In the 1960s, the Soviet mathematician Olga Ladyzhenskaya showed that two-dimensional Navier-Stokes solutions do not blow up.
“the Soviet mathematician, uh, Ladyzhenskaya, she in the sixties showed in two dimensions there was no blowup”
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Terence Tao
1 episodes · 21 quoted claims
Terence Tao thanked his wife, who trained as an electrical engineer, for the circuit-style thinking behind his averaged Navier-Stokes construction.
“So I I actually thanked my wife for this because she was trained as a electrical engineer.”
In Conway's Game of Life, a glider is a small configuration of about four or five cells that moves in a fixed direction.
“they discovered this thing called a glider. So a glider is a very tiny configuration of like four, five cells, which evolves and it just moves at a certain direction.”
Szemerédi's theorem, which finds arithmetic progressions in sets of positive density, was proven in the 1970s.
“there's this old theorem in mathematics called Szemerédi's theorem, uh, proven in the 1970s.”
Carl Friedrich Gauss used tables of primes, including help from hired human computers, to conjecture the prime number theorem.
“to compute the first 100,000 primes or something and made tables and made a prediction.”
Tao says the standard dark matter model has about fourteen parameters yet fits petabytes of astronomical observations.
“the dark matter model, I think it has like fourteen parameters, and it explains petabytes of data.”
Tao says models of mortgage defaults assumed bell-curve behavior and did not fully account for systemic shocks that push many borrowers to default together, as in 2008.
“if there are systemic shocks in the economy, uh, that can push everybody to default at the same time, uh, that's very non-Gaussian behavior.”
As a Princeton graduate student, Tao attended a talk by John Conway on 'extreme proofs', which treated the space of proofs as having optimal extremes.
“I remember one of the very first research talks I I went to was a talk by Conway on what he called extreme proof.”
In the 19th century William Rowan Hamilton reformulated Newtonian mechanics as Hamiltonian mechanics, making energy the central object.
“Hamilton, eventually, in the 19th century, reformulated Newton's laws of physics into what's called Hamiltonian mechanics, where the energy, which is now called the Hamiltonian, was the dominant object.”
Tao found a gauge transformation for the wave maps equation while visiting his aunt in Australia around 2000.
“I found this transformation while visiting my aunt in Australia, and I was trying to understand the dynamics of all these theories.”
Kevin Buzzard has a five-year grant to formalize the proof of Fermat's Last Theorem in Lean.
“Kevin Buzzard's goal, actually, he has a five year grant to formalize Fermat's last theorem.”
Mathlib is a Lean library of tens of thousands of formalized mathematical facts built by the Lean developers.
“Mathlib, a a collection of tens of thousands of useful facts, ah, about mathematical objects.”
Tao estimates formalizing a proof in Lean takes about ten times as much effort as writing it on paper.
“I estimate that the effort time and effort taken to formalize a proof is about ten times the amount taken to to write it out.”
Formalizing one theorem with the constant 12 took about three weeks and around twenty people, after which updating it to 11 took only a day or two.
“So that took like three weeks to formalize, uh, and and like twenty people to formalize this, this, this original proof.”
The Green–Tao theorem states that the prime numbers contain arithmetic progressions of any length.
“it's a statement that the primes contain arithmetic progressions of any length.”
In Tao's Equational Theories Project, all but two of about 22 million implication questions about algebraic laws had been settled at the time of the interview.
“Um, so of these 22 million, all but two have been settled.”
A crowdsourced Polymath project published under the pseudonym D. H. J. Polymath, in the spirit of the Bourbaki group.
“So we created this fictional character called D H J Polymath, in the spirit of Bourbaki.”
The proven bound is that infinitely many pairs of primes differ by at most 246.
“the infinite pairs of primes which differ by at most, um, 246, actually is is is a is the current record.”
Richard Hamilton proposed attacking the Poincaré conjecture with a partial differential equation, now called Ricci flow.
“Richard Hamilton proposed a, um, partial differential equations approach.”
The Collatz procedure divides even numbers by two and replaces odd numbers with three times the number plus one.
“So if it's even, you divide it by two, and if it's odd, you multiply it by three and add one.”
Tao proved that, statistically, almost all starting values in the Collatz process drift down to much smaller values.
“that statistically, like, 99% of all inputs would would drift down to, no, maybe not all the way to one, but to be much, much smaller than what you started.”