.

Pythagoras using Hilbert 90

Sat, 08 Jun 2024 12:33:01 +0000

Dummit and Foote has a fun way of classifying Pythagorean triples. It is a classic fact that if $(a, b, c)$ is a reduced Pythagorean triple then there exists integers $m, n$ such that

$$ \begin{align*} a &= m^2 - n^2, \ b &= 2mn, \ c &= m^2 + n^2. \end{align*} $$

To prove this using Hilbert’s 90, first we change the problem to rational numbers i.e. let $a, b\in \mathbb{Q}$ be such that $a^2 + b^2 = 1$. We want to show that there exists $m, n \in \mathbb{Q}$ such that $$ \begin{align*} a &= \frac{m^2 - n^2}{m^2 + n^2}, \ b &= \frac{2mn}{m^2 + n^2}. \end{align*} $$

Phistomephel Ring in Sudoku

Mon, 08 Jan 2024 00:37:57 +0000

Came across this really cool short video about identities in a Sudoku square:

It is such a simple identity yet something that I had not noticed myself before. Makes me wonder what other subset sum identities are there in a Sudoku? Is this a linear algebra problem? Are we looking for interesting elements in the kernel of some maps?

Closed Convex Cones in LP

Tue, 02 Jan 2024 00:13:03 +0000

I was formalizing some linear programming in Lean and came across one of the most surprisingly difficult to prove theorems.

::: {#thm-closed-convex-cones}

Let $A : \mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. Then the set $$ { A x : : : x \ge 0 } $$ is closed in $\mathbb{R}^n$. :::

The naive topological approach doesn’t work as this theorem is no longer true if we replace $A$ by an arbitrary continuous map or if we replace $x \ge 0$ by an arbitrary closed set.

Combinatorial Nullstellensatz

Sun, 26 Nov 2023 12:27:01 +0000

I came across this fun theorem in a talk about tree colorings:

Combinatorial Nullstellensatz. Let $f \in F[x_1, x_2, \ldots, x_n]$ be a polynomial of degree $t_1 + \cdots + t_n$. If $S_1, S_2, \ldots, S_n$ are nonempty subsets of $F$ such that $\left| S_i \right| \geq t_i + 1$ for all $i$, then there exists $s_1 \in S_1$, $s_2 \in S_2$, $\ldots$, $s_n \in S_n$ for which $$ f(s_1, s_2, \ldots, s_n) \neq 0 $$ as long as the coefficient of $x_1^{t_1} x_2^{t_2} \cdots x_n^{t_n}$ is nonzero.

Logarithms and primes

Thu, 26 Oct 2023 09:01:26 +0000

I came across this video by NumberPhile and 3Blue1Brown about the density of primes that are 1 mod 4 vs the density of primes that are 3 mod 4. The main fact is quite cool but there was something that he said during in the middle of the video that I found more interesting.

Define $\chi : \mathbb{N} \to {-1, 0, 1}$ as $$ \chi(n) = \begin{cases} 1 & \text{ if } n \equiv 1 \mod 4 \ -1 & \text{ if } n \equiv 3 \mod 4 \ 0 & \text{ otherwise. } \end{cases} $$ The first interesting fact that I do not know how to prove: $$ \sum \limits_{n \in \mathbb{N}} \dfrac{\chi(n)}{n} = \dfrac{\pi}{4}. $$ Note that the series is only conditionally convergent so the order of summation matters.

How They Fool Ya (live) | Math parody of Hallelujah

Wed, 18 Oct 2023 16:58:33 +0000

The Tau Manifesto

Sun, 24 Sep 2023 12:37:29 +0000

$\tau$ manifesto

The Tau Manifesto is dedicated to one of the most important numbers in mathematics, perhaps the most important: the circle constant relating the circumference of a circle to its linear dimension. For millennia, the circle has been considered the most perfect of shapes, and the circle constant captures the geometry of the circle in a single number. Of course, the traditional choice for the circle constant is ($\pi$)—but, as mathematician Bob Palais notes in his delightful article “$\pi$ is Wrong!”, $\pi$ is wrong. It’s time to set things right.

The L-functions and modular forms database (LMFDB)

Sun, 24 Sep 2023 12:22:43 +0000

LMFDB Github

The LMFDB is a database of mathematical objects arising in number theory and arithmetic geometry that illustrates some of the mathematical connections predicted by the Langlands program.

I love this website. I don’t understand anything on it. But I’m amazed that this exists and I marvel at the amount of effort the writers must have put into making this happen. I’ve always thought that number theory is unique in how it manages to straddle the concrete and the abstract worlds so effortlessly. This website is a proof of this.

Automated Mathematics and the Reconfiguration of Proof and Labor

Sun, 24 Sep 2023 12:14:03 +0000

I came across this essay by Rodrigo Ochigame exploring possible future impact of math formalization and automation.

Automated Mathematics and the Reconfiguration of Proof and Labor

Abstract:

This essay examines how automation has reconfigured mathematical proof and labor, and what might happen in the future. It discusses practical standards of proof, distinguishes between prominent forms of automation in research, provides critiques of recurring assumptions, and asks how automation might reshape economies of labor and credit.

When Math Becomes an Art

Sat, 16 Sep 2023 20:15:39 +0000

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Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares

Sun, 10 Sep 2023 15:35:44 +0000

I came across this “one sentence” proof that every prime congruent to 1 mod 4 can be written as a sum of two squares:

https://doi.org/10.2307/2323918

The proof involves constructing the following cryptic involution $$ (x, y, z) \mapsto \begin{cases}(x+2 z, z, y-x-z) & \text { if } x<y-z \ (2 y-x, y, x-y+z) & \text { if } y-z<x<2 y \ (x-2 y, x-y+z, y) & \text { if } x>2 y\end{cases} $$ on the set $S=\left{(x, y, z) \in \mathbb{N}^{3}: x^{2}+4 y z=p\right}$ and showing that the involution has exactly one fixed point $(1, 1, k)$ if and only if $p$ is a prime of the form $4k + 1$.

Etymology of Integer

Tue, 29 Aug 2023 09:13:31 +0000

We use integers so much in math that it never occurred to me before yesterday that this is a weird word with a weirder symbol associated to it. Wikipedia explains this very nicely.

The words integer and entire share the same Latin root: in (“not”) plus tangere (“to touch”). Google says that the first known use of the word integer was in 1571 by Thomas Digges in a book called A Geometrical Practise Named Pantometria. The word origin still doesn’t make complete sense to me but then I don’t know any Latin. The symbol $\mathbb{Z}$ comes from Zahlen, which is German for numbers, and was first used by Hilbert.

History-of-Math Resources

Mon, 21 Aug 2023 22:30:11 +0000

I’ve always been fascinated by the history of mathematics. Some of the classic and dramatic discoveries I read a kid were those of non-Euclidean geometry, Galois theory, the proof of Fermat’s last theorem, as well as Cantor and Gödel’s breakthroughs in logic. However, some of the more commonplace achievements, such as the invention of calculus by Newton and its subsequent formalization a century later through $\epsilon \delta$ proofs, also possess a very intriguing narrative that involves not only mathematicians but also philosophers and theologians¹.

A Brief History of Time

Wed, 14 Mar 2018 00:00:00 +0000

Someone had gifted me this book as a kid, I do not remember who or when, I think it was my maths teacher.

I remember Hawking says in his book, that adding equations to books decreases their popularity. I remember not particularly liking this.

But I also remember reading, for the first time in my life, about light cones, about causality, about the big bang, black holes and their event horizons, entropy and the arrow of time, about Hubble’s discovery, the multitude of galaxies and the ever expanding universe, about the electrons with their dual natures and their half spins, bosons and fermions, Heisenberg’s uncertainty principle, Pauli’s exclusion principle, about supernovas and the creation of heavy elements, the spinning neutron stars, the Chandrasekhar limit, the spontaneous creation and annihilation of particles, the Hawking radiation, about quarks and their funny color names, about gluons, about the strong and weak nuclear forces, radioactivity, mesons, and muons, and pions, about quasars and gamma rays, about relativity and the space-time, the precession of mercury, the ultraviolet catastrophe, Einstein’s genius, about his folly, about mass and energy, the Planck’s constant, about gravitational lensing, about strings, about mirrors, about the vastness of the universe, and our place in it.

Spoken Language

Mon, 22 Jan 2018 20:00:00 +0000

I was attending the Science of Learning Symposium at JHU, which is usually quite fascinating, but was very boring this year. These are the things that frustrate me about formal events:

People take extended time to introduce the speakers (nobody wants to hear you talk, go away).
People in suits, usually senior professors who have stopped being scientists, who think they’re funny because nobody is honest to them, make shitty jokes and laugh at their own jokes.
People make a huge preamble when asking questions, dude the question time is very limited and there are several others who also have things to ask.

It is at formal events like these that I sympathize strongly with Salinger’s protagonist from Catcher in the Rye.

Popular Science on PostDoc Problems

Mir Publications

Pythagoras using Hilbert 90

Phistomephel Ring in Sudoku

Closed Convex Cones in LP

Combinatorial Nullstellensatz

Logarithms and primes

How They Fool Ya (live) | Math parody of Hallelujah

The Tau Manifesto

The L-functions and modular forms database (LMFDB)

Automated Mathematics and the Reconfiguration of Proof and Labor

When Math Becomes an Art

Every Prime $\equiv 1 \mod 4$ Is a Sum of Two Squares

Etymology of *Integer*

History-of-Math Resources

A Brief History of Time

Spoken Language

Etymology of Integer