Notices by Terence Tao (tao@mathstodon.xyz)

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 been fully automated, although spherical trigonometry classes are still offered at naval academies, and ships still carry printed navigational tables in case of emergency instrument failures.

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 electronic computers, although human mathematicians and programmers are still required to direct these machines. (1/3)

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.)

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.

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 ending up becoming the only known source of 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 disseminated, due to the aforementioned reporting bias, and perhaps also due to some stigma regarding the topic). So I decided to report this (rather routine) rejection as a token gesture towards more accurate disclosure. (1/2)

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 prove the full conjecture without the epsilon loss, 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)

The Bernoulli Center for fundamental studies (based in EPFL in Lausanne, Switzerland) has issued a call for research programs (of two to nine weeks) in the areas of mathematics, theoretical physics, and computer science. https://bernoulli.epfl.ch/bernoulli-programs/

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.)

I may have posted this link before, but Matt Might's short visual essay "Illustrated Guide to a PhD" is a pretty reasonable starting model for what a #PhD contributes to the sum of human knowledge. https://matt.might.net/articles/phd-school-in-pictures/

Over on the #CosmicDistanceLadder instagram, my co-author Tanya Klowden has a new post on the constellations on the ecliptic (most familiar to western astrology as the signs of the Zodiac), as viewed by native Americans prior to European colonization. https://www.instagram.com/p/DC8gvXXxtfw/?img_index=1

In a conversation today with one of my graduate students, my student observed that there are various folklore tips about applying for postdoctoral positions that are passed down from student to student in an ad hoc fashion, that are not formally recorded in any shareable form. As an example, one tip he received was to look for math postdoctoral positions not just under the "postdoc" listing in Mathjobs, but also under "Assistant Professor" positions, since many postdocs are technically classified under the latter category and not listed as official postdocs.

I myself last applied for postdoc positions almost thirty years ago, so I had little current advice to give on these matters, but I wonder if there already exist some standard venues where these sorts of tips are shared (e.g., academia stackexchange), and if not, whether it might make sense to try to create one. I'd be interested to know of any efforts in this area (if nothing else, it would be an obvious additional set of links to add to my career advice pages).

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.

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)

Tim Gowers, Ben Green, Freddie Manners and I just uploaded a paper https://arxiv.org/abs/2311.05762 proving the Polynomial Freiman-Ruzsa conjecture. More details at https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/ .

I am planning to formalize this paper in #Lean4 also; more on this later.

Finally completed the #Lean4 formalization of my Maclaurin-type inequality paper at https://github.com/teorth/symmetric_project . Specifically, if \(s_k\) denotes the \(k^{th}\) symmetric mean of \(n\) real numbers, the inequality
\[ |s_l|^{1/l} \leq C \max ( (l/k)^{1/2} |s_k|^{1/k} \] \[, (l/(k+1))^{1/2} |s_{k+1}|^{1/(k+1)})\]
is established for \( 1 \leq k < l \leq n\). In fact as a byproduct of the formalization I now get the explicit value \(C = 160 e^7\) for the constant \(C\).

Was surprisingly satisfied to get the blue "No goals" message in the Lean infoview.

In the end it took perhaps a hundred hours of effort, spread out over a month. It was about 20x slower to formalize this paper than it was to write it in LaTeX. Initially this was due to my own inexperience with Lean, though near the end I was hitting the limitations of existing Lean tools, as well as the speed of the Lean compiler (though this was in part due some inefficiencies in my coding). However, Lean is a living language still in development, and several helpful people provided me during this project with upgraded Lean tactics (such as an improved `positivity` tactic that could handle finite sums and products, a `rify` tactic to easily convert natural number inequalities to real number ones, and a `rw_ineq` tactic to implement the rewriting of inequalities I had noted was previously missing). These definitely helped expedite the formalization process, particularly in the last week or so. 1/3

Happily, apart from the one bug previously noted, no other errors (other than very small typos) were detected in the paper. I found some mild simplifications to some steps in the proof as a consequence of formalization, but nothing particularly notable. (But this was a rather elementary paper, and I was not really expecting to glean deep mathematical insights from the formalization process.)

The Sapir-Whorf hypothesis https://en.wikipedia.org/wiki/Linguistic_relativity is controversial for natural languages, but for the specific case of proof formalization languages I think the effect is real. The Lean syntax and library of lemmas and tools certainly pushed me to plan ahead and abstract away many small lemmas to a far greater extent than I would ordinarily do; long "stream of consciousness" type arguments, which I tend to favor personally, are possible in Lean, but often not the most efficient way to proceed. I have also started noticing myself mentally converting results I read in regular mathematical literature to a sort of "Lean pseudocode", though this is probably just a version of the "Tetris effect" https://en.wikipedia.org/wiki/Tetris_effect and will likely dissipate now that my project has concluded.

A combination of automated tools, AI tools, and interactions with human Lean experts proved invaluable. GPT greatly sped up the initial learning of syntax, tools such as `exact?`, `apply?`, and Moogle https://moogle-morphlabs.vercel.app/ helped locate key lemmas, and Copilot had an often uncanny ability to autocomplete one or more lines of code at a time. But there was certainly a lot of "folklore" tips and tricks that I had to go to the Lean Zulip to get answers to. 2/3

Every five to ten years, I make the effort to switch my #TeXLaTeX editor to a more modern one. This process has now iterated several times back from when I was a graduate student in the mid-1990s using vi from a UNIX shell to write in plain TeX. On the suggestion of a reader here, I installed #VSCode + #TeXLive + #LaTeXworkshop + #GithubCopilot as an upgrade from my current setup of #TeXnicCenter + #MiKTeX
which I had been using for almost a decade, and am recording my first impressions here (which will most likely be quite naive for existing VSCode users).

The installation had no problems (other than the four hours needed to download TeX live on a slow internet connection). I began experimenting with various features. So far I have mostly played with the user-defined code snippets feature, which can allow me for instance to create an entire corollary environment by typing in a trigger word (I chose "cor") and pressing tab (see enclosed screenshots). Strangely enough I had a version of this functionality 20 years ago during a brief period when I experimented with using Microsoft Word as a LaTeX editor purely for the ability to use Visual Basic macros (though I abandoned this shortly after due to the lack of other LaTeX-friendly features). I could certainly see myself using this feature frequently as a time-saver.

So far the AI-powered Copilot suggestions have been mainly useful for filling out the snippet functionality: after giving a few examples of the snippets I wanted, it was able to suggest more that I could accept, again with the single click of the tab key.

(Incidentally, the screenshots are displaying a paper which I will be putting on the arXiv shortly. Stay tuned...)

I found #ChatGPT useful for creating more complicated snippets involving regex expressions. Specifically, I wanted a snippet that could take a LaTeX expression such as (x+y=z) and add \left and \right modifiers to the delimiter to create \left(x+y=z \right). It took two tries on GPT's part, but the second solution worked to give exactly what I wanted (and even helpfully formatted it in the VScode JSON format that I needed, even though I didn't ask for this). https://chat.openai.com/share/35a5eee7-9318-4bdf-809a-a77ba9afba81

As an experiment, I asked #ChatGPT to write #Python code to compute, for each 𝑛, the length 𝑀(𝑛) of the longest subsequence of \(1,\dots,n\) on which the Euler totient function ϕ is non-decreasing. For instance, 𝑀(6)=5, because ϕ is non-decreasing on 1,2,3,4,5 (or 1,2,3,4,6) but not 1,2,3,4,5,6. Interestingly, it was able to produce an extremely clever routine to compute the totient function (that I had to stare at for a few minutes to see why it actually worked), but the code to compute \(M(n)\) was slightly off: it only considered subsequences of consecutive integers, rather than arbitrary subsequences. Nevertheless it was close enough that I was able to manually produce the code I wanted using the initial GPT-produced code as a starting point, probably saving me about half an hour of work. (and I now have the first 10,000 values of \(M\)). The results were good enough that I would likely turn to GPT again to provide initial code for similar calculations in the future. https://chat.openai.com/share/a022e1d6-dddc-4817-8bbd-944a3e742d9f

Earlier this year, I was offered the opportunity by Eric Horvitz at Microsoft to gain access to the (then unreleased) #GPT4 model and write about my experiences with it for an anthology of short #AI essays. The essays are now in the process of being published at https://unlocked.microsoft.com/ai-anthology/ , with my own essay just released today. The topic is still developing rapidly; it wil be interesting to revisit these essays in a year or so and see how the reality of this new AI technology compared with these early expectations.