Math@BklynInstitute

Timeline of @BklynInstitute mathematics courses we have offered over the past 10 (!) years: 1/ Gödel incompleteness, spring 2013 (back when BISR courses were (6 weeks)*(2h/week) - met in that sci fi bookstore in Dumbo

In the 1940s, Subrahmanyan Chandrasekhar was committed to his teaching role at the University of Chicago, despite being based at the Yerkes Observatory. Each week, he traveled 80 miles to teach a special course attended by only two students. The students were Tsung-Dao Lee and Chen-Ning Yang. They proved their mentor's faith was well-placed when they both won the Nobel Prize in Physics in 1957, years before Chandrasekhar received the same honor in 1983. Remarkably, this course went down in history as the only one where every attendee received a Nobel Prize, underscoring the extraordinary impact of Chandrasekhar's dedication and teaching. 📷 AIP Emilio Segrè Visual Archives, Physics Today Collection

In 1966, mathematician Alexander Grothendieck was awarded the Fields Medal for his revolutionary work in algebraic geometry, especially the development of schemes, étale cohomology, and the concept of toposes. The ceremony was held in Moscow. Grothendieck chose not to travel to the Soviet Union due to political reasons. His refusal was a political protest against the Soviet Union's imprisonment of two writers, rather than a refusal of the award itself.

To all math fans: This is going to be an incredible movie when it comes out. Can't wait! It's about Maryam Mirzakhani, the first woman to win the Fields Medal. MARYAM: The Mirror and the Map

i follow AI adoption pretty closely, and i have never seen such a yawning inside/outside gap. people in SF are putting multi-agent claudeswarms in charge of their lives, consulting chatbots before every decision, wireheading to a degree only sci-fi writers dared to imagine. people elsewhere are still trying to get approval to use Copilot in Teams, if they're using AI at all. it's possible the early adopter bubble i'm in has always been this intense, but there seems to be a cultural takeoff happening in addition to the technical one. not ideal!

Even if you take the conservative defense of each ICE killing, you’re left with “We haphazardly scaled up a poorly trained police force to storm into neighborhoods that voted against the president, where we antagonize the local population until someone resists arrest, and then we kill them,” which is morally horrendous on top of being an absurd way to do immigration policy.

I was expecting to hate Tim Hartford's piece, which essentially says that the math in economics "feels wrong" for the subject. But I liked it! He's right. Today's economics does invest a lot of its mathematical effort in the wrong stuff. Thread 1/

AI economists and AI researchers: this is *excellent*. Details below, but as I feel like I've said in every talk on this topic since my slides said "GPT-2", 1) AI technical capabilities are better and improving quicker than you think, 2) impact on economy *much* slower. 1/13

Green's theorem is probably the most important result that multivariable calculus students need to grasp. A very solid understanding of Green's theorem makes its higher-dimensional counterparts (the divergence theorem, Stokes' theorem) feel almost trivial.

Multivariable Calculus Course I gave at @UCBerkeley in 2009 is apparently still being watched. Cool!

The Fourier series is a way to represent any periodic function as a sum of simple sine and cosine waves. In essence, it breaks down complex signals into a combination of basic trigonometric functions, making it a powerful tool in mathematics, physics, and engineering for analyzing waveforms, heat transfer, vibrations, and more. 📹 Credit : mathswithmuza

the self-aggrandizement is overwhelming. we are no position to do either with current technology.

Real Analysis Lecture 2 is up. youtu.be/U2oCgeVgd9M We went over Newton’s computation of Pi, both because it’s super cool (first and foremost), and also because it’s a quintessential calculation encompassing everything that’s “wrong” with plain old Calculus, needing rigorous justification: - What does it mean for a sequence of real numbers a₀, a₁, a₂, ... to converge? - What does it mean for a series (that is, sequence of partial sums of some sequence) a₀ + a₁ + a₂ + ··· to converge, and can we sum these numbers in any order we like? - What does it mean for a series involving a variable, like a power series a₀ + a₁x + a₂x² + ··· to converge, and if it does, what kind of function does it converge to? - When can we interchange limits with integrals, like integrating term by term, ∫(a₀ + a₁x + a₂x² + ...)dx ··· ?=? ∫ a₀ dx + ∫ a₁x dx + ∫ a₂x² dx + ···? These are all exactly the kinds of questions that Real Analysis answers, but I find most presentations of these vital issues unmotivated. Hopefully the Newton example brings them to life...

• The "3 weeks vs 18 months" comparison is exaggerated. That they could do this in 3 weeks *is* extremely impressive, and surely significantly shorter than what a team of volunteers could do. But we announced PNT+ in Feb ’24, and had a complete proof of the asymptotic (no error term) 9 weeks later. Then the pace slowed significantly, and more and more parts were handed off to grad students and postdocs, letting them learn Lean and the math. The project became as much about education and an experiment in large scale collaboration as it was in formalization, and we were in no hurry. At the Simons workshop this past June, the participants spent a week making a significant push on Medium PNT, and two weeks later, the goal was closed. (In fact, in our latest "Outstanding Tasks" list, a student had staked a claim to Borel-Caratheodory, promising to work on it... Oh well, sometimes you get scooped; that's life!) • As mentioned above, if I understand correctly, the modus operandi was: write a blueprint; see what the model can prove and where it gets stuck. For any theorem on which it’s stuck, decompose further and try again; iterate until success. The result is that the *natural language* decomposition reads like a formal proof (!); that is, extremely basic things are added explicitly as lemmas. So sandwiched between “real” lemmas are complete trivialities like |x|=x for positive x. So there’s a sense in which some of the “hard” parts of the work of formalization have been shifted over to the natural language side (which was allowed to have as much human interaction as necessary), to claim a win for autoformalization. I still think it *is* a win, especially because AI should be able to do this part of the job, too (and I’m sure their next iteration will try to use humans less and less, until Christian’s longstanding dream of fully AI formalization is realized).

Today we're announcing Gauss, our first autoformalization agent that just completed Terry Tao & Alex Kontorovich's Strong Prime Number Theorem project in 3 weeks—an effort that took human experts 18+ months of partial progress.

AI and math. Geometry and symbolic reasoning. Amazing recent developments and stellar line up of speakers. It is going to be an exciting week! The Geometry of Machine Learning @ Harvard Center for Mathematical Sciences and Applications (CMSA) cmsa.fas.harvard.edu/event/mlgeomet…

There is a lot about Euclid’s Elements that is easily misunderstood. Some proofs seem to have logical gaps. Some constructions seem pointless, others seem needlessly convoluted. Each of these provides a window into how the ancient Greeks thought about math and the philosophical role that geometry played. In the fifth and final of a series of guest videos I've been posting, @BenSyversen delves into a question anybody who has had to do ruler and compass constructions in a geometry class may have wondered: What's the point? youtu.be/M-MgQC6z3VU

I'm in NYC for the first time in almost 10 years and ... a couple quick observations: Anyone who thinks it takes too long to transform a city is wrong. This city is utterly transformed. It is faster to change a city than to change a suburb. And: The U.S. needs two dozen NYC's.

The 3rd edition of my book Deep Learning with Python is being printed right now, and will be in bookstores within 2 weeks. You can order it now from Amazon or from Manning. This time, we're also releasing the whole thing as a 100% free website. I don't care if it reduces book sales, I think it's the best deep learning intro around, and more people should be able to read it.

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