Greg Kuperberg is having a short visit in Israel and yesterday he gave a fantastic lecture on an improved bound for the Solovay-Kitaev theorem. Here is a videotaped lecture of Greg on the same topic in QIP2023. Solovay-Kitaev theorem from 1995 (in a stronger version by Kitaev-Shen-Viyalyi) asserts that Theorem:  If is a finite subset of  that densely generates with the property that , then there is an efficient algorithm to -approximate every element by a word   of length where we can take , for every . Greg mentioned several improvements and related results over the year ..read more

ICM 2022 is running virtually and you can already watch all the videos of past lectures at the IMU You-Tube channel, and probably even if you are not among the 7,000 registered participants you can see them “live” on You-Tube in the assigned date and hour.  I landed back from Helsinki on Wednesday night and I devoted Thursday to watch lectures, while in later days other tasks and obligations gradually took over part of my time. I plan to catch up during the summer. The three plenary lectures on Thursday, July 7 were around the Langlands program. David Kazhdan, Marie-France Vignéras and Fr ..read more

Hugo Duminil-Copin, June Huh, James Maynard and Maryna Viazovska were awarded the Fields Medal 2022 and Mark Braverman was awarded the Abacus Medal 2022. I am writing from Helsinki where I attended the meeting of the General Assembly of the IMU and yesterday I took part in the moving award ceremonies of ICM2022 hosted by Aalto University.  This will be the first post about the ICM 2020 award ceremonies. The opening day of ICM2022 was exciting. Hugo Duminil-Copin, June Huh, James Maynard and Maryna Viazovska were awarded the Fields Medals 2022. Mark Braverman was awarded the Abacus Medal ..read more

This is my fifth and last report from ICM 2018 at Rio. I will talk a little about the three Wednesday plenary talks by Assaf Naor, Geordie Williamson, and Christian Lubich. See here for other posts about ICM2018. (For the three talks, I relied on a recent viewing (Dec. 2021) of the videotaped lectures rather than on my incomprehensible earlier notes.)     While at the Rio conference, I met, for the first time, quite a few people including  the (then) current and previous IMU president Shigefumi Mori and Ingrid Daubechies, and the current IMU secretary Holge Holdem. I went m ..read more

Before getting to the main topic of this post, first, Happy New Year 2022 and Merry Christmas to all readers, and second, a quick update: A community blog to discuss open problems in algebraic combinatorics was created. Everybody is invited to check out the problems posted there, to subscribe, and to contribute their own problems. Virtual Poll: What will be the next term in the sequence AGC-GTC-TGC-GTC-TGC-GAC-GATC- ??? GTC GATC TAGC GCTC C:GAT Another answer Oberwolfach, Geometric, algebraic and topological combinatorics, 2019. Giulia Codenotti, Aldo Conca, Sandra Di Rocco, Bruno Benedetti ..read more

One of the exciting directions regarding applications of computers in mathematics is to use them to experimentally form new conjectures. Google’s DeepMind launched an endeavor for using machine learning (and deep learning in particular) for finding conjectures based on data. Two recent outcomes are toward the Dyer-Lusztig conjecture (Charles Blundell, Lars Buesing, Alex Davies, Petar Veličković, Geordie Williamson) and for certain new invariants in knot theory (Alex Davies, András Juhász, Marc Lackenby, Nenad Tomasev). There is also a Nature article Advancing mathematics b ..read more

“Lean is a functional programming language that makes it easy to write correct and maintainable code. You can also use Lean as an interactive theorem prover.” (See Lean’s homepage and see here for an introduction to lean.) Kevin Buzzard’s blog XENA is largely devoted to lean. “The Xena project is an attempt to show young mathematicians that essentially all of the questions which show up in their undergraduate courses in pure mathematics can be turned into levels of a computer game called Lean.” (Perhaps this could also appeal to old mathematicians especially those interested in computer games ..read more

Topology Quasi-polynomial algorithms for telling if a knot is trivial Marc Lackenby announced a quasi-polynomial time algorithm to decide whether a given knot is the unknot! This is a big breakthrough. This question is known to be both in NP and in coNP. See this post, and updates there in the comment section. Topology seminar, UC Davis, February 2021 and Oxford, March 2021 and today, May 20! 16:00 CET, in the Copenhagen-Jerusalem combinatorics seminar (see details at the end of the post). Unknot recognition in quasi-polynomial time (The link is to the slides) NP hardness for related questions ..read more

There is a very famous conjecture of Irving Kaplansky that asserts that the group ring of a torsion free group does not have zero-divisors. Given a group G and a ring R, the group ring R[G] consists of formal (finite) linear combinations of group elements with coefficients in the ring. You can easily define additions in R[G], and can extend the group multiplication to R[G], which makes the group ring a ring. (And if R is a field, R[G] is an algebra, called group-algebra.) Kaplansky’s zero divisor conjecture asserts that if G is torsion-free and K is a field then K[G] has no zero-divisors. Irv ..read more

Stavros Argyrios Papadakis, Vasiliki Petrotou, and Karim Adiprasito In 2018, I reported here about Karim Adiprasito’s proof of the g-conjecture for simplicial spheres.  This conjecture by McMullen from 1970 was considered a holy grail of algebraic combinatorics and it resisted attacks by many researchers over the years. Today’s post  reports on two developments. The first from December 2020 is a second proof for the -conjecture for spheres by Stavros Argyrios Papadakis and Vasiliki Petrotou. (Here is a link to the arXive paper.) A second proof to a major difficult theorem is always ..read more