Terence Tao | What's new Note

Terence Tao | What's new

Terry Tao's blog is a valuable resource for anyone interested in mathematics. Terry Tao is a highly respected mathematician, and his blog features a range of content from detailed mathematical expositions to discussions on various topics in the field. He often shares insights into his research, addresses questions from the mathematical community, and provides explanations of complex concepts. If you're looking to deepen your understanding of advanced mathematics or keep up with current developments in the field, his blog is definitely worth exploring.

Thread Of Notes

On the proposed rule changes to the administration of federal grants

The United States Office of Management and Budget (OMB) has proposed a vast and radical set of rule changes to how federal grants from all funding agencies are administered.  (A summary of the key changes, by a former Senior Program Officer at the National Institutes for Health, can be found here.) This is no mere […]

Modular Arithmetic Challenge

A couple months ago, Damek Davis and I launched the first mathematical challenge at the SAIR Foundation, aimed at “distilling” the ability to solve 22 million problems in universal algebra into a condensed form. Stage one of that challenge has now been completed, with several effective “cheat sheets” generated to guess the truth or falsity […]

Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond

Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, and I have just uploaded to the arXiv our paper Primitive sets and von Mangoldt chains: Erdős Problem \#1196 and beyond. This paper (which is a work in progress) represents our efforts to digest and document the recent flurry […]

Products of consecutive integers with unusual anatomy

I’ve just uploaded to the arXiv my paper “Products of consecutive integers with unusual anatomy“. This paper answers some questions of Erdős and Graham which were initially motivated by the study of the Diophantine factorial equation where and are positive integers. Writing , one can rewrite this equation as where denotes the squarefree part of […]

Mathematical methods and human thought in the age of AI

Tanya Klowden and I have uploaded to the arXiv our preprint “Mathematical methods and human thought in the age of AI“. This is an unabridged version of a solicited article for a forthcoming Blackwell Companion to the Philosophy of Mathematics. I rarely write article-length essays of a philosophical nature (perhaps the last one was in […]

Local Bernstein theory, and lower bounds for Lebesgue constants

I’ve just uploaded to the arXiv my paper “Local Bernstein theory, and lower bounds for Lebesgue constants“. This paper was initially motivated by a problem of Erdős} on Lagrange interpolation, but in the course of solving that problem, I ended up modifying some very classical arguments of Bernstein and his contemporaries (Boas, Duffin, Schaeffer, Riesz, […]

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 do mathematics, in which one works with a broad community of mathematically minded people on problems which may not be as deep as the problems one traditionally works […]

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 fundamental concepts in mathematics — numbers, algebra, geometry, probability, analysis, and dynamics — and how they connect with our real-world intuition, the history of math […]

IPAM industrial short course in generative AI algorithms – deadline for applications closing soon

(Sharing this in my capacity of director of special projects at IPAM.) IPAM is holding an Industrial Short Course on Generative AI Algorithms on March 5-6, 2026. The short course is aimed at people from industry or government who want to get started in deep learning, apply deep learning to their projects, learn how to […]

A crowdsourced repository for optimization constants?

Thomas Bloom’s Erdös problem site has become a real hotbed of activity in recent months, particularly as some of the easiest of the outstanding open problems have turned out to be amenable to various AI-assisted approaches; there is now a lively community in which human contributions, AI contributions, and hybrid contributions are presented, discussed, and […]

The integrated explicit analytic number theory network

Like many other areas of modern analysis, analytic number theory often relies on the convenient device of asymptotic notation to express its results. It is common to use notation such as or , for instance, to indicate a bound of the form for some unspecified constant . Such implied constants vary from line to line, […]

Polynomial towers and inverse Gowers theory for bounded-exponent groups

Asgar Jamneshan, Or Shalom and I have uploaded to the arXiv our paper “ Polynomial towers and inverse Gowers theory for bounded-exponent groups“. This continues our investigation into the ergodic-theory approach to the inverse theory of Gowers norms over finite abelian groups . In this regard, our main result establishes a satisfactory (qualitative) inverse theorem […]

The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree

I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree. This paper asymptotically resolves an old question about the polynomial lemniscates attached to monic polynomials of a given degree , and specifically the question of bounding the arclength of such lemniscates. For instance, when , the lemniscate […]

The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan Kocsis, Bruno Le Floch, Amir Livne Bar-on, Lorenzo Luccioli, Douglas McNeil, Alex Meiburg, Pietro Monticone, Pace P. Nielsen, Emmanuel Osalotioman Osazuwa, Giovanni Paolini, Marco Petracci, Bernhard Reinke, David Renshaw, Marcus […]

Quantitative correlations and some problems on prime factors of consecutive integers

Joni Teravainen and I have uploaded to the arXiv my paper “Quantitative correlations and some problems on prime factors of consecutive integers“. This paper applies modern analytic number theory tools – most notably, the Maynard sieve and the recent correlation estimates for bounded multiplicative functions of Pilatte – to resolve (either partially or fully) some […]

Growth rates of sequences governed by the squarefree properties of its translates

Wouter van Doorn and I have uploaded to the arXiv my paper “Growth rates of sequences governed by the squarefree properties of its translates“. In this paper we answer a number of questions of Erdős} (Problem 1102 and Problem 1103 on the Erdős problem web site) regarding how quickly a sequence of increasing natural numbers […]

Call for industry sponsors for IPAM’s RIPS program

Over the last 25 years, the Institute for pure and applied mathematics (IPAM) at UCLA (where I am now director of special projects) has run the popular Research in Industrial Projects for Students (RIPS) program every summer, in which industry sponsors with research projects are matched with talented undergraduates and a postdoctoral mentor to work […]

Climbing the cosmic distance ladder: another sample chapter

Five years ago, I announced a popular science book project with Tanya Klowden on the cosmic distance ladder, in which we released a sample draft chapter of the book, covering the “fourth rung” of the ladder, which for us meant the distances to the planets. In the intervening time, a number of unexpected events have […]

Sum-difference exponents for boundedly many slopes, and rational complexity

I have uploaded to the arXiv my paper “Sum-difference exponents for boundedly many slopes, and rational complexity“. This is the second spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about. One of the many problems we experimented using the AlphaEvolve tool with was that of […]

New Nikodym set constructions over finite fields

I have uploaded to the arXiv my paper “New Nikodym set constructions over finite fields“. This is a spinoff of my previous project with Bogdan Georgiev, Javier Gómez–Serrano, and Adam Zsolt Wagner that I recently posted about. In that project we experimented with using AlphaEvolve (and other tools, such as DeepThink and AlphaProof) to explore […]

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 the experiments we did in collaboration with Google Deepmind with their AlphaEvolve tool, which is in the process of being made available for broader use. Some of […]

A crowdsourced project to link up erdosproblems.com to the OEIS

Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been […]

SLMath announces new research programs

The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing three new research initiatives, whose applications open on Sep 1 2025: (Disclosure: I am vice-chair of the board of trustees at SLMath.)

Rough numbers between consecutive primes

First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. […]

Salem Prize now accepting nominations for 2025

The Salem prize was established in 1968 and named in honor of Raphaël Salem (1898-1963), a mathematician famous notably for his deep study of the links between Fourier series and number theory and for pioneering applications of probabilistic methods to these fields. It was not awarded from 2019-2022, due to both the COVID pandemic and the death […]

Decomposing a factorial into large factors (second version)

Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew Sutherland, Markus Uhr, Kevin Ventullo, and I have uploaded to the arXiv a second version of our paper “Decomposing a factorial into large factors“. This is a completely rewritten and expanded version of a previous paper of the same name. Thanks to many additional theoretical and numerical contributors […]

On the number of exceptional intervals to the prime number theorem in short intervals

Ayla Gafni and I have just uploaded to the arXiv the paper “On the number of exceptional intervals to the prime number theorem in short intervals“. This paper makes explicit some relationships between zero density theorems and prime number theorems in short intervals which were somewhat implicit in the literature at present. Zero density theorems […]

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 foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic […]

Some variants of the periodic tiling conjecture

Rachel Greenfeld and I have just uploaded to the arXiv our paper Some variants of the periodic tiling conjecture. This paper explores variants of the periodic tiling phenomenon that, in some cases, a tile that can translationally tile a group, must also be able to translationally tile the group periodically. For instance, for a given […]

A tool to verify estimates, II: a flexible proof assistant

In a recent post, I talked about a proof of concept tool to verify estimates automatically. Since that post, I have overhauled the tool twice: first to turn it into a rudimentary proof assistant that could also handle some propositional logic; and second into a much more flexible proof assistant (deliberately designed to mimic the […]

Orders of infinity

Many problems in analysis (as well as adjacent fields such as combinatorics, theoretical computer science, and PDE) are interested in the order of growth (or decay) of some quantity that depends on one or more asymptotic parameters (such as ) – for instance, whether the quantity grows or decays linearly, quadratically, polynomially, exponentially, etc. in […]

A proof of concept tool to verify estimates

This post was inspired by some recent discussions with Bjoern Bringmann. Symbolic math software packages are highly developed for many mathematical tasks in areas such as algebra, calculus, and numerical analysis. However, to my knowledge we do not have similarly sophisticated tools for verifying asymptotic estimates – inequalities that are supposed to hold for arbitrarily […]

Stonean spaces, projective objects, the Riesz representation theorem, and (possibly) condensed mathematics

A basic type of problem that occurs throughout mathematics is the lifting problem: given some space that “sits above” some other “base” space due to a projection map , and some map from a third space into the base space , find a “lift” of to , that is to say a map such that […]

A suspicious conference

I have received multiple queries from colleagues who have been invited (from a non-academic email address) to speak a strange-sounding conference that is allegedly supported by major mathematical institutions, allegedly hosted at a prestigious university, and allegedly having myself (and two other Fields Medalists) as plenary speakers. The invitees are asked to pay “registration fees” […]

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 , defined as the largest quantity such that it is possible to factorize into factors , each of which is at least . The first few values of this sequence are (OEIS A034258). For instance, we […]

The three-dimensional Kakeya conjecture, after Wang and Zahl

There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture! This conjecture asserts that a Kakeya set – a subset of that contains a unit line segment in every direction, must have Minkowski and […]

Closing the “green gap”: from the mathematics of the landscape function to lower electricity costs for households

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 learned (from Svitlana Mayboroda, the director of the collaboration as well as one of the principal investigators) of a remarkable statistic: net electricity consumption by residential customers […]

Cosmic Distance Ladder video with Grant Sanderson (3blue1brown): commentary and corrections

Grant Sanderson (who creates the website and Youtube channel 3blue1brown) has been collaborating with myself and others (including my coauthor Tanya Klowden) on producing a two-part video giving an account of some of the history of the cosmic distance ladder, building upon a previous public lecture I gave on this topic, and also relating to […]

New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach

Timothy Trudgian, Andrew Yang and I have just uploaded to the arXiv the paper “New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach“. This paper launches a project envisioned in this previous blog post, in which the (widely dispersed) literature on various exponents in classical analytic number theory, as well […]

Quaternions and spherical trigonometry

Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and are anti-commuting square roots of with , , . While they are non-commutative, they do keep many other properties of the complex numbers: Being non-commutative, the quaternions do not form a field. […]

On the distribution of eigenvalues of GUE and its minors at fixed index

I’ve just the arXiv the paper “On the distribution of eigenvalues of GUE and its minors at fixed index“. This is a somewhat technical paper establishing some estimates regarding one of the most well-studied random matrix models, the Gaussian Unitary Ensemble (GUE), that were not previously in the literature, but which will be needed for […]

AI for Math fund

Renaissance Philanthropy and XTX Markets have announced the launch of the AI for Math Fund, a new grant program supporting projects that apply AI and machine learning to mathematics, with a focus on automated theorem proving, with an initial $9.2 million in funding. The project funding categories, and examples of projects in such categories, are: […]

On several irrationality problems for Ahmes series

Vjeko Kovac and I have just uploaded to the arXiv my paper “On several irrationality problems for Ahmes series“. This paper resolves (or at least makes partial progress on) some open questions of Erdös and others on the irrationality of Ahmes series, which are infinite series of the form for some increasing sequence of natural […]

Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

Kaisa Matomäki, Maksym Radziwill, Fernando Xuancheng Shao, Joni Teräväinen, and myself have (finally) uploaded to the arXiv our paper “Higher uniformity of arithmetic functions in short intervals II. Almost all intervals“. This is a sequel to our previous paper from 2022. In that paper, discorrelation estimates such as were established, where is the von Mangoldt […]

Climbing the cosmic distance ladder: launching an instagram

I have been collaborating for many years with a long-time personal friend and polymath, Tanya Klowden, on a popular book on astronomy tentatively entitled “Climbing the cosmic distance ladder“, which was initially based on a public lecture I have given on this topic for many years, but which we have found to be a far […]