A “Feature” article in Nature (21 May 2026) and a long article in the New York Time (June 8, 2026) are both covering the sudden intrusion of AIs in proving or disproving mathematical conjectures… Genuine, hard, conjectures likes Erdös problems. Specialised AIs like Alethia, Math Inc., AlphaProof are hacking at existing open problems and a benchmark collection of problems whose solutions are not available (yet) called First Proof has been recently gathered, solved by OpenAI, and is soon to be extended. In an earlier interview with Nature (27 April 2026), Terry Tao acknowledges the tectonic shift represented by this intrusion and how a mathematician’s skill need evolve. (This is also the starting point of the NYT story. ) A side issue of this is that reviewing the proofs thus produced by an AI may prove inhumane if it reaches hundreds of pages, which also relates to the un-academic (?) tendency of AIs companies to over-blow the results produced by their machines. For the times they are a-Changin…
Archive for conjecture
AIs rushing to the proofs
Posted in Books, Kids, University life with tags AIs, Alethia, AlphaProof, Bob Dylan, computer-based proof, conjecture, Erdös problems, Fields medal, Math Inc., mathematics, Nature, NTY, OpenAI, Terry Tao, The New York Times on June 23, 2026 by xi'anNature snapshots
Posted in Books, Kids, pictures, Statistics, University life with tags AI, Bank of Sweden, computer-based proof, conjecture, cows, Daniel Kahneman, Emmanuel Macron, French politics, mathematics, Nature, Paris-Saclay campus, Pink Floyd, religious persecutions, SARS-CoV-2, Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, Vichy régime, WW II on July 13, 2024 by xi'an
- One of the editorials is about AI-assisted design of mathematical conjectures. Reminding me of grant panels I took part in where candidates in pure maths were often focussing on solving one or many conjectures, as opposed to more applied branches (like statistics).
- Two entries about France, one about Macron’s idea of a European DARPA, unlikely to convince European partners after his calling most unnecessarily an election and bringing the extreme-right to its apex since the Vichy régime. Another one about the super mega campus Paris-Saclay unable to settle on a form of leadership and hence elect a president. Federalist versus centralised… Once again, thanks to Macron’s reckless gamble, his current Ministry for Higher Education may be in need of a position next month and could return to heading the campus.
- Worrying coverage of bird flu in US cows (not yet pigs on the wing but worrying enough) and of zombie cells that should have died and that new treatments can fight way better.
- A reflection on coming up with a treaty against AI weapons with autonomous kill decisions but isn’t it too late?!
- Building design that would avoid total collapse of a failing building, as in South Florida four years ago. But solely for new ones, unfortunately. (And the connection with lizards escapes me, except for managing to escape by loosing a tail…)
- A biography of Daniel Kahneman, decision-theorist, 2002 Nobel economics (aka Bank of Sweden) Prize, author of Thinking, Slow and Fast, and spending the WW II in occupied France hiding from round-ups by the Vichy police.
complex Cauchys
Posted in Books, pictures, Statistics, Travel, University life with tags Augustin Cauchy, Cauchy distribution, complex numbers, confidence distribution, conjecture, Don Fraser, Nancy Reid, Peter McCullagh, Sceaux, seminar, Université Paris Dauphine, William Feller on February 8, 2018 by xi'an
During a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as
f(x;θ¹,θ²) = |θ²| / |x-θ|²
when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,
C(aθ+b/cθ+d)
[when X is C(θ)] follows rather readily. But then comes the remark that
“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”
which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).
bounded normal mean
Posted in R, Statistics, University life with tags Bayesian decision theory, bounded normal mean problem, conjecture, EuroBayes, La Sapienza, least favourable priors, minimaxity, MLE, Roma, Statistical decision theory on November 25, 2011 by xi'an
A few days ago, one of my students, Jacopo Primavera (from La Sapienza, Roma) presented his “reading the classic” paper, namely the terrific bounded normal mean paper by my friends George Casella and Bill Strawderman (1981, Annals of Statistics). Even though I knew this paper quite well, having read (and studied) it myself many times, starting in 1987 in Purdue with Mary Ellen Bock, it was a pleasure to spend another hour on it, as I came up with new perspectives and new questions. Above are my scribbled notes on the back of the [Epson] beamer documentation. One such interesting question is whether or not it is possible to devise a computer code that would [approximately] produce the support of the least favourable prior for a given bound m (in a reasonable time). Another open question is to find the limiting bounds for which a 2 point, a 3 point, &tc., support prior is the least favourable prior. This was established in Casella and Strawderman for bounds less than 1.08 and for bounds between 1.4 and 1.6, but I am not aware of other results in that direction… Here are the slides used by Jacopo:
