In my previous post, I walked through the task of formally deducing one lemma from another in Lean 4. The deduction was deliberately chosen to be short and only showcased a small number of Lean tactics. Here I would like to walk through the process I used for a slightly longer pr … | Continue reading
Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “On a conjecture of Marton“. This paper establishes a version of the notorious Polynomial Freiman–Ruzsa conjecture (first proposed by Katalin Marton): Theorem 1 (Polynomial Freiman–Ruzsa conje … | Continue reading
The National Academies of Science, Engineering, and Mathematics are hosting a virtual workshop on the topic of “AI to Assist Mathematical Reasoning” from June 12-14. The tentative program can be found here. I am one of the members of the organizing committee for this workshop, to … | Continue reading
This post is an unofficial sequel to one of my first blog posts from 2007, which was entitled “Quantum mechanics and Tomb Raider“. One of the oldest and most famous allegories is Plato’s allegory of the cave. This allegory centers around a group of people chained to a wall in a c … | Continue reading
An unusual lottery result made the news recently: on October 1, 2022, the PCSO Grand Lotto in the Philippines, which draws six numbers from $latex {1}&fg=000000$ to $latex {55}&fg=000000$ a… | Continue reading
Prodded by several comments, I have finally decided to write up some my thoughts on time management here. I actually have been drafting something about this subject for a while, but I soon realise… | Continue reading
In everyday usage, we rely heavily on percentages to quantify probabilities and proportions: we might say that a prediction is $latex {50\%}&fg=000000$ accurate or $latex {80\%}&fg=000000$ … | Continue reading
This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechan… | Continue reading
This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechan… | Continue reading
In this post I would like to collect a list of resources that are available to mathematicians displaced by conflict. Here are some general resources: The IIE Scholar Rescue FundScholars at Risk Sta… | Continue reading
About a year ago, I was contacted by Masterclass (a subscription-based online education company) on the possibility of producing a series of classes with the premise of explaining mathematical ways… | Continue reading
As math educators, we often wish out loud that our students were more excited about mathematics. I finally came across a video that indicates what such a world might be like: | Continue reading
A popular way to visualise relationships between some finite number of sets is via Venn diagrams, or more generally Euler diagrams. In these diagrams, a set is depicted as a two-dimensional shape s… | Continue reading
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao | Continue reading
(Originally written, Apr 8, 2012.) Anonymity on the internet is a very fragile thing; every anonymous online identity on this planet is only about $latex 31$ bits of information away from being com… | Continue reading
In everyday usage, we rely heavily on percentages to quantify probabilities and proportions: we might say that a prediction is $latex {50\%}&fg=000000$ accurate or $latex {80\%}&fg=000000$ … | Continue reading
Every mathematician worthy of the name has experienced … the state of lucid exaltation in which one thought succeeds another as if miraculously… this feeling may last for hours at a tim… | Continue reading
Better beware of notions like genius and inspiration; they are a sort of magic wand and should be used sparingly by anybody who wants to see things clearly. (José Ortega y Gasset, “Notes on t… | Continue reading
I’ve just uploaded to the arXiv my paper “Sendov’s conjecture for sufficiently high degree polynomials”. This paper is a contribution to an old conjecture of Sendov on the z… | Continue reading
Several years ago, I developed a public lecture on the cosmic distance ladder in astronomy from a historical perspective (and emphasising the role of mathematics in building the ladder). I previous… | Continue reading
Starting on Oct 2, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence at the math departmeht here at UCLA. This first course covers much of the … | Continue reading
Vaughan Jones, who made fundamental contributions in operator algebras and knot theory (in particular developing a surprising connection between the two), died this week, aged 67. Vaughan and I gre… | Continue reading
I’ve just uploaded to the arXiv my paper The Ionescu-Wainger multiplier theorem and the adeles”. This paper revisits a useful multiplier theorem of Ionescu and Wainger on “major a… | Continue reading
The square root cancellation heuristic, briefly mentioned in the preceding set of notes, predicts that if a collection $latex {z_1,\dots,z_n}&fg=000000$ of complex numbers have phases that are … | Continue reading
I was greatly saddened to learn that John Conway died yesterday from COVID-19, aged 82. My own mathematical areas of expertise are somewhat far from Conway’s; I have played for instance with … | Continue reading
Next quarter, starting March 30, I will be teaching “Math 247B: Classical Fourier Analysis” here at UCLA. (The course should more accurately be named “Modern real-variable harmon… | Continue reading
Nothing is particularly hard if you divide it into small jobs. (Henry Ford) A typical argument in modern mathematics is often quite intricate, requiring many different steps, ingredients, and notat… | Continue reading
I have always found that plans are useless, but planning is indispensable. (Dwight Eisenhower, quoted in “Six crises”) Some thought should be given as to the logical layout of the paper… | Continue reading
The best performance improvement is the transition from the nonworking state to the working state. (John Ousterhout) Discovering a solution to a mathematical problem is only half of the battle. Ac… | Continue reading
There are three rules for writing the novel. Unfortunately, no one knows what they are. (W. Somerset Maugham) Everyone has to develop their own writing style, based on their own strengths and weakn… | Continue reading
As part of social distancing efforts to slow down the spread of the novel coronavirus, several universities have now transitioned, or begun transitioning, to online teaching models. (My home unive… | Continue reading
The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Wh… | Continue reading
Asgar Jamneshan and I have just uploaded to the arXiv our paper “concrete measurable maps as concrete measurable spaces. (One could also call $latex {X = (X,{\mathcal X}, \mu)}&fg=000000$… | Continue reading
Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues”. This note gives two proofs of a general eige… | Continue reading
I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values”, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I re… | Continue reading
The Riemann zeta function $latex {\zeta(s)}&fg=000000$ is defined in the region $latex {\hbox{Re}(s)>1}&fg=000000$ by the absolutely convergent series $latex \displaystyle \zeta(s) = \su… | Continue reading
The AMS and MAA have recently published (and made available online) a collection of essays entitled “Living Proof: Stories of Resilience Along the Mathematical Journey”. Each author contribu… | Continue reading
Given three points $latex {A,B,C}&fg=000000$ in the plane, the distances $latex {|AB|, |BC|, |AC|}&fg=000000$ between them have to be non-negative and obey the triangle inequalities $latex … | Continue reading
(This post is mostly intended for my own reference, as I found myself repeatedly looking up several conversions between polynomial bases on various occasions.) Let $latex {\mathrm{Poly}_{\leq n}}&a… | Continue reading
I have just learned that Jean Bourgain passed away last week in Belgium, aged 64, after a prolonged battle with cancer. He and Eli Stein were the two mathematicians who most influenced my early ca… | Continue reading
In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Inte… | Continue reading
About six years ago on this blog, I started thinking about trying to make a web-based game based around high-school algebra, and ended up using Scratch to write a short but playable puzzle game in … | Continue reading