Terence Tao

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

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Last Notes

2025-03-26 23:35:00

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As with previous advances in mathematics automation, students will still need to know how to perform these operations manually, in order to correctly interpret the outputs, to craft well-designed and useful prompts (and follow-up queries), and to able to function when the tools are not available. This is a non-trivial educational challenge, and will require some thoughtful pedagogical design choices when incorporating these tools into the classroom. But the payoff is significant: given that such tools can free up the significant fraction of the research time of a mathematician that is currently devoted to such routine calculations, a student trained in these tools, once they have matured, could find the process of mathematical research considerably more efficient and pleasant than it currently is today. (3/3)

2025-03-26 23:34:24

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Today, it is increasingly commonplace for human mathematicians to also outsource symbolic tasks in such fields as linear algebra, differential equations, or group theory to modern computer algebra systems. We still place great emphasis in our math classes on getting students to perform these tasks manually, in order to build a robust mathematical intuition in these areas (and to allow them to still be able to solve problems when such systems are unavailable or unsuitable); but once they have enough expertise, they can profitably take advantage of these sophisticated tools, as they can use that expertise to perform a number of "sanity checks" to inspect and debug the output of such tools.
With the advances in large language models and formal proof assistants, it will soon become possible to also automate other tedious mathematical tasks, such as checking all the cases of a routine but combinatorially complex argument, searching for the best "standard" construction or counterexample for a given inequality, or performing a thorough literature review for a given problem.
To be usable in research applications, though, enough formal verification will need to be in place that one does not have to perform extensive proofreading and testing of the automated output. (2/3)

2025-03-26 23:34:12

In the first millennium CE, mathematicians performed the then-complex calculations needed to compute the date of Easter. Of course, with our modern digital calendars, this task is now performed automatically by computers; and the older calendrical algorithms are now mostly of historical interest only.
In the Age of Sail, mathematicians were tasked to perform the intricate spherical trigonometry calculations needed to create accurate navigational tables. Again, with modern technology such as GPS, such tasks have again been fully automated, although spherical trigonometry classes are still offered at naval academies.
During the Second World War, mathematicians, human computers, and early mechanical computers were enlisted to solve a variety of problems for military applications such as ballistics, cryptanalysis, and operations research. With the advent of scientific computing, the computational aspect of these tasks has been almost completely delegated to modern computers, although human mathematicians and programmers are still required to direct these machines. (1/3)

2025-03-24 04:41:16

A new #CosmicDistanceLadder post on why lunar and solar eclipses tend to come in pairs (for instance, the solar eclipse next week is paired with the lunar eclipse from last week). https://www.instagram.com/p/DHkS3EcA40L

2025-03-21 23:04:07

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Even in the realm of pure mathematics, where the notion of inerrancy is the most developed (and nowadays even able to be automatically verified by formal proof assistants), it is still true that mathematics *as practiced by mathematicians* still contains important components of accuracy-based thinking rather than inerrancy-based thinking. In line with the "three stages of mathematical development" I discussed in https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ , while it is vitally important to be able to use the inerrant mechanism of formal (or semi-formal) mathematical proof to verify one's arguments, in order to actually *locate* these proofs it is far faster and more efficient to adopt looser, but still reasonably accurate, heuristic models in order to get an early sense of the feasibility or infeasibility of different ways to attack the problem, and to glimpse what the potential outcome of a complex calculation might be. Nowadays one even can employ completely non-rigorous techniques such as numerical simulations or machine learning to map out the landscape of intermediate results (which may be true or false, and whose proof or disproof may be routine, difficult, or hopeless) in order to select the most promising strategy (or to select a more tractable problem to study first). (3/3)

2025-03-21 23:03:52

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In such contexts, errors in which one's model deviates from reality, while undesirable in the long term, become quite valuable in the short term, as important clues to locate ways to improve one's models. Another consequence of keeping one's models distinct from reality is the recognition that it can be advantageous at times to select a simplified model which may be less accurate at a fundamental level than a more sophisticated one, but which is so much more convenient for calculations and analysis that its *effective* accuracy in applications may end up being far superior to the complicated alternative. Knowing the appropriate level of sophistication of a model to select for a given application is part of the skill set of being an effective scientist.
Unfortunately, this adaptive model selection process does have the side effect of making each stage of one's scientific education (through primary, secondary, tertiary, and postdoctoral training) come with the admission (either implicit or explicit) that the simplified models one was taught in the previous stage contained certain "white lies" which one now needs to correct for in order to do more advanced scientific analysis. From the perspective of an observer who has adopted an inerrancy-based mindset, this looks extremely suspicious, and possibly even incriminating; but it is in fact a natural and rational consequence of an accuracy-based approach to the subject. (2/3)

2025-03-21 23:03:27

One subtle difference between the scientific approach to truth and non-scientific approaches is that the former places (or aims to place) a high value on *accuracy*, while the latter often places a high value instead on *inerrancy*. The two objectives, though superficially similar, are actually not the same, and in fact opposed to each other in some fundamental ways.
By valuing inerrancy, one seeks to claim authority through never making, or at least never admitting, an error or mistake. If there is a conflict between one's authoritative statements and external observations or events, there is thus the incentive to find a rationalization that retains the former at the expense of the latter. To quote from "The Simpsons": "Am I so out of touch? No, it is the children who are wrong".
Inerrancy is a binary concept - one is either inerrant, or one is not (and almost always, the latter is actually the case). In contrast, accuracy is a continuous concept - perfect accuracy is generally impossible outside of purely mathematical situations, and so the realistic objective is instead to achieve gradually higher, but still imperfect, levels of accuracy through better use of observations, models, and theory. This attitude is perhaps exemplified by George Box's well known dictum: "All models are wrong. Some are useful."
(1/4)

2025-03-21 23:02:56

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@nprofile…0ae2 Lean's Mathlib is perhaps in this spirit: https://arxiv.org/abs/1910.09336 . However, it is not aimed towards unrestricted submission; there is a coherent vision for what types of results should be in the library, what format the statements and proofs should be in, and how they should integrate with the rest of the library.

2025-03-17 18:37:21

“This invention will produce forgetfulness in the minds of those who learn to use it, because they will not practice their memory.” - Socrates, circa 5th century BCE, lamenting the then-recent invention of the book.
Plus ça change...

2025-03-17 01:49:28

A new #CosmicDistanceLadder post, on how the recent lunar eclipse from the vantage point of the Earth becomes a solar eclipse from the vantage point of the Moon: https://www.instagram.com/p/DHR1tuWonDR/

2025-03-16 19:12:28

This morning I gave an answer to a MathOverflow question using traditional pen-and-paper analysis: https://mathoverflow.net/a/489533/766 . The answer was not in closed form, so I wanted to simulate it approximately. At this point I asked o3-mini-high for some code for this. Interestingly, it first declared that the quantity I was trying to compute was infinite (it wasn't), but nevertheless provided numerical code which did give a rough approximation to the quantity I wanted (to one decimal place): https://chatgpt.com/share/67d71204-3510-800e-8bca-11bfbf53fc3d . At that point I figured out how to get a more precise answer (using the theory of Markov chains) and asked o3-mini-high again for more code for this. Interestingly, it was able to correct a basic error in the prompt (I had written max instead of min when writing a truncation), and gave me perfectly good code, which I was able to adapt to then give a more numerically precise answer to the MO question.
So all in all a pretty good assist from o3; it made a mistake that I corrected, but I also made a mistake that it corrected, and code that would have taken perhaps an hour of my time on my own was generated, tested, modified, and reported in maybe ten minutes.

2025-03-15 22:08:03

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Finally, mathematical negative results can provide theoretical upper bounds for various desirable outcomes, which can then provide targets for practical designs. Energy conservation prevents energy efficiency from exceeding 100%, but instead sets the target goal of getting as close to 100% as possible. Information theory sets theoretical limits on the communication capacity or error-correcting capacity of a channel, but again this sets the very useful target goal of developing compression and encoding protocols that come close to such limits. And so forth. It is only through the combination of both positive and negative results that we can fully understand and exploit the space of possibilities. (7/7)

2025-03-15 22:07:48

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Another role that mathematical negative results can play is to focus attention on the minimal component of a proposed technology that is in conflict with such results. For instance, regardless of whether the law of conservation of energy holds or not, it is a mathematical theorem (assuming additivity of energy) that if a machine fails to conserve energy, then at least one of its components must do so as well. And so, if forced to evaluate some complex "Rube Goldberg" design of a perpetual motion machine, it becomes natural to insist that the inventor identify at least one atomic component of this machine that violates this law. (There is a similar technique when reading papers claiming to prove a difficult result: see https://terrytao.wordpress.com/advice-on-writing-papers/on-the-strength-of-theorems/ ) (6/7)

2025-03-15 22:07:37

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But also, by mathematically identifying the precise hypotheses and axioms needed to establish a negative result, one can gain valuable clues as to how to find a substitute positive result. For instance, by identifying the role of conservation of money axiom in preventing the global profitability a pyramid scheme, one can see how it is possible, in economies where wealth can be created over time via long-term investment, to create social insurance schemes (such as Social Security in the US) that can generate broad social benefits despite having some superficial resemblance to a pyramid scheme. (This is not to say that such schemes are guaranteed to be sustainable over the long term, especially if mismanaged or designed badly; but it is not mathematically impossible for this to be the case.) Or: mathematical compression of all data is impossible; however, this reveals the possibility that compression of specific types of data, such as text, is still feasible, but must somehow utilize the structural properties of that data. (5/7)

2025-03-15 22:05:47

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2025-03-15 22:05:21

But also, by mathematically identifying the precise hypotheses and axioms needed to establish a negative result, one can gain valuable clues as to how to find a substitute positive result. For instance, by identifying the role of conservation of money axiom in preventing the global profitability a pyramid scheme, one can see how it is possible, in economies where wealth can be created over time via long-term investment, to create social insurance schemes (such as Social Security in the US) that can generate broad social benefits despite having some superficial resemblance to a pyramid scheme. (This is not to say that such schemes are guaranteed to be sustainable over the long term, especially if mismanaged or designed badly; but it is not *mathematically* impossible for this to be the case.) Or: mathematical compression of all data is impossible; however, this reveals the possibility that compression of specific types of data, such as text, is still feasible, but must somehow utilize the structural properties of that data. (5/7)

2025-03-15 22:04:55

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The value of a positive technological breakthrough is obvious. Negative results however are also valuable and important, though in significantly less intuitive fashions. Firstly, they allow for more efficient allocation of resources: with the knowledge of negative results, scarce R&D funding can be allocated away from, say, the design of perpetual motion machines and towards more feasible goals, such as increasing the fuel efficiency of engines (though given the residual uncertainties and the potential payoffs, it is still rational to allow for a very small and lightly funded amount of research activity into proper mathematical and scientific analysis of possible loopholes in the conservation of energy argument). (4/7)

2025-03-15 22:04:41

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Mathematical arguments come the closest to providing a completely bulletproof negative result, although even mathematical reasoning must rely on some axioms. For instance, it is mathematically impossible to design a pyramid scheme in which everyone involved profits monetarily at the end of the scheme, if one assumes the axiom that the total amount of
money in the system is preserved. The "strategy stealing argument" similarly shows that, with some basic axioms concerning the free market, it is not possible to create a publically describable strategy to play the stock market with guaranteed positive returns, based only on publicly available information. Information theory shows that it is not possible to design a compression algorithm that can compress every single type of data input into a smaller size. And so forth. (3/7)

2025-03-15 22:04:21

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There are a small number of physical laws that are considered so fundamental and universally applicable that they provide much stronger negative results. Famously, the law of conservation of energy prohibits the existence of a perpetual motion machine that produces more energy output than energy input over extended periods of time, while Einstein's theory of special relativity (together with basic axioms of causality) prohibits faster than light travel for any object capable of carrying information. Even in such cases, though, some loopholes remain. For instance, there are mathematically viable (though practically impossible) "warp drives" that exploit the spacetime-distorting effects of extremely massive objects to achieve travel that is faster than light at global scales, despite not exceeding the light speed limit at any local point in the trajectory. (2/7)

2025-03-15 22:04:04

Science and technology can produce positive results that can be extremely convincing: a successful demonstration of heavier-than-air flight, for instance, is hard to refute (assuming of course that no fraudulent manipulation of the presentation was made, such as editing of the video recordings). Negative results, by their nature, are much more provisional in nature. Suppose for instance that a prototype heavier-than-air machine fails to take flight. This can be useful experimental data for future designs, but it does not provide strong evidence that such flight is physically impossible.
Empirical laws of nature can provide stronger negative evidence, but they are still not ironclad, as many such laws have limitations on their range of applicability. For instance, Archimedes law of bouyancy is an empirically valid law which does seem to provide a strong obstruction to heavier-than-air flight; but of course such powered flight is still possible, as one needs to recognize that Archimedes' law only applies to fluids in static equilibrium, as opposed to dynamic fluid flows where additional effects such as aerodynamic lift are in play. (1/7)

2025-03-12 01:32:49

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@nprofile…7afh Most questions on MathOverflow cannot be solved through code, as they often involve infinite spaces, or unbounded numerical quantities. In the long term, when formal proof assistants improve and become more integrated with AI, it may become possible for AI to generate answers to MathOverflow type questions written in such languages; but this is still several years away from being feasible at current trends.

2025-03-11 23:38:59

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@nprofile…7afh According to the original post, Claude and Gemini performed strictly worse than o1 for this experiment.

2025-03-11 17:38:20

An interesting experiment on #MathOverflow, where a user gave 15 different MO problems for o-1 to answer, with the aim of verifying and then rewriting the answer into a presentable form if the AI generated answer was correct. The outcome was: one question answered correctly, verified, and rewritten; one question given a useful lead, which led the experimenter to find a more direct answer; one possibly correct answer that the experimenter was not able to verify; and the remainder described as "a ton of time consuming chaos", in which the experimenter spent much time trying to verify a hallucinated response before giving up. https://meta.mathoverflow.net/questions/6114/capabilities-and-limits-of-ai-on-mathoverflow
I found the discussion for possible AI disclosure policies for MO in the post to also be interesting.

2025-03-11 17:04:25

The National Academies is now taking nominations for the 2025 Eric and Wendy Schmidt Awards for Excellence in Science Communications, which honor exceptional science communicators, journalists, and research scientists who have developed creative, original work to communicate issues and advances in science, engineering, or medicine for the general public. Applications close Mar 31. https://www.nationalacademies.org/news/2025/02/submission-period-now-open-for-the-2025-eric-and-wendy-schmidt-awards-for-excellence-in-science-communications-640-000-to-be-awarded

2025-03-10 19:48:28

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@nprofile…g23r Actually, I think early-era Wikipedia offers an interesting analogy with current-era LLMs. There are parallels: back then, crowdsourced encyclopedias was the upstart paradigm challenging expertly curated encyclopedias, similar to how LLM-generated content challenges human-generated content today; and there were even similar concerns about students blindly using these new resources for their coursework. The skepticism concerns about reliability, and the visible failings, of current-era LLMs are echoed in similar concerns about crowdsourced content two decades ago. Nevertheless, with the passage of time, Wikipedia has thrived to be a usable and largely trustable resource (at least outside of highly politicized topics), and academia has adapted to incorporating Wikipedia into its workflows.
That said, the process of addressing quality concerns was far from automatic: most other wikis have proven far less successful and reliable than Wikipedia, even if they use identical software. In the case of Wikipedia, strongly enforced policies and a lively community of stakeholders played a major role in allowing it to improve over time, though not without its own internal controversies and drama. Perhaps one also needs similar "soft" infrastructure around the purely technical platforms of modern AI tools in order to guide it to a state similar to where current-era Wikipedia stands today.

2025-03-10 17:46:18

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@nprofile…g23r Actually, I think the problem of untrustworthy sources on the internet (particularly on social media) predates generative AI, though certainly AI bots and "deepfake" images exacerbate the issue. With or without generative AI, it has become increasingly important to know how to independently verify information.
In the specific realm of pure mathematics, though, there is a potential solution to this problem by directing generative AI output to pass through a formal proof assistant to obtain a guarantee of correctness. At present, the experiments in this direction are only capable of resolving low-level undergraduate problems (such as computing a definite integral) by this approach, and it is still not clear whether the high-level conceptual component of a LLM-generated answer to a mathematical question can be captured by such formal languages; but I would imagine that requiring the LLM to formally verify at least some of the finer details of their output would significantly increase their broader reliability. (A similar phenomenon has already been observed in LLM-based solutions to Math Olympiad type challenges, in which models which do not directly attempt to answer the question, but instead create code in a more reliable language such as Python to solve the problem, significantly outperform pure LLM models.)

2025-03-10 17:04:10

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@nprofile…g23r Currently, I rate the best-case outcome of LLM queries to be superior to web search due to the speed and appropriateness of the response; it requires less mental effort to create a prompt than to create a search query, scan through a half-dozen hits to determine which ones are the most promising, and then mentally translate the answer in the best hit into the notation and context one actually wants for one's application. However, in the worst case, these queries require more mental bandwidth than a traditional search. So right now the two options are roughly comparable in utility. But I expect the quality of LLM queries to continue to improve, whereas for web search it seems that the main potential sources for further possible improvement also come from AI integration.

2025-03-10 16:45:39

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My general sense is that for research-level mathematical tasks at least, current models are somewhere between "genuinely useful with only broad guidance from user" and "only useful after substantial detailed user guidance", with the most powerful models having a greater proportion of answers in the former category. They seem to work particularly well for questions that are so standard that their answers can basically be found in existing sources such as Wikipedia or StackOverflow; but as one moves into increasingly obscure types of questions, the success rate tapers off, and the more user guidance (or higher compute resources) one needs to get the LLM output to a usable form. (2/2)

2025-03-10 16:40:39

Today I was trying to reconstruct the proof of a standard (to experts) result in graph theory, namely that the "triangle removal lemma" of Ruzsa and Szemeredi implied that a graph on n vertices composed of n induced matchines contained only o(n^2) edges. My main three options here were: try to work it out by pen and paper; do a traditional web search; or ask a large language model. In this case, I chose the third option, and in a few seconds I received a perfectly good answer that explained the implication correctly: https://chatgpt.com/share/67cf13cf-53dc-800e-a382-e4ece8341a6d
This satisfied my immediate request - and indicated a good use case for LLMs for quickly providing details of some arguments standard in one's field, that one can then verify for correctness - but then I got curious and asked the model to also explain another standard consequence of the triangle removal lemma, namely the (6,3) theorem of Ruzsa and Szemeredi concerning the size of 3-uniform hypergraph with certain forbidden configurations. Here the results were decidely more mixed. The initial answer had the right general strategy - use the hypergraph to encode a graph - but lacked all the key details. When I pressed it further, it did not mention the two most essential ideas - the use of the (6,3) condition to limit unwanted triangles in the encoded graph, or the initial reduction to linear hypergraphs - but with additional prompting, it was able to reconstruct these components and eventually provide an essentially correct proof of the derivation. But I had to guide it with rather explicit hints, which I could only do because I had looked up the proof first via a traditional web search. (1/2)

2025-03-05 06:20:20

A new post on my #CosmicDistanceLadder Instagram with Tanya Klowden on the parallels (but also differences) between ancient Greek and ancient Indian astronomy. https://www.instagram.com/p/DGzJs02AbBA

2025-03-01 22:44:19

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@nprofile…307x My general impression is that any topic that already has a reasonable Wikipedia page can be accurately summarized by LLM tools, in particular being able to succinctly answer precise basic questions without requiring detailed parsing of longer partly relevant texts, which is particularly important in a lecture environment where one still wants to keep most of one's attention on the talk in real time.

2025-02-27 23:00:08

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@nprofile…4f8c I have not done systematic testing, but I think at this point (roughly speaking) any modern model would give satisfactory results for any topic that has a non-trivial Wikipedia page.

2025-02-27 19:41:03

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@nprofile…937m Should be fixed now, sorry.

2025-02-27 19:29:16

Yesterday I attended a lecture on a topic outside of my own area of expertise - the Langlands program and the cohomology of Shimura varieties - and found that it was actually quite helpful to discreetly ask some questions from a large language model during the lecture to help keep up. (A log of my questions can be found here at https://chatgpt.com/c/67bf9fd3-193c-800e-999c-05408cfb8845 .) There is still some cultural resistance to having audience members access their computers and other devices during a lecture, but I think there is some non-trivial value in permitting this sort of augmentation of the lecture so long as it is not disruptive or distracting, and I wonder if cultural norms about this sort of assistance will evolve over time.

2025-02-26 04:48:49

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

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

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

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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

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

2025-02-15 20:48:09

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@nprofile…uwy3 The restriction phenomenon, first discovered by Stein, asserts that the Fourier transform of a function, when restricted to a curved surface such as a sphere or paraboloid, is surprisingly well-behaved in size compared to if one had restricted to a flat plane instead. This phenomenon helps explain the dispersive nature of waves, whose Fourier transform is often concentrated on such surfaces; through geometric optics approximations (the wave packet decomposition), it is also related to geometric incidence problems such as the Kakeya needle problem. In particular, the central conjecture in restriction theory - appropriately named the restriction conjecture - implies one of the major open problems in geometric measure theory, namely the Kakeya conjecture.

2025-02-14 19:03:53

Some progress in restriction theory: the Stein conjecture, a weighted L^2 version of the restriction conjecture that would imply both the restriction and Kakeya conjectures, has shown by Hannah Cairo to be false (at least if no epsilon losses are allowed)! https://arxiv.org/abs/2502.06137 . In fact a slightly weaker, but still very interesting, version of the conjecture, known as the Mizohata-Takeuchi conjecture, was also been disproven. The counterexamples are concentrated near a lattice, and the failure is only by a logarithmic factor; so it is still possible that versions of this conjecture with "epsilon losses" are still true.
Incidentally, there will soon be an announcement of another major advance in this area (not by myself), but I cannot reveal the details at this stage. Stay tuned...

2025-02-12 19:28:14

Nominations for the Riemann prize in mathematics are now open https://www.rism.it/rp2025-nomination : nominations close May 10, 2025.
More details about the prize can be found at https://www.rism.it/riemann-prize

2025-02-08 17:49:30

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 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://discord.com/channels/816415436161417308/816415436684918796/1337838845697462312 .
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. (See also our instagram https://www.instagram.com/cosmic_distance_ladder/ for this project.)

2025-02-07 20:38:53

#DARPA just announced a forthcoming program entitled "Exponentiating Mathematics", focusing on challenges to auto-formalization (and auto-decomposition of large proofs into small lemmas). The preliminary announcement (still short on many details) can be found at https://sam.gov/opp/4def3c13ca3947069b1779e7ff697c6a/view
[As mentioned in previous posts, there is significant uncertainty in the federal funding environment at present, nevertheless there is still activity in launching new proposals.]

2024-11-08 01:26:53

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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)

2023-08-05 18:49:49

I don't have domain expertise to directly comment on the potential #LK99 room temperature superconductor, except to say that the experts that I have talked to are currently quite skeptical. But I can draw one analogy with my experience with mathematical research. A typical math research project consists of months of proposed attacks on a problem, resulting in all sorts of failures or partial successes, until enough experience and intuition is gained to locate the correct approach (or to realize that one needs to modify the problem, or work on a completely different project). However, when the time comes to write up the work, usually the failed or partial attempts are not mentioned at all, except perhaps as brief motivation for the final successful approach. This has some sense to it - a reader is likely to be more interested in the approach that worked than the approaches that didn't quite work - but can give the mistaken impression that good mathematics consists entirely of correct arguments, and that disclosing the failures one had to attempt before locating the correct approach is somehow shameful. But such failures are in fact enormously instructive, and I wish our culture was more open to sharing them.
With LK99, I have seen it reported that the initial announcements were released prematurely, while the research was still in the "partial success at best" stage. As such, the work fares poorly if judged by the usual standard of "successful, completed research", and criticism is due if one or more of the authors were presenting it as such. But as "research in progress, accidentally revealed to the public", I am inclined to be charitable, and wait for the science to play out.