Blog on Applied Mathematics | Blog divulgativo di matematica applicata .
Joined January 2015
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I curated 125 free math textbooks, written by the mathematicians who teach the courses and shared on their own university pages. No sign-up, no paywall, no pirated copies. From high school algebra to graduate analysis. 👇
abakcus.com/book-lists/free-…
Math is in the Air retweeted
The idea of an intelligence explosion caused by recursive self improvement has been around for a long time but until very recently it did not seem imminent. Now many leading researchers think it may happen quite soon. You can read our paper about it here:
casp.ac/reports/intelligence…
Math is in the Air retweeted
Nobel ödüllü Thomas J. Sargent tarafından yazılan, yapay zeka ile makine öğrenmesinin kökenleri ve bunların istatistik ve ekonomi ile ilişkisi üzerine oldukça iyi bir çalışma…
🔗 tomsargent.com/research/AI_S…
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A free sample of BIG MATH (my new book with Alex Townsend) is now up on Amazon. Hope you like it!
read.amazon.com/sample/03165…
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Beautiful article about math and AI by Dan Rockmore. The connection to art and photography via Walter Benjamin was very fresh -- definitely worth reading, even if you're tired of reading about math and AI.
“What we are witnessing is a spectacle of computational power staged to acquire market share, and with it control of a landscape that extends well beyond mathematics.” —Dan Rockmore go.nybooks.com/4hzk2OK
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Since all sorts of people have asked me, I decided to write down my current thoughts on "What's the Future for Pure Math Research in the Age of AI?"
writings.stephenwolfram.com/…
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Another "coincidence", like the one that happened to Tristan Buckmaster with the solution of the Navier-Stokes problem? Except now it's in biology: Did Anthropic’s A.I. Really Make a Scientific Discovery on Its Own? nytimes.com/2026/09/27/scien… via @NYTimes
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you don’t hate calculus. you just never saw what the gradient was doing.
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How are prices determined? While I slept, Claude made me "History of economic thought since Aristotle." I asked it to use my favourite textbook (Blaug, Economic Theory in retrospect). The result will astonish of you. Even top economists here will learn something, I promise.
Math is in the Air retweeted
Hey Steve, I'd suggest checking out Mario Krenn's work on "On Scientific Understanding with Artificial Intelligence" (nature.com/articles/s42254-0…) and also "Philosophy of Autonomous Science" (direct.mit.edu/daed/article/…)- the basic argument is that if were an oracle that could solve all scientific problems, scientists would still be unsatisfied as they'd want to understand things... then they borrow a definition of scientific understanding from the philosophy of science to explain what that even means. I think AI systems should be pushed to the absolute limit for "scientific discovery" and human ego should not stand in the way of that. After that, scientific understanding by humans should be undertaken which will be expensive and valuable and then the knowledge can be assimilated in human-centered knowledge databases. So if we decouple "discovery" from "understanding", we could have this "human-centered AI model" for the "future of science and mathematics".
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Why should we care whether humans still understand mathematics if AI can solve the problems for us? Fair question. But we can say this at least: Historically, understanding has been a route to predictive power — with enormous benefits for our lives.
nytimes.com/2026/09/22/scien…
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I just published: Evolution of LLM Architecture
RNN -> Attention -> Transformer -> Scaling -> MoE -> New Directions
Read here: outcomeschool.com/blog/evolu…
Math is in the Air retweeted
“Ekonomistlerin Bilmesi Gereken Makine Öğrenme Yöntemleri”
Susan Athey ve Guido W. Imbens
🔗annualreviews.org/content/jo…
Math is in the Air retweeted
[461-page PDF eBook] Physics-based Deep Learning: arxiv.org/abs/2109.05237
+
Learn more at: physicsbaseddeeplearning.org…
——————
#AI #MachineLearning #ML #DataScience #NeuralNetworks #Algorithms #Mathematics #DataScientist
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[Download 585-page PDF eBook]
Game Theory: arxiv.org/abs/1512.06808
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#GameTheory #Gamification #Mathematics #Statistics #Probability
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Google's Jeff Dean just released the best 1-hour lecture on AI engineering: from basics to Graphs
1:45 - LLM from scratch
17:22 - how to use AI models
30:03 - prompt engineering
52:35 - one human coordinating 100 agents
1:02:40 - where the coordination actually lives
27 years of building AI at Google, compressed into one hour
Prompts → Agents → Loops → Graphs
most people will stop at the prompt engineering chapter and call it learning
he spends the last twenty minutes on the part that is still true next year
same model, same tokens, completely different week
watch it today
the full guide on graph engineering is below, save it while it is still early ↓
Math is in the Air retweeted
Things are heating up on Terry Tao’s blog. In “If Math Is More Than Proof, We Need to Better Celebrate the Rest of It,” Grant Sanderson of @3blue1brown proposes “open exposition problems” -- rewarding the work of making math genuinely understandable.
terrytao.wordpress.com/2026/…
The secret lecture from Gilbert Strang that MIT almost never released.
The harder one. The one that separates the engineers earning $180K from the ones earning $500K.
Senior simulation engineers at SpaceX, Boeing, and NVIDIA who can debug numerical instability make $400K to $600K a year.
The ones who cannot hand the problem to someone who can.
This lecture is the difference.
His name is Gilbert Strang. MIT, fifty years of teaching. Textbooks in 300 universities. The most watched mathematics professor on OpenCourseWare.
This is lecture 1 of 18.086. The course engineers actually need when the math stops being clean.
He opens with a question every simulation engineer faces and almost none of them can answer cleanly: when does a numerical method blow up?
Not approximately. Exactly. And why.
Two methods dominate. Forward Euler and backward Euler. One is fast. One is stable. You cannot have both.
At 16:35 he explains stiff equations — the class of problems that destroys explicit methods. Two decay rates in the same system. One slow. One a hundred times faster. The fast one forces you to take tiny time steps even when the slow one is what actually matters.
Every chemical simulation. Every financial model with multiple timescales. Every control system. Stiff problems everywhere.
At 31:41 he builds a matrix with eigenvalues minus 1 and minus 99. Then shows exactly what happens to forward Euler — why the fast eigenvalue kills your time step even when it contributes nothing to the answer you care about.
At 35:08 he derives the stability limit in one line. Go past it by a single percent and the solution explodes.
Then backward Euler. One change. The growth factor is always below 1, for any time step, for any negative eigenvalue. Absolutely stable. No limit.
ODE45 in MATLAB — running inside every simulation at Boeing, SpaceX, and every pharmaceutical lab computing drug kinetics — is built on exactly this tradeoff.
A simulation engineer I know says this lecture is what got him from $160K to $480K in three years. Said he was the only person on his team who could explain why the solver was diverging.
The lecture was recorded in 2006 and barely noticed. The engineers who found it anyway are the ones debugging simulations everyone else gave up on.
The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can.
His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head.
He opens with one confession.
The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all.
Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency.
Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back.
Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications.
Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time.
Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all.
A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize.
Free on YouTube, MIT OpenCourseWare.
bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated
