«Великий государь, Царь и Великий Князь Пётр Алексеевич […] указал Именным Своим Великого Государя повелением в государстве Богохранимой Своей Державы Всероссийского Самодержавия на славу Всеславного Имени Всемудрейшего Бога и Своего Богосодержимого храбропремудрейшего царствования, во избаву же и пользу Православного Христианства, быть Математических и Навигацких, то есть мореходных хитростно наук учению»

Это уже было сказано тысячу раз, но позвольте мне добавить свой голос: эпоха людей, пишущий код, закончилась
Это тревожно для тех из нас, кто идентифицирует себя как SWE, но это факт
Это не значит, что SWE теперь лишены работы, но это уже точно не написание синтаксиса напрямую

«I have little doubt that AI tools will substantially change the way mathematics is done, likely pretty soon
But communicators like this are doing their audiences a disservice

In this particular example the “new” contribution from the AI tool was plugging (n,m)=(5,3) into the expression 2n-m. I’m not exaggerating
The “number of multiplications” in the result in question was shown by Winograd ~45 years ago to be 2n-m, where the matrices in question are nxn, and m is the number of factors of the polynomial x^n-1

The “result” here is that this polynomial has 3 factors over Q(sqrt(5)), and then that 2*5-3=7

That’s it
Winograd’s result itself is an application of the Chinese Remainder Theorem
(…)
That automated tools can make progress on minor open questions is incredibly impressive
But that doesn’t excuse inaccuracy about current capabilities»

— (…) combinatorics has undergone a remarkable development over the past, say 50 years
Still in the 1960s and even 1970s, it had not been recognised as a subject of its own (…). I get the impression that this has changed now, even if combinatorics may not be ranked among the “most important” (whatever that may mean) subjects by many people working in these “most important” subjects
On the surface, this change in perception of combinatorics is witnessed by the Fields Medal of June Huh, or the Abel Prizes of Endre Szemeredi and Laszlo Lovasz
However, it seems to go deeper as it is now more and more acknowledged that also in combinatorial theory there are deep results whose impact goes far beyond combinatorics, even if there is no streamlined theory as, say, in analysis
(On the side: I consider this variety as part of the attraction and fascination of combinatorics)
(…)
— What three results do you consider the most influential in combinatorics during the last thirty years?

— Obviously, my answer can only be very subjective

I would begin with Greg Kuperberg’s “Symmetry classes of alternating-sign matrices under one roof” where he provides proofs of four of the long-standing conjectures on the enumeration of symmetry classes of alternating sign matrices
He does that by interpreting the relevant classes of alternating sign matrices as instances of the six-vertex model with certain boundary conditions and by finding determinantal or Pfaffian formulae for the corresponding partition functions (…)

Next Gilles Schaeffer’s bijections between planar maps and trees that first appeared in his thesis come to my mind
(…)
Moreover, these bijections have been the driving engine for the fantastic results on scaling limits of planar maps, with the limiting object being a very fractal random sphere which is called Brownian map

Another article that I consider extremely influential is “The cyclic sieving phenomenon” by Vic Reiner, Dennis Stanton, and Dennis White
(…)
On the probabilistic side, the arctic circle theorem for domino tilings of the Aztec diamond due to Bill Jockusch, Jim Propp and Peter Shor is a milestone, exhibiting the first such scaling limit result in the area
Many more results of this kind were proved subsequent to that article, culminating in “Dimers and amoebae” by Rick Kenyon, Andrei Okounkov and Scott Sheffield

In that article, a general theory for such limiting phenomena for the dimer model on periodic graphs is outlined (even if not everything here may be completely rigorous)
Another breakthrough that has to be mentioned here is the theorem by Jinho Baik, Percy Deift and, Kurt Johansson that shows that the length of the longest increasing subsequence in a random permutation, if suitably rescaled, behaves like the celebrated Tracy–Widom distribution
Although, on the outset, this seems to be quite different from the other results that I have mentioned in this paragraph, it implicitly concerns very similar processes

In another direction, I must mention the work by June Huh and the theory of Lorentzian polynomials due to Petter Branden and June Huh that paved the way to prove many combinatorial inequalities that had been conjectured for a very long time