Terence Tao

Professor of #Mathematics at the University of California, Los Angeles #UCLA (he/him).

Home page: https://www.math.ucla.edu/~tao
Blog: https://terrytao.wordpress.com/
Bluesky: https://bsky.app/profile/teorth.bsky.social
Cosmic distance ladder: https://www.instagram.com/cosmic_distance_ladder/

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Last Notes

2026-09-07 17:40:55 UTC

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However, as observed in Section 14.1, there were noticeable opportunity costs with moving too rapidly. We noted a few cases where a key implication stumped us for a while, and forced us to come up with interesting new techniques to construct infinite counterexamples to that implication; but subsequently it was found via an automated theorem prover that the implication in question actually had a relatively small finite counterexample. But if that counterexample had been found first, we might not have discovered the more interesting method. We in fact suspect that some of our early automated sweeps to resolve implications may have accidentally obscured some potentially fruitful problems that would have revealed even more constructions or insights, although we do not have the counterfactual data to confirm these suspicions definitively. For better or for worse, the ETP problem set is now "solved", and there is no practical way to undo this and recreate the unsolved status of this problem to encourage further exploration.
For the first time in the history of mathematics, we are faced with the situation that open problems in pure mathematics -- one of the primary fuels for mathematical discovery -- are being mined as a non-renewable resource; and if the solution is extracted poorly from that resource, then the net value to mathematics gained as a result can be significantly less than if it were obtained in a responsible and sustainable fashion. (2/3)

2026-09-07 17:40:45 UTC

While working on the revision of the Equational Theories Project (ETP) report at https://arxiv.org/abs/2512.07087 I came across an observation that we quietly made at the back of that report (see Section 14.1) but has become more relevant today.
The ETP was a successful project, running over several months, to resolve over 22 million implications in universal algebra (and formalize them in Lean) through a variety of methods, both human and automated, although in our case the automation was much more in the ``good old-fashioned'' form of utilizing automated theorem provers (ATP) and simple scripts, rather than modern LLMs. If ETP were run today, it is likely that modern LLMs would very quickly resolve almost all of the implications studied, and the few remaining holdouts would likely then fall in a matter of days with additional LLM compute. (1/3)

2026-08-09 23:59:37 UTC

I decided to convert one of the points in my ICM 2026 slides https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf into a meme format.
https://media.mathstodon.xyz/media_attachments/files/117/068/251/170/802/187/original/528f90f546bcd025.jpg

2026-04-09 21:09:24 UTC

RE: https://mastodon.social/@ZachWeinersmith/116375447678097912
This was a fun collaboration! See also the crypto challenge at the end of the final strip in the 5-part series.
#note1ywn…0gym

2026-03-08 00:25:04 UTC

After a nine month hiatus, I have recorded another Lean formalization video, this time using the Claude Code agent to formalize the same informal proof I had recorded doing with Github Copilot and canonical last year. https://youtu.be/JHEO7cplfk8

2026-02-23 01:43:08 UTC

This is an expansion of a point I initially wrote in the context of a MathOverflow question https://mathoverflow.net/questions/487041/collaborative-repositories-on-open-problems/487065#487065, but also has relevance to the role of AI in activities such as mathematics where ideation is important.
It is commonly accepted that one of the impediments to further progress in mathematics is a shortage of new ideas. Naively, one can model this hypothesis by proposing that
(number of new ideas) (*)
is the key factor determining the rate of progress, and then try to support efforts to maximize the quantity (*).
However, in the era of increasingly large amounts of AI-generated mathematics, the _quality_ of these ideas becomes increasingly relevant. Only a small fraction of new ideas tend to be good and fruitful ones; a bad idea can actually impede progress by wasting more time than it saves. So, a more realistic model would be that it is the actually the product
(number of good new ideas) * (signal-to-noise ratio of the idea pool) (**)
that is the important factor which is worth maximizing. (This is still a massive oversimplification - for instance, it assumes a binary classification of ideas into "good" and "bad" - but will serve as a minimal toy model that suffices to illustrate the broader points that I wish to make here.) (1/3)

2026-02-08 23:21:06 UTC

A few days ago, I noted the revival of one archaic mathematical practice, namely that of encrypting one's proofs (or announcements). Today, as part of the ongoing Integrated Explicit Analytic Number Theory Network project https://www.ipam.ucla.edu/news-research/special-projects/integrated-explicit-analytic-number-theory-network/ , we found ourselves reviving another archaic piece of mathematical infrastructure: the logarithm table.
These tables, pioneered by Napier in the 17th century, were a mainstay of mathematical computation until eventually supplanted first by calculators and then by modern computers. But we are finding that verifying in Lean that, say, ln 2 is equal to 0.693147 to six decimal places is somewhat fiddly and computationally expensive to formally verify (one has to use a Taylor series with explicit remainder and figure out where to cut off the series).
Eventually we settled on what is basically the 17th century solution, modernized for the era of formal proof verification: the project now sports a file `LogTables.lean` which systematically gathers formally verified calculations of logarithms via a new interval arithmetic package. Similar to a precomputed log table, this file is intended to be typechecked once (as a laborious calculation), and then imported as needed by all other files.
It is a fascinating paradox that cutting edge technology can sometimes make obsolete practices relevant again, albeit with a modern spin. (Yet another example: the capability of current AI tools has revived the in-person class exam, which we had just started to move away from in the COVID era.)

2026-02-06 06:14:11 UTC

This experiment (authored by several well-known mathematicians) revives an archaic practice (from the era of Gauss) of posting encrypted proofs before revealing them: https://arxiv.org/abs/2602.05192 . Here, the challenge is to see whether 10 research-level problems (that arose in the course of the authors research) are amenable to modern AI tools within a fixed time period (until Feb 13).
The problems appear to be out of reach of current "one-shot" AI prompts, but were solved by human domain experts, and would presumably a fair fraction would also be solvable by other domain experts equipped with AI tools. They are technical enough that a non-domain-expert would struggle to verify any AI-generated output on these problems, so it seems quite challenging to me to have such a non-expert solve any of these problems, but one could always be surprised. It will be interesting to see if there were any notable outcomes to this experiment by the expiration of the time linit.

2026-02-04 19:19:50 UTC

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No, although they supply the compute resources for the AlphaEvolve experiments I have run.

2026-02-03 18:50:51 UTC

It will still be a few months before AlphaEvolve is fully released to the public, but we at least now have a small gallery of AlphaEvolve experiments one can showcase, including some of my own: https://alphaevolve-examples.web.app/ae/gallery

2026-01-17 18:41:05 UTC

One of the biggest statistical biases one encounters when trying to assess the true success rate of AI tools is the strong reporting bias against disclosing negative results. If an individual or AI company research group applies their AI tool to an open problem, but makes no substantial progress, there is little incentive for the user of that tool to report the negative statement; furthermore, even if such results are reported, they are less likely to go "viral" on social media than positive results. As a consequence, the results one actually hears about on such media is inevitably highly skewed towards the positive results.
With that in mind, I commend this recent initiative of Paata Ivanisvili and Mehmet Mars Seven to systematically document the outcomes (both positive and negative) of applying frontier LLMs to open problems, such as the Erdos problems: https://mehmetmars7.github.io/Erdosproblems-llm-hunter/index.html
As one can see, the true success rate of these tools for, say, the Erdos problems is actually only on the level of a percentage point or two; but with over 600 outstanding open problems, this still leads to an impressively large (and non-trivial) set of actual AI contributions to these problems, though overwhelmingly concentrated near the easy end of the difficulty spectrum, and not yet a harbinger that the median Erdos problem is anywhere within reach of tehse tools.

2025-11-30 18:08:38 UTC

As with many other real-world distributions, the class of open problems in mathematics has a "long tail" - a large number of problems which would be relatively easy to prove or disprove, but which have not recieved significant attention from the (limited) number of expert mathematicians available. To switch metaphors, this tail can thus contain a large amount of "low-hanging fruit" of new mathematical results that could be obtained if there was some way to automatically attack these problems at scale.
I saw this first-hand when running the Equational Theories Project last year, in which 22 million implications in universal algebra were attacked (and ultimately resolved). Initial passes with low-tech automated methods resolved a large percentage of these implications within days, with increasingly sophisticated methods brought to bear to pick off the more stubborn holdouts that resisted earlier sweeps. The final few implications took months of human effort to settle: https://github.com/teorth/equational_theories/wiki/Terence-Tao's-personal-log
(1/3)

2025-11-06 03:42:20 UTC

A new paper with Bogdan Georgiev, Javier Gomez-Serrano, and Adam Zsolt Wagner: "Mathematical exploration and discovery at scale" https://arxiv.org/abs/2511.02864 , in which we record our experiments using the LLM-powered optimization tool #AlphaEvolve to attack 67 different math problems (both solved and unsolved), improving upon the state of the art in some cases and matching preivous literature in others. The data for these experiments can be found at https://github.com/google-deepmind/alphaevolve_repository_of_problems and further discussion is at https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/

2025-10-22 05:40:25 UTC

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The author's account of this story in Sections 1 and 7 of https://borisalexeev.com/pdf/erdos707.pdf make for fascinating reading. Some takeaways from this story:
1. Both human and AI-powered literature reviews can still fail to turn up relevant hits. There is still room for improvement in this area.
2. The Lean formalization allowed for "instant refereeing": the Lean community at https://leanprover.zulipchat.com/#narrow/channel/113488-general/topic/Lean.20in.20the.20wild/with/546368325 were able to verify the proof within thirty minutes of the announcement.
3. There appears to be one very specific use case in which "vibe coding" can actually be used responsibly and effectively: generating a formal proof artefact for a statement that has already been formalized by human experts. But even then, there is the potential for human-generated error in the statement formalization.
4. This also appears to be one of the very few use cases where LLM output can be used responsibly in a research paper. Importantly, no LLM-generated output was directly placed into the main body of the text (other than when quoting an excerpt from the LLM-generated Lean code for illustrative purposes); instead, such output was only used in completely verifiable contexts (in this case, in generating code that can be type checked by Lean).
Given recent experience, I can imagine that there will be some breathless reports that "LLMs actually solved an Erdos problem for real this time!". The truth however is extremely nuanced, and really requires a detailed study of the situation before jumping to any conclusions.

2025-10-22 05:40:02 UTC

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* But neither of the human authors had much experience in Lean, so they decided to "vibe code" a proof via ChatGPT. In contrast to most vibe coding use cases, this actually worked after several days of effort, giving a ~3000 line Lean proof that has been fully verified, using the formal statement of the theorem that was already present in the Formal Conjectures Repository at https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/707.lean
* But in doing so, they located a typo in that formalization (a condition that a certain modulus had to be nonzero was omitted).
Due to these latter developments, both ChatGPT and Lean are also listed as co-authors on the paper. (2/3)

2025-10-22 05:39:47 UTC

Another interesting example of modern computer assistance in mathematics, again involving the https://www.erdosproblems.com/ site: Problem #707 (https://www.erdosproblems.com/707), previously marked as "open", is now "disproved" - with the disproof formalized in Lean : https://borisalexeev.com/papers/erdos707.html . But the path towards that disproof was quite unusual, not fitting neatly into any of the standard narratives about AI in mathematics:
* The authors' initial proof used some numerical computer experiments, but did not get much help from LLMs initially, with even the code generated by hand.
* The authors then found a disproof by conventional human arguments (and also obtained some stronger results of a similar nature), and did not turn up any prior solution to this problem in their literature searches. Even the modern AI deep research tools did not locate any hits.
* Nevertheless, as part of the process of writing the paper, they found a solution to the problem (slightly different from theirs) had been obtained by Hall, thirty years before Erdos even posed the problem!
* Perturbed by this, the authors then decided to formalize both their result and Hall's result in Lean. (1/3)

2025-10-16 17:28:31 UTC

I am increasingly of the opinion that the most productive near-term adoptions of AI in mathematics will primarily come not from applying the most powerful models to the most challenging problems (although we will see a few isolated examples of progress along those lines, especially when large amounts of computational resources and expert attention are applied), but from using medium-powered tools to accelerate and scale up more mundane and time-consuming, but still essential, research tasks, using the accumulated human experience with (and understanding of) such tasks to guide, verify, and safely incorporate the AI output into one's workflows. In such use cases, the output of the AI tool could also have been produced (with increased expenditure of time and attention) by a human expert - but this is actually a feature rather than a bug, as it allows for the AI output to be readily and reliably assessed, confirmed, and converted to a format that such experts are already comfortable working with.
An example of such a mundane task is literature review: locating relevant prior literature on a given problem. If the problem already has a commonly agreed upon name, as well a well-established community of researchers working on it, then existing web search and bibliographic search tools are already more than adequate to find both past and current literature on the problem: in particular, the citation graph between the literature will be dense enough that one can start with one key paper in the subject and perform both forward and backward citation searches to obtain a reasonably complete picture of the current state of knowledge on the problem. (1/4)

2025-10-02 20:20:12 UTC

I was able to use an extended conversation with an AI https://chatgpt.com/share/68ded9b1-37dc-800e-b04c-97095c70eb29 to help answer a MathOverflow question https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t/501125#501125 . I had already conducted a theoretical analysis suggesting that the answer to this question was negative, but needed some numerical parameters verifying certain inequalities in order to conclusively build a counterexample. Initially I sought to ask AI to supply Python code to search for a counterexample that I could run and adjust myself, but found that the run time was infeasible and the initial choice of parameters would have made the search doomed to failure anyway. I then switched strategies and instead engaged in a step by step conversation with the AI where it would perform heuristic calculations to locate feasible choices of parameters. Eventually, the AI was able to produce parameters which I could then verify separately (admittedly using Python code supplied by the same AI, but this was a simple 29-line program that I could visually inspect to do what was asked, and also provided numerical values in line with previous heuristic predictions).
Here, the AI tool use was a significant time saver - doing the same task unassisted would likely have required multiple hours of manual code and debugging (the AI was able to use the provided context to spot several mathematical mistakes in my requests, and fix them before generating code). Indeed I would have been very unlikely to even attempt this numerical search without AI assistance (and would have sought a theoretical asymptotic analysis instead).

2025-09-24 15:20:54 UTC

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Much of the current debate on societal issues is then framed as conflicts between large organizations (e.g., opposing political parties, or extremely powerful or wealthy individuals with a status comparable to such organizations), conflicts between large organizations and average individuals, or a yearning for a return to a more traditional era where legacy small organizations recovered their former role. While these are valid framings, I think one aspect we could highlight more is the valuable (though usually non-economic) roles played by emerging grassroots organizations, both in providing "softer" benefits to individuals (such as a sense of purpose, and belonging) and as a way to meaningfully connect with larger organizations and systems; and be more aware of what the tradeoffs are when converting such an organization to a larger one (or component of a larger organization).
(5/5)

2025-09-24 15:20:32 UTC

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My tentative theory is that the systems, incentives, and technologies in modern world have managed to slightly empower the individual, and massively empower large organizations, but at the significant expense of small organizations, whose role in the human societal ecosystem has thus shrunk significantly, with many small organizations either weakening in influence or transitioning to (or absorbed by) large organizations. While this imbalanced system does provide significant material comforts (albeit distributed rather unequally) and some limited feeling of agency, it has led at the level of the individual to feelings of disconnection, alienation, loneliness, and cynicism or pessimism about the ability to influence future events or meet major challenges, except perhaps through the often ruthless competition to become wealthy or influential enough to gain, as an individual, a status comparable to a small or even large organization. And larger organizations have begun to imperfectly step in the void formed by the absence of small communities, providing synthetic social or emotional goods that are, roughly speaking, to more authentic such products as highly processed "junk" food is to more nutritious fare, due to the inherently impersonal nature of such organizations (particularly in the modern era of advanced algorithms and AI, which when left to their own devices tend to exacerbate the trends listed above).
(4/5)

2025-09-24 15:19:53 UTC

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Large organized groups can offer substantially more economies of scale, and so can outcompete small organizations based on the economic goods they offer. They also have more significant impact on global systems than either average individuals or small organizations. But the social and emotional services they provide are significantly less satisfying and authentic. And unless an individual is extremely wealthy, well-connected, or popular, they are unlikely to have any influence on the direction of such a large organization, except possibly through small organizations acting as intermediaries. In particular, when a large organization becomes dysfunctional, it can be an extremely frustrating task to try to correct its course (and if it is extremely large, other options such as escaping it or leaving it to fail are also highly problematic).
(3/5)

2025-09-24 15:19:38 UTC

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An individual human without any of the support provided by larger organized groups is only able to exist at quite primitive levels, as any number of pieces of post-apocalyptic fiction can portray. Both small and large organized groups offer significant economies of scale and division of labor that provide most of the material conveniences that we take for granted in the modern world: abundant food, access to power, clean water, internet; cheap, safe and affordable long distance travel; and so forth. It is also only through such groups that one can meaningfully interact with (and even influence) the largest scale systems that humans are part of.
But the benefits and dynamics of small and large groups are quite different. Small organized groups offer some economy of scale, but - being essentially below Dunbar's number https://en.wikipedia.org/wiki/Dunbar%27s_number in size - also fill social and emotional needs, and the average participant in such groups can feel connected to such groups and able to have real influence on their direction. Their dynamics can range anywhere from extremely healthy to extremely dysfunctional and toxic, or anything in between; but in the latter cases there is real possibility of individuals able to effect change in the organization (or at least to escape it and leave it to fail on its own).
(2/5)

2025-09-24 15:19:23 UTC

Some loosely organized thoughts on the current Zeitgeist. They were inspired by the response to my recent meta-project mentioned in my previous post https://mathstodon.xyz/@tao/115254145226514817, where within 24 hours I became aware of a large number of ongoing small-scale collaborative math projects with their own modest but active community (now listed at https://mathoverflow.net/questions/500720/list-of-crowdsourced-math-projects-actively-seeking-participants ); but they are from the perspective of a human rather than a mathematician.
As a crude first approximation, one can think of human society as the interaction between entities at four different scales:
1. Individual humans
2. Small organized groups of humans (e.g., close or extended family; friends; local social or religious organizations; informal sports clubs; small businesses and non-profits; ad hoc collaborations on small projects; small online communities)
3. Large organized groups of humans (e.g., large companies; governments; global institutions; professional sports clubs; large political parties or movements; large social media sites)
4. Large complex systems (e.g., the global economy; the environment; the geopolitical climate; popular culture and "viral" topics; the collective state of science and technology).
(1/5)

2025-09-15 23:32:23 UTC

#SLMath and #ICARM are hosting a joint hybrid workshop on Lean for PDEs, Oct 6-9. Registration is now open: https://www.slmath.org/workshops/1180

2025-09-13 12:12:48 UTC

Any complex project tends to have both explicitly stated and implicitly unstated target goals. For instance, a Lean formalization project may have as its explicit goal the task of obtaining a formal proof of some mathematical claim X; but there are often unstated goals, such as also formalizing key subclaims and definitions X_1, X_2, ... to X in a fashion that would be suitable for upstreaming to the Mathlib library; learning how to use various collaboration tools and distribute tasks; organically discovering insights to the finer structure of the proof of X that might not be emphasized in previous informal proofs; giving real-world training and experience to novice formalizers; and more generally building a community of humans expert in the art of formalization.
In the past, it has generally not been necessary to state these implicit goals because of a strong empirical correlation between the achievement of these goals and the achievement of the explicit goals. In the example of the formalization project, pretty much any human-centric effort to accomplish the explicit goal will end up naturally also achieving most of the implicit goals stated above. So the explicit goal effectively becomes a viable proxy for the broader range of actual goals. (1/2)

2025-09-06 02:17:16 UTC

Earlier this year, the second #AIMO (artificial intelligence mathematical olympiad) concluded, with the winning team solving 34/50 in the final set of math problems (that had been selected to be harder for AI than the first AIMO).
The competition was restricted to open source models and run with a limited amoutn of compute. The AIMO has now conducted a retest of these problems both for the top two teams from that competition (NemoSkills and imagination research), as well as OpenAI's o3 model, both with comparable levels of compute resources, and with high resources. Unsurprisingly, the high resource models did better, with the high resource o3 model scoring as high as 47/50, or even 50/50 if given two tries at each question. On the other hand, the gap between the open source models and the commercial models for a fixed amount of compute was relatively slight.
More details of this experiment are available at https://aimoprize.com/updates/2025-09-05-the-gap-is-shrinking

2025-09-01 07:25:43 UTC

Thomas Bloom and I are launching a crowdsourced project to link up Thomas's erdosproblems.com site with the #OEIS, by systematically calculating the various integer sequences associated with the Erdos problems and crosschecking them against the OEIS database: https://terrytao.wordpress.com/2025/08/31/a-crowdsourced-project-to-link-up-erdosproblems-com-to-the-oeis/ https://github.com/teorth/erdosproblems

2025-08-30 16:58:52 UTC

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:
* AxIOM (Accelerating Innovation in Mathematics) is a new, month-long research program at SLMath, designed to accelerate innovation and introduce transformative ideas into the mathematical sciences. Programs begin in Spring 2027.
* PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints. Programs begin June-July 2026.
* Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE) is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. Programs begin June-July 2026.
https://www.slmath.org/news-and-events/new-research-initiatives-starting-fall-2025
(Disclosure: I am vice-chair of the board of trustees at SLMath.)

2025-08-28 22:58:48 UTC

Recently, the erdosproblems.com site run by Thomas Bloom has enabled a discussion forum for each of the problems on the site.
As a result of some discussions I had with Stijn Cambie and Vjeko Kovac on this forum, we were able to come up with a short elementary solution to a previously open problem on the site: https://www.erdosproblems.com/forum/thread/379 . In fact the proof is sufficiently short and elementary that it was straightforward to formalize in Lean! https://github.com/teorth/analysis/blob/main/analysis/Analysis/Misc/erdos_379.lean

2025-08-18 15:45:37 UTC

I wrote an op-ed on the world-class STEM research ecosystem in the United States, and how this ecosystem is now under attack on multiple fronts by the current administration: https://newsletter.ofthebrave.org/p/im-an-award-winning-mathematician

2025-08-14 02:19:12 UTC

In response to a recent court ruling, the NSF has unsuspended its various grants to UCLA, including the one to IPAM (and my own personal grant). There is however the possibility of an appeal; and the (somewhat larger) funding from tne NIH to UCLA remains suspended at this time.
Over the last week or so I have been involved in emergency fundraising (mostly on behalf of IPAM), and am heartened by the broad and significant support we have received, both for immediate funding needs, as well as offers to sponsor some future events at the institute. It is a small silver lining to know that there was a reservoir of goodwill and private resources that one could tap into in times of crisis, although the time and energy required to perform this fundraising was significant.
In any event, this is no way to govern a country with a world-class research ecosystem, with quite unnecessary injections of uncertainty and chaos into what had been a predictable funding enviroment conducive to long-term planning.

2025-08-09 16:39:48 UTC

I have recently learned that there was a cryptocurrency launched with my name attached to it. For the record, I am not involved with any such crypto project.

2025-08-02 02:37:23 UTC

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But perhaps the biggest positive contributions of institutes such as IPAM is not on senior faculty such as myself (who often have other resources and connections to draw upon), but on early career researchers, especially those from less well known institutions who might not otherwise have many opportunities to interact with the researchers at the emerging interface between two or more fields that were just beginning to become interconnected. When I was a postdoc, I experienced that opportunity myself in 1997 at a different NSF-funded institute (MSRI, now known as SLMath) in a program https://www.slmath.org/programs/60 on harmonic analysis, which was instrumental in setting up several extremely productive collaborations in my early career. Many of my colleagues and collaborators can also testify to positive early-career experiences in such institute programs as similarly unlocking their own research potential. Losing one of these institutes would have major negative impacts on the next generation of mathematical scientists.
(Disclosure: I am scheduled this year to become Director of Special Projects at IPAM, taking over from Stanley Osher.) (4/4)

2025-08-02 02:37:15 UTC

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Some accounts claim that Emmanuel and I actually started collaborating at the preschool that both of our children attended at the time, but the truth is that our main collaboration actually started at IPAM; the fact that we met on a near-daily basis at the preschool was very useful to continue the collaboration, but it was not exactly an ideal environment to initiate it.
I have been involved in several other very interesting IPAM programs since then; for instance, in 2023 I was the lead organizer in an IPAM-hosted workshop on Machine Assisted Proof https://www.ipam.ucla.edu/programs/workshops/machine-assisted-proofs/, which turned out to be a very well-timed event (occurring a few months after the launch of ChatGPT, for instance), bringing together pure mathematicians, computer scientists, and several people from industry and opening important channels of communication between researchers in such topics as proof formalization, machine learning, large language models, computer algebra solvers, and satisfiability solvers. (I previously posted on my experiences at that workshop at https://mathstodon.xyz/@tao/109858184238417737 .) My experiences at that workshop, as well as the connections made, permitted me to get up to speed on the latest developments in all of these areas, which now encompass a large portion of my current research interests. (3/4)

2025-08-02 02:37:05 UTC

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IPAM (pictured here in a photo I took today), as one of the six NSF-funded math institutes, has been a great success since its founding in 2000. Its specialty is creating three-month programs where participants (both junior and senior) from two or more fields of mathematics, science, or industry interact through workshops, participant-driven seminars, and informal interactions, centered around a theme that had been identified as particularly fertile for bringing together two or more otherwise disparate communities.
One well-known example that I was involved many years ago was the 2004 program https://www.ipam.ucla.edu/programs/long-programs/multiscale-geometry-and-analysis-in-high-dimensions/ on Multiscale geometry and analysis in high dimensions, where the organizers had identified the potential for bringing together pure mathematicians whose work involved geometry at multiple scales with scientists interested in such applied topics as signal processing or the accurate modeling of materials. I participated extensively in this program, and in particular interacted quite a bit with one of the organizers (Emmanuel Candes) as well as Justin Romberg, leading to several foundational papers in the field now known as "compressed sensing", which permits (in certain circumstances) the rapid acquisition of high-resolution images or other information from a relatively small number of measurements. (Perhaps the most well known applications of the compressed sensing algorithms that came out of the work of Emmanuel, myself, Justin, David Donoho, and others was the ability to speed up the time required for a medical-grade MRI scan by up to an order of magnitude.) (2/4)
https://media.mathstodon.xyz/media_attachments/files/114/956/842/222/491/398/original/aabc63ba52fb02b8.jpg

2025-08-02 02:36:27 UTC

The current administration in the US has, through various funding agencies such as the NSF and NIH, has recently suspended virtually all federal grants to my home university, UCLA (including my own personal grant, although that is far from the most serious impact of this decision), on the grounds that UCLA was “failing to promote a research environment free of antisemitism and bias”. One can certainly debate whether these grounds were justified, or whether they merit the extremely draconian damage to the very research environment that this decision is claiming to protect, but if nothing else this unprecedented decision does not appear to have followed the usual standards of due process for actions of this nature; for instance, there appears to have been no good faith effort by the administration to receive a response from UCLA to its allegations before implementing its decision.
The suspension of my personal grant has a non-trivial impact on myself (in particular, my summer salary, which I had already deferred in order to allow the previously released NSF funds to support several of my graduate students over this period, is now in limbo), and now gives me almost no resources to support my graduate students going forward; but this is only a fraction of a percentage of the entire amount being suspended. A far greater concern is the impact on the Institute for Pure and Applied Mathematics (IPAM) https://www.ipam.ucla.edu/, which despite receiving preliminary approval earlier this year for a new five-year round of funding (albeit at significantly reduced levels) from the NSF, now only has enough emergency funding for a few months of further operation at best if the suspension is not lifted. (1/4)

2025-07-25 19:49:59 UTC

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Many of the proposed use cases for AI tools try to place such tools in the "blue team" category, such as creating code, text, images, or mathematical arguments in some semi-automated or automated fashion, that is intended for use for some external application. However, in view of the unreliability and opacity of such tools, it may be better to put them to work on the "red team", critiquing the output of blue team human experts but not directly replacing that output; "blue team" AI use should only be permitted up to the capability of one's "red team" to catch and correct any errors generated. This approach not only plays to current AI strengths, such as breadth of exposure and fast feedback, but also mitigates the risks of deploying unverified AI output in high-stakes settings.
In my own personal experiments with AI, for instance, I have found it to be useful for providing additional feedback on some proposed text, argument, code, or slides that I have generated (including this current text). I might only agree with a fraction of the suggestions generated by the AI tool; but I find that there are still several useful comments made that I do agree with, and incorporate into my own output. This is a significantly less glamorous or intuitive use case for AI than the more commonly promoted "blue team" one of directly automating one's own output, but one that I find adds much more reliable value. (3/3)

2025-07-25 19:49:47 UTC

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The blue team / red team distinction extends beyond cybersecurity to many other disciplines as well. In software engineering, for instance, "blue teaming" might correspond to the generation of new computer code, while "red teaming" would consist of such tasks as quality assurance and testing of such code. In mathematics, "blue teaming" could involve coming up with speculative ideas to solve a math problem, while "red teaming" checks the arguments for formal errors, and also raises heuristic objections to a blue team approach being viable. (See also my discussion about "local" and "global" errors in mathematics at https://terrytao.wordpress.com/advice-on-writing-papers/on-local-and-global-errors-in-mathematical-papers-and-how-to-detect-them/ ).
I like to refer to these two teams in mathematics as the "optimists" and "pessimists"; in my experience, the strongest collaborations arise when there is a roughly equal split between the optimists and pessimists in the collaborations. (Depending on the collaboration, I myself have sometimes played the optimist, sometimes the pessimist, and sometimes a mixture of both.) (2/3)

2025-07-25 19:49:34 UTC

In the field of cybersecurity, a distinction is made between the "blue team" task of building a secure system, and the "red team" task of locating vulnerabilities in such systems. The blue team is more obviously necessary to create the desired product; but the red team is just as essential, given the damage that can result from deploying insecure systems.
The nature of these teams mirror each other; mathematicians would call them "dual". The output of a blue team is only as strong as its weakest link: a security system that consists of a strong component and a weak component (e.g., a house with a securely locked door, but an open window) will be insecure (and in fact worse, because the strong component may convey a false sense of security). Dually, the contributions to a red team can often be additive: a red team report that contains both a serious vulnerability and a more trivial one is more useful than a report that only contains the serious issue, as it is valuable to have the blue team address both vulnerabilities. (But excessive low-quality reports can dilute attention from critical issues.)
Because of this, unreliable contributors may be more useful in the "red team" side of a project than the "blue team" side, though the blue team can still accommodate such contributors provided that the red team is competent enough to catch almost all of the errors that the contributor to the blue team might make. Also, unreliable red team contributions only add value if they _augment_ the output of more reliable members of that team, rather than _replace_ that output, and if their output can be effectively filtered or triaged by more experienced red team members. (1/3)

2025-07-05 15:57:53 UTC

"In the end, the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?" - George Orwell, "Nineteen Eighty-Four". (1/6)

2025-05-25 14:47:08 UTC

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But when cellphone adoption became widespread, it became necessary to figure out how to efficiently encode the signals of multiple cellular devices in the wireless spectrum in such a way that they do not interfere with each other. As it turns out, many of the mathematical techniques and insights generated by exploring these discrete and high-dimensional versions of the sphere packing problem have been of immense value for this problem - not just in the "positive" sense of designing efficient signal encoding methods, but also in the "negative" sense of also giving theoretical upper bounds on such efficiency, thus setting the right benchmarks to evaluate progress, and to avoid wasting resources on attempting encodings that are mathematically impossible.
(As a side note, the successful formalization of the proof of the Kepler conjecture has also inspired and informed many further collaborative formal projects, including my own experiments in this area, even if those projects do not directly involve sphere packing.)
Such contributions to tangible technological advances are subtle and indirect; but without such basic research, many such advances would have taken far longer to be developed, and some may not have been pursued at all. The cuts to funding for such reseearch - which will particularly impact the next generation of researchers - may save a few cents a year in the short term, but greatly reduce the capacity to solve many challenging technological problems of significant real-world impact in the future. (3/3)

2025-05-25 14:43:05 UTC

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Basic mathematical research pursues questions that are often quite far from actual practical application; but they contribute, in a largely invisible way, to the broader research ecosystem that eventually does generate such application. For instance, consider the problem of packing spheres into space as efficiently as possible - a question first proposed by Kepler in 1611. At a practical level, the solution to this problem has been "known" to greengrocers for centuries - one should stack the spheres in a hexagonal close packing. But mathematicians spent decades to work out how to establish the optimality of this packing, culminating in a formally verified proof in 2012.
Mathematicians also explored variants of this sphere packing problem in other geometries than three-dimensional Euclidean space: for instance in higher dimensions, with the famous recent breakthroughs of Viazovska in 8 and 24 dimensions, or in more discrete geometries over finite fields. Such curiosity-driven questions appear to lack immediate application - nobody had a need to pack eight-dimensional oranges together, for instance. (2/3)

2025-05-25 14:42:37 UTC

The NSF has reduced spending on the basic sciences by 50% or more in 2025, with similar cuts proposed for next year as well. For instance, through to May 21 of this year, the funding awarded to the mathematical sciences is currently $32 million, compared to the 10-year average of $113 million. (These are of course large numbers for to an individual, but with the US population of 340 million, this amounts to spending less than 22 cents per American per year on basic mathematical research, compared to the 10-year average of 80 cents per American per year.) https://www.nytimes.com/interactive/2025/05/22/upshot/nsf-grants-trump-cuts.html
I myself have been fortunate to be supported by a small fraction of this 80 cents for almost the entirety of my professional career, allowing me to conduct research in the summer, invite speakers to my department, and to support graduate students. For now, I can continue these activities at a minimal level using by existing (and relatively modest) NSF grant https://www.nsf.gov/awardsearch/showAward?AWD_ID=2347850 , but already I do not have the resources to put into any future long-term projects. For instance, my experiments with using new technologies for mathematical workflows is being conducted purely by myself and contributions from unpaid online volunteers; I am applying for funding from several sources for these projects, but I am expecting the process to be extremely competitive. (1/3)

2025-05-02 06:05:31 UTC

The details are unclear, but it may be that the NSF has been directed to not only halt approval of new grants, but to halt the funding of ongoing ones that have already been approved. If so, this is going to be hugely disruptive (for instance, to the mathematics institutes), and in conflict with the congressional authorization of funds appropriated for the NSF. https://www.nature.com/articles/d41586-025-01396-2

2025-04-10 00:03:26 UTC

Perhaps this does not need to be said, but
\[ \Delta \tau_i = \frac{x_i - m_i}{\epsilon * \varphi * m_i }\]
is, in no shape or form, anything remotely resembling a solution to a mean field equation in a highly non-perturbative and non-linear regime. (See my previous post https://mathstodon.xyz/@tao/113910070146861518 for more discussion of trying to solve the real-world mean field games of non-perturbative economics.)
Incidentally, Matt Parker (a fellow Aussie) has a good breakdown of the equation over at https://www.youtube.com/watch?v=j04IAbWCszg

2025-02-26 04:48:49 UTC

I am happy to announce that the Kakeya set conjecture, one of the most sought after open problems in geometric measure theory, has now been proven (in three dimensions) by Hong Wang and Joshua Zahl! https://arxiv.org/abs/2502.17655 I discuss some ideas of the proof at https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/

2025-02-24 18:51:10 UTC

I wrote a blog post on how a piece of pure mathematics - the development of the landscape function in PDE - played a part in realizing noticeable savings in household energy bills due to improved LED lighting technology: https://terrytao.wordpress.com/2025/02/23/closing-the-green-gap-from-the-mathematics-of-the-landscape-function-to-lower-electricity-costs-for-households/

2025-02-23 17:06:15 UTC

The second part of Grant Sanderson's video interview with myself on the cosmic distance ladder is now out: https://www.youtube.com/watch?v=hFMaT9oRbs4
I wrote a blog post with additional commentary and corrections on both videos at https://terrytao.wordpress.com/2025/02/13/cosmic-distance-ladder-video-with-grant-sanderson-3blue1brown-commentary-and-corrections/

2025-02-21 17:04:22 UTC

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The American Mathematical Society has also started a page to coordinate support for professional mathematics, so far focusing on executive orders impacting the National Science Foundation: https://www.ams.org/government/getinvolved-dc#/

2025-02-20 22:11:29 UTC

The current administration has proposed to cap indirect costs (which covers laboratory maintenance, equipment provision and salaries for support and administrative staff) at the National Institute for Health at 15%, from the current levels of 50%-70%, which will likely make many ongoing research projects impossible to continue in their current form.
My own mathematics department receives very little direct funding from the #nih, but already the mere *possibility* of such a drastic cap has impacted us; we have been unable to make a job offer to an outstanding candidate because the budget uncertainty created by this proposal has forced my university to close off almost all discretionary sources of funding for faculty hiring at all departments, regardless of their level of NIH support.
My university, as well as many others, are circulating a letter of support for NIH funding of biomedical research, for which the US is currently an international leader. The letter is available for anyone to sign at https://docs.google.com/forms/d/1Agzz5mQgfEMpLlBhoE4ZyHdiTvdc9O84zF4vNEHyKTI/viewform?edit_requested=true

2025-02-08 17:49:30 UTC

Many years ago, I met with Grant Sanderson (best known for his Youtube channel 3Blue1Brown) to discuss a possible topic suitable for his channel. I proposed the #CosmicDistanceLadder, and presented a version of my public #astronomy lecture on this subject for him, but for various technical reasons (and the fact that I was basically just reciting my slides), the footage was not suitable for a presentation. But a few months ago, we met up again, and this time we did manage to record enough content that he could apply his signature editing, graphics illustration, and narration to bring the story of the cosmic distance ladder to life, well beyond what I was able to do with my much cruder graphical skills. Part one of the final video can now be found at https://www.youtube.com/watch?v=YdOXS_9_P4U , and a "deleted scene" is available at https://x.com/3blue1brown/status/1888222450841722954 .
I am also working on a book project on the same topic with Tanya Klowden, who also helped Grant with the historical background for the ladder; Grant mentions some of his discussisions with Tanya in a "secret vlog" https://www.youtube.com/watch?v=uq83hprtpGE . (See also our instagram https://www.instagram.com/cosmic_distance_ladder/ for this project.)

2025-01-27 18:55:21 UTC

There are definitely many real-world issues and problems where we could benefit from significantly more quantitative (and mathematical) analysis and thinking than we do currently. But, at the same time, there are certain dimensions of our modern society where we overcorrected for this, and allowed quantitative reasoning to dominate at the expense of other important modes of thought, or to be deployed in a highly imbalanced fashion.
As anyone who has encountered a word problem in their high school math classes knows, the first step in quantitative reasoning is to assign numerically precise metrics as proxies for one's goals, parameters, and variables. Most of our wants and needs are quite qualitative in nature: happiness, comfort, security, companionship, and the like. But these are too fuzzy to be optimized and analyzed by the mathematics of quantitative reasoning. Which, to oversimplify things, leaves us with basically two options: either use more qualitative modes of thinking, such as "gut feelings", emotional responses, or drawing on past experiences of similar situations, accepting any cognitive biases that result from doing so; or to create quantitative proxies for these goals, and then optimize those proxies in a more dispassionate (and hopefully more objective) fashion. (1/5)

2025-01-22 17:05:35 UTC

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Meanwhile, the trend in recent years has been to prefer models of the world that are as simplified or low dimensional as possible - in particular, short enough to describe in a meme or short video to circulate on social media. For instance, regarding the Los Angeles fires, there were several comments on such media suggesting that one simply use the water from the nearby Pacific Ocean to douse the fires, suggesting an oversimplified firefighting model in which complex issues of logistics, wind, salt corrosion, etc. were reduced to the mere proximity of any source of water, regardless of quality or supporting infrastructure. As H.J. Mencken wrote back in 1920, "Explanations exist; they have existed for all time; there is always a well-known solution to every human problem—neat, plausible, and wrong."
Complex dynamics are going to require complex solutions, in which simplified models are still partially used when appropriate, but with as much awareness of their limitations as possible; long-standing assumptions are tested and updated as new information about the current state of the world comes in; and human expertise and perspectives on the many different aspects of the problem are combined in a constructive fashion. These are all achievable practices - indeed, in my own field of mathematics, they are all completely standard parts of our workflow. Despite recent trends, I do hope that they will also adopted more broadly. (4/4)

2025-01-22 17:05:19 UTC

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In many aspects of the modern world (economic, political, environmental, technological, etc.), we are arguably transitioning from a predictable regime of effective dynamics to a more uncertain regime without such effective dynamics. As such, empirical laws that were trustworthy in the past, may become significantly less accurate, and can in fact become dangerous to rely on too blindly. (An example from my own recent personal experience: as the world's climate departs from its historical equilibrium state, the empirical observation that dangerous fires mainly occur in summer rather than winter is no longer safe to rely upon.) The world is becoming more non-perturbative, more nonlinear, and more coupled; developments in distant countries or demographics, or in distant disciplines, have more impact on one's personal life than before, in ways that cannot be easily described by a small number of macro variables. (Example: the COVID-19 pandemic.) (3/4)

2025-01-22 17:05:08 UTC

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In many cases, the effective dynamics essentially decouple many of the degrees of freedom, allowing one to study simpler subsystems almost in isolation, with only a few bulk variables remaining to represent all the exogenous factors. (This is for instance the case in modern economics, in which a complex economy of a large number of independent agents can be decoupled, as an initial approximation, into independent microeconomic systems, interacting with some background macroeconomic variables such as inflation, interest rates, or unemployment.)
However, there is also complementary regime of "no effective dynamics", where the hypotheses on the system state that permit simpler approximations to the dynamics to be effective break down, because some key variables become non-perturbative, or so correlated with other variables that statistical laws such as the law of large numbers are no longer accurate. For instance, if one pulls a spring too far from its equilibrium, then the internal structure of the spring can be impacted, and the restoring force can become nonlinear, or cease to exist entirely (or in more plain language, the spring can break if pulled too hard). (2/4)

2025-01-22 17:04:56 UTC

The behavior of a high-dimensional dynamical system can, very roughly speaking, be divided into two regimes. The first is what one might call the "effective dynamics" regime, in which the complex, high-dimensional dynamics can be well approximated (in the observables that one particularly cares about, at least) by lower-dimensional effective equations or models that emerge from the more foundational laws of motion, and are easier to understand and analyze. A classical example is the laws of thermodynamics, which can effectively govern (some of the) macroscopic behavior of a large number of interacting particles, due to mixing effects that greatly simplify the impact of most of the degrees of freedom. Another example from physics is Hooke's law, that asserts that an elastic object, such as a spring, exerts a linear restoring force to push it in the direction of its equilibrium. Similar linear restoring force phenomena can be seen across the sciences (such as climate science, biology, economics, or even political science), not as fundamental laws of nature, but as empirically observable laws that emerge from more fundamental ones. Such effective laws can provide a valuable amount of long-term stability, predictability, and simplification to the dynamical understanding of many real-world complex systems. (1/4)

2025-01-21 00:15:27 UTC

As of today, the reports of the #PCAST committee I served on are no longer available on the whitehouse.gov web page. As a temporary solution, my former PCAST colleague, Eric Horvitz, has made them available on his page: https://www.erichorvitz.com/PCAST_reports_2020-2025.htm

2025-01-09 15:28:40 UTC

An illustration of survivor bias: yesterday the speaker in our seminar asked of anyone was affected by the ongoing fires. Everyone present assured the speaker that they were not significantly impacted.
Attendance was about half of the normal size.

2024-12-26 21:19:04 UTC

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With hindsight, some of my past rejections have become amusing. With a coauthor, I once almost solved a conjecture, establishing the result with an "epsilon loss" in a key parameter. We submitted to a highly reputable journal, but it was rejected on the grounds that it did not resolve the full conjecture. So we submitted elsewhere, and the paper was accepted.
The following year, we managed to finally resolve the conjecture, and decided to try submitting to the highly reputable journal again. This time, the paper was rejected for only being an epsilon improvement over the previous literature!
(This paper was also submitted elsewhere, and accepted; and I have subsequently published in that highly selective journal since. Being an editor myself, and having had to decline some decent submissions for a variety of reasons, I find it best not to take these sorts of rejections personally, and move on to other journals, of course after revising the paper to address any issues brought up by the rejection.) (2/2)

2024-12-26 21:14:41 UTC

One of my papers got declined today by the journal I submitted it to, with a polite letter saying that while they found the paper interesting, it was not a good fit for the journal. In truth, I largely agreed with their conclusions, and the paper is now submitted to a different (and hopefully more appropriate) journal.
Rejection is actually a relatively common occurrence for me, happening once or twice a year on average. I occasionally mention this fact to my students and colleagues, who are sometimes surprised that my rejection rate is far from zero. I have belatedly realized our profession is far more willing to announce successful accomplishments (such as having a paper accepted, or a result proved) than unsuccessful ones (such as a paper rejected, or a proof attempt not working), except when the failures are somehow controversial. Because of this, a perception can be created that all of one's peers are achieving either success or controversy, with one's own personal career being the only known examples of "mundane" failure. I speculate that this may be a contributor to the "impostor syndrome" that is prevalent in this field (though, again, not widely reported, due to the aforementioned bias). So I decided to report this (rather routine) rejection as a token gesture towards more accurate disclosure. (1/2)

2024-12-13 01:04:42 UTC

I have now had the following experience with at least three graduate students in the last ten years: in our weekly meeting, they mention that they needed to locate a key article or book for their research project, but despite searching all over the internet, they are unable to find it. I then ask if they have checked our local Science and Engineering Library, which is literally in the same building as the Math Department. Ten years ago, the response would be embarrassment that this option did not occur to them; but now, the response is surprise that a library containing physical copies of math journals and textbooks even existed.
Perhaps physical libraries are a vestigial remnant of a pre-digital era, but I do have fond memories as a graduate student of randomly browsing books next to the ones I had been looking for, or articles after or preceding the one I was initially locating. The current technological paradigm of being able to near-instantly locate nearly any article one desires (assuming one's university has a subscription to the relevant journals) is undeniably convenient, but has reduced the opportunity for serendipitious discovery. (On the other hand, there are other ways now to make such discoveries, for instance through browsing math question-and-answer sites or math-oriented social media.)

2024-11-28 06:25:04 UTC

Vjeko Kovac and I just uploaded a preprint to the arXiv entitled "On several irrationality problems for Ahmes series" https://arxiv.org/abs/2406.17593 , making progress on several open problems of Erdos and of Stolarsky regarding when sums of reciprocals of natural numbers are rational or not. Perhaps the most striking result is that we can find an increasing sequence 𝑎ₙ of natural numbers with the property that ∑ₙ1/(𝑎ₙ+𝑡) converges to a rational for all rational 𝑡 for which the denominators are non-zero, answering a question of Stolarsky in the negative. More discussion at https://terrytao.wordpress.com/2024/11/27/on-several-irrationality-problems-for-ahmes-series/

2024-11-12 23:43:02 UTC

Today, one of the graduate students I was mentoring asked me what target audience to have in mind when writing their first research paper, which combined ideas from two different mathematical fields - for instance, how much detail to spell out for a lemma which would be basic and well known to experts in one of the fields, but not the other. After discussing some hypothetical readers with strengths in one field but not the other, the framework that seemed to generate a "lightbulb" moment in the student's head was the suggestion to take as the model reader a version of the student's past self from about twelve months ago when first learning the subject, and to try to write to specifically address any confusions that the student had that could have had been resolved if only there was a text that had explained these basic points clearly. It was clear that the student had several such points come to mind immediately, and almost instantly had a much clearer plan for how to write the paper.

2024-11-11 21:06:17 UTC

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So, somewhat in desperation, I lay on the floor, closed my eyes and tried to imagine _being_ the plane being rotated. I imagined the base point being moved in the x-direction, and rolled accordingly; then in the y-direction. With this, I was able to figure out that if I could find a rotation matrix depending on both time and space whose derivatives behaved in a certain way, then I could find my good change of variables and solve the problem. (In retrospect, I was trying to flatten a connection by choosing an appropriate gauge.)
Miraculously, the algberaic form of the equation was such that a natural choice of gauge that did nearly flatten everything was evident. (Again, in retrospect: the connection attached to this problem happened to have a small curvature.) After some frenzied calculations with pen and paper, I was able to get a plan to solve my problem, and then write the paper. (Which actually ended up earning me a Bocher prize.)
I was visiting my aunt in Australia at the time, and she managed to walk in on me while I was rolling around on the floor with my eyes closed. In the grand scheme of things, not the most embarrassing situation to be in, but I am not sure how satisfied she was by the explanation that I was "thinking about math". In a strange way, though, my profession can occasionally benefit from our reputation for eccentricity; she did not ask any further questions. (3/3)

2024-11-11 19:08:10 UTC

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I had the idea to try to perform a change of variables to rotate these tangent planes to line up and become parallel, in the hope of simplifying the equation. At the time, I did not have the required background in differential geometry to recognize that I was actually trying to perform a gauge transformation; but in fact this project finally set me on the path to learn differential geometry properly (I had not fared well with it as an undergraduate, and was slightly intimidated by it as a graduate student), which has definitely helped my mathematical career since.
The problem was that the coordinate changes I wanted to implement were not unique. If I wanted to rotate a tangent plane on the sphere to be tangent to the north pole, say, I could easily visualize rotating the sphere so that the point of tangency was on the pole, but then I could also twist the plane around that pole arbitrarily. So, for each point in space and time, there was an additional twist I could perform, and this transformed the equation that I was trying to solve in a way that could either make the equation more like the free wave equation (and thus easier to solve), or less like the wave equation.
If I focused only on how these twists evolved in a single spatial direction - such as the x-axis, it was like trying to smooth out a chain of disks to all be parallel, and I could see how to perform a change of variables to do that. (Now that I know differential geometry, I know that I was integrating a connection along this direction.) But then this operation made things curve in a funny way in the y-axis. I couldn't visualize all this in my head.
(2/3)

2024-11-11 19:07:54 UTC

An anecdote that I shared about rolling around on the floor back in 2000 to solve a math problem, both in my #Masterclass at https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking/chapters/transforming-problems , and on #MathOverflow at https://mathoverflow.net/a/38882/766 , as well as the #NewYorkTimes https://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html , has for some reason recently gone viral on various social media. Just for the record, I wanted to add some mathematical background behind the story, which eventually led to my paper https://arxiv.org/abs/math/0010068 . At the time, I was trying to construct solutions to an equation known as the wave maps equation on the sphere: the solution was like a solution to the wave equation, except being forced to take values in a sphere rather than in a vector space.
I was trying to solve the equation iteratively, breaking up the solution to a low frequency base solution and a high frequency correction. As a first approximation, the low frequency base could also be assumed to stay on the sphere and solve the wave maps equation, so the main problem was to work out what the high frequency correction was doing.
Because the high frequency correction also had to keep the solution on the sphere, one could assume as a first approximation that the high frequency correction was tangent to the low frequency base. So, at any given point in space and time, the low frequency base solution was located on some point on the sphere, and the high frequency correction basically lived on the tangent plane to the sphere at that point. But because the base solution evolved (slowly) in space and time, this tangent space kept rotating around the sphere.
(1/3)

2024-11-08 01:26:53 UTC

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For most of my professional career, I have tried to keep my personal life separate from my professional one. While I did occasionally post on non-mathematical topics, such as some notorious posts on politics, my social media has largely been restricted to purely mathematical ones (or reporting activities that I have been involved in, such as my work at PCAST). I will continue doing so, but I have decided to also share here more of my thought processes and beliefs, as I have realized how helpful this exercise is for myself, and sometimes also for those around me. When times are good and normal, and people have most of their needs met, this would be unnecessary, and arguably rather pretentious; but these are not normal times.
So, expect some more personal thoughts and reflections on this Mastodon account going forward. But in the meantime, I can highlight the most recent post of my coauthor, Tanya Klowden, on our "Climbing the cosmic distance ladder" instagram at https://www.instagram.com/p/DCFoHqMomPx/?img_index=1 , on the perspective one can gain from being in darkness, whether in astronomy or here on Earth. (3/3)

2024-11-08 01:26:43 UTC

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But, in what was close to the first time ever for me, I got knocked down a few levels of the pyramid. I have had the honor of serving for the last few years on the President's Council of Advisors on Science and Technology (PCAST), and in particular co-chaired our report on AI which we sent to President Biden. I continue to serve on PCAST for the next two months or so, but it is clear my term will not continue after that, and with that (and with the broader impacts of the election), some sense of self-worth and belonging was lost. As such, I felt a loss of purpose, to the extent that I had to seriously debate with myself whether to show up to teach my scheduled class yesterday. (Which I did.)
However, as one would predict from Maslow's model, I then sought out the connections to restore the levels of the pyramid I had been taking for granted. I reconnected with distant friends and acquaintances. In these conversations, we discussed what gives us purpose, and what we envision for the future, and what role our interests (for instance, math and AI in my case) could play in that. And through these reflections and conversations, I felt restored, and in fact strangely even more energized than I had been in the past. (2/3)

2024-11-08 01:26:22 UTC

Maslow's hierarchy of needs https://en.wikipedia.org/wiki/Maslow's_hierarchy_of_needs is a popular framework for conceiving human needs. Some of its finer details have received both popular and academic criticism, but with the understanding that it is a hugely oversimplified approximation, it can still be a good initial starting point.
In this framework, needs are arranged in a "pyramid", and as a general rule one has to have one's needs lower in the pyramid met before one has the ability to pursue needs higher in the pyramid. One needs food and water for instance above all else, but once one has those, one can seek out safety, then friendships, then senses of self-worth and purpose, and eventually the ability to pursue creative activities.
I have been fortunate for my entire life to have largely avoided any serious trauma, and have had almost all of my needs met on this pyramid throughout my childhood and later career. As such, I have been able to spend a significant fraction of my time on what might be called "self-actualization", which for me largely involves working on mathematics, and lately the intersection of mathematics with modern computer tools such as AI. (1/3)

2024-10-29 15:05:28 UTC

I remember when this story of two high school students discovering one or more new proofs of the Pythagorean theorem came out some years ago, but (frustratingly) without any substantive details on what these proofs were and why they were new. Now there is a published paper: https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2370240
The question of defining precisely when two proofs are "the same" (or whether a proof is "new" compared to existing proofs) is actually a very subtle and interesting one (see some discussion at https://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/ or https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs). Here the authors take a largely syntactic approach: they are considering a proof to be "trigonometric" if it avoids the use of circles (or coordinates), but uses angles in an essential way. With these restrictions, they do find at least five proofs that do not obviously resemble any of the standard known proofs, for instance one that involves summing a geometric series.
In ptinciple there may be "semantic" ways to distinguish these proofs from other proofs, in that there may be exotic variants of Euclidean geometry in which one of the proofs in this paper is valid but other proofs are not, or vice versa. But even without having such a semantic way to make this distinction, this was a fun read and a reminder that even the most ancient and well-established foundational results in mathematics can sometimes be revisited from a fresh perspective.

2024-09-24 19:54:25 UTC

The new #NSF program "
Artificial Intelligence, Formal Methods, and Mathematical Reasoning (AIMing)" is now in its second call for proposals, with a deadline of Feb 5 2025. https://new.nsf.gov/funding/opportunities/aiming-artificial-intelligence-formal-methods-mathematical . I have heard from some of the program directors that they are particularly interested in proposals from the mathematical community on ways to incorporate formal methods into mathematical research.

2024-09-16 04:26:28 UTC

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I am belatedly realizing that in my attempts to describe my evaluation of the capability of an AI tool, I inadvertently gave the incorrect (and potentially harmful) impression that human graudate students could be reductively classified according to a static, one dimensional level of “competence”. This was not my intent at all; and I would therefore like to make the following clarifying remarks.
Firstly, the ability to contribute to an existing research project is only one aspect of graduate study, and a relatively minor one at that. A student who is not especially effective in this regard, but excels in other dimensions such as creativity, independence, exposition, professionalism, work ethic, organization, or social skills can in fact end up being a far more successful and impactful mathematician than one who is proficient at assigned technical tasks but has weaknesses in other areas.
Secondly, and perhaps more importantly, human students learn and grow during their studies, and areas in which they initially struggle with can become ones in which they are quite proficient at after a few years; and personally I find being able to assist students in such transitions to be one of the mmost rewarding aspects of my profession. In contrast, while modern AI tools have some ability to incorporate feedback into their responses, each individual model does not truly have the capability for long term growth, and so can be sensibly evaluated using static metrics of performance. However, I believe such a fixed mindset is not an appropriate framework for judging human students, and I apologize for conveying such an impression.

2024-09-14 15:40:41 UTC

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My hope is that a diverse ecosystem of AI tools emerges to handle a variety of useful research tasks, including the ones you listed. The extremely large, general-purpose proprietary LLMs are the ones attracting the most attention now, but at some point I believe the marginal cost in data and compute to improve these models further (or to fine tune them for specific applications) will become prohibitively expensive, and that more lightweight and open source models (and data sets) developed by the research community for tailored needs will also begin playing an important role, perhaps with the general-purpose models serving as a user-friendly interface to coordinate these narrower tools.

2024-09-13 22:03:25 UTC

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In https://chatgpt.com/share/94152e76-7511-4943-9d99-1118267f4b2b I gave the new model a challenging complex analysis problem (which I had previously asked GPT4 to assist in writing up a proof of in https://chatgpt.com/share/63c5774a-d58a-47c2-9149-362b05e268b4 ). Here the results were better than previous models, but still slightly disappointing: the new model could work its way to a correct (and well-written) solution *if* provided a lot of hints and prodding, but did not generate the key conceptual ideas on its own, and did make some non-trivial mistakes. The experience seemed roughly on par with trying to advise a mediocre, but not completely incompetent, graduate student. However, this was an improvement over previous models, whose capability was closer to an actually incompetent graduate student. It may only take one or two further iterations of improved capability (and integration with other tools, such as computer algebra packages and proof assistants) until the level of "competent graduate student" is reached, at which point I could see this tool being of significant use in research level tasks. (2/3)

2024-09-13 22:03:15 UTC

I have played a little bit with OpenAI's new iteration of #GPT, GPT-o1, which performs an initial reasoning step before running the LLM. It is certainly a more capable tool than previous iterations, though still struggling with the most advanced research mathematical tasks.
Here are some concrete experiments (with a prototype version of the model that I was granted access to). In https://chatgpt.com/share/2ecd7b73-3607-46b3-b855-b29003333b87 I repeated an experiment from https://mathstodon.xyz/@tao/109948249160170335 in which I asked GPT to answer a vaguely worded mathematical query which could be solved by identifying a suitable theorem (Cramer's theorem) from the literature. Previously, GPT was able to mention some relevant concepts but the details were hallucinated nonsense. This time around, Cramer's theorem was identified and a perfectly satisfactory answer was given. (1/3)

2024-09-13 21:43:58 UTC

The Mathematics in Open Access (MathOA) foundation has announced a fund to support the creation of new Diamond Open Access journals (or conversion of existing journals to a Diamond OA model), covering costs of up to €10k for the transition. https://www.mathoa.org/

2024-09-01 01:22:40 UTC

The two most time-saving uses of #AI I am finding in my work are not in the obvious use case of asking AI about my math problems, but rather either in creating computer code for various scientific computations, or in mundane data formatting tasks. A case in point is that of putting together a bibliography for a paper (with or without BibTeX). Often, I need to grab a dozen or so references from other sources (e.g., PDF documents) and format them to a specific bibliography style. This is not difficult, but is certainly tedious. With Github Copilot, it has become quite trivial; I paste in the raw reference in some random format as a comment, and Copilot usually understands without any additional prompting that I want to convert it to precisely the same format that all other references are in, usually correcting for any typos and other misprints in the original comment This is far faster than any previous method of collecting references that I have used. (And I have started taking joint papers in Overleaf offline to edit, solely to take advantage of these sort of autocompletions.)
https://media.mathstodon.xyz/media_attachments/files/113/059/660/092/233/764/original/03b614b62be5a168.png