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John Conway
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 …
Terence Tao
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao
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Visit terrytao.wordpress.com → RSS feedOn Hacker News Jan 2010 – Nov 2026
Math & Algorithms 41Science & Space 29Life, Learning & Writing 10AI & Machine Learning 8Career & Management 6Society, Policy & Culture 3
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Eigenvectors from eigenvalues
Peter Denton, Stephen Parke, Xining Zhang, and I have just uploaded to the arXiv the short unpublished note “Eigenvectors from eigenvalues”.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…
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If math is more than proof, we need to better celebrate the rest of it
[This is a guest post by Grant Sanderson. This blog post was initially written in a different file format and converted using AI. — T.] A sentiment echoing throughout the mathematics communit…
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Mathematics has changed. Let's act like it
[This is a guest post by Frank Calegari. This blog post was initially written in a different file format and converted using AI. — T.] Why we should not be sad The release by OpenAI on Octobe…
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A digestion of the Jacobian conjecture counterexample
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let $latex {F:{\bf C}^n \rightarrow {\bf C}^n}&fg=000000$ …
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Six Math Essentials
Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled “Six Math Essentials”, which will cover six of the fundament…
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A Lean companion to Analysis I
Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I”. It was intended to complement the many good available analysis textbooks out there by focusing more on foun…
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Mathematical exploration and discovery at scale
Bogdan Georgiev, Javier Gómez-Serrano, Adam Zsolt Wagner, and I have uploaded to the arXiv our paper “Mathematical exploration and discovery at scale”. This is a longer report on…
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Problem solving strategies in a graduate real analysis course (2010)
This is going to be a somewhat experimental post. In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a…
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Palomar: A registry of Lean verified mathematics
In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking tha…
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Gamifying propositional logic: QED, an interactive textbook
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 …
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A pilot project in universal algebra to explore new ways to collaborate
Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that…
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A mathematical formalization of dimensional analysis (2012)
Mathematicians study a variety of different mathematical structures, but perhaps the structures that are most commonly associated with mathematics are the number systems, such as the integers $late…
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How to assign partial credit on an exam of true-false questions?
Note: the following is a record of some whimsical mathematical thoughts and computations I had after doing some grading. It is likely that the sort of problems discussed here are in fact well studi…
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Erdős Problem #1026
Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try…
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Will it be a golden age or a dark age for mathematics (and science)?
[This is a guest post by Francesco Maggi. This blog post was initially written in a different file format and converted using AI. — T.] Large language models (LLMs) and the research systems b…
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What Are the Odds?
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…
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Decomposing a Factorial into Large Factors
I’ve just uploaded to the arXiv the paper “Decomposing a factorial into large factors”. This paper studies the quantity $latex {t(N)}&fg=000000$, defined as the largest quanti…
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Bill Thurston has died
Bill Thurston, who made fundamental contributions to our understanding of low-dimensional manifolds and related structures, died on Tuesday, aged 65. Perhaps Thurston’s best known achievement…
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The Future of Mathematics
[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.] “Mathematics underwent, in the nineteenth century, …
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Quaternions and spherical trigonometry
Hamilton’s quaternion number system $latex {\mathbb{H}}&fg=000000$ is a non-commutative extension of the complex numbers, consisting of numbers of the form $latex {t + xi + yj + zk}&f…
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Closing the “green gap”: energy savings from the math of the landscape function
I recently returned from the 2025 Annual Meeting of the “Localization of Waves” collaboration (supported by the Simons Foundation, with additional related support from the NSF), where I…
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Special Relativity and Middle-Earth
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&…
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Mathematics Distillation Challenge – Equational Theories
Mathematical research traditionally involves a small number of professional mathematicians working closely on difficult problems. However, I have long believed that there is a complementary way to …
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Terry Tao on how to compute non-converging infinite sums (2010)
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…
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Climbing the cosmic distance ladder: Terence Tao book announcement
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…
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Illustration Beyond Proof
[This is a guest post by Kate Stange and Edmund Harriss. This blog post was initially written in a different file format and converted using AI. — T.] In January a group of mathematicians, hi…
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A review of probability theory
In preparation for my upcoming course on random matrices, I am briefly reviewing some relevant foundational aspects of probability theory, as well as setting up basic probabilistic notation that we…
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Jean Bourgain
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…