Category Archives: Number theory

Absolutely Sensational Morning News – Zander Kelley and Raghu Meka proved Behrend-type bounds for 3APs

What is the density of a subset of that guarantees that contains a 3-term arithmetic progression? And, more generally, if the density of is what is the minimum number of 3-terms AP that contains? These problems and the more general … Continue reading

Posted in Combinatorics, Number theory, Updates | Tagged , | 1 Comment

ICM 2022. Kevin Buzzard: The Rise of Formalism in Mathematics

In this post I would like to report on Kevin Buzzard’s spectacular lecture on moving mathematics toward formal mathematical proofs.  (Here are the slides.) The picture above is based on images from the other spectacular Saturday morning lecture by Laure … Continue reading

Posted in Computer Science and Optimization, ICM2022, Logic and set theory, Mathematics and Computers, Number theory, What is Mathematics | 12 Comments

ICM 2022: Langlands Day

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 … Continue reading

Posted in Algebra, ICM2022, Number theory | Tagged , , , | Leave a comment

To cheer you up in difficult times 25: some mathematical news! (Part 2)

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 … Continue reading

Posted in Algebra, Combinatorics, Geometry, Number theory | 3 Comments

To cheer you up in difficult times 20: Ben Green presents super-polynomial lower bounds for off-diagonal van der Waerden numbers W(3,k)

What will be the next polymath project? click here for our post about it.  New lower bounds for van der Waerden numbers by Ben Green Abstract: We show that there is a red-blue colouring of [N] with no blue 3-term … Continue reading

Posted in Combinatorics, Number theory | Tagged , | 4 Comments

Péter Pál Pach and Richárd Palincza: a Glimpse Beyond the Horizon

  Prologue Consider the following problems: P3: What is the maximum density of a set A in without a 3-term AP? (AP=arithmetic progression.) This is the celebrated Cap Set problem and we reported here in 2016 the breakthrough results by … Continue reading

Posted in Combinatorics, Geometry, Number theory, Open problems | Tagged , | 9 Comments

To cheer you up in difficult times 10: Noam Elkies’ Piano Improvisations and more

For the previous post “quantum matters” click here. Noam D. Elkies piano improvisations Every day since March 27, 2020 Noam Elkies (Noam’s home page) uploaded a new piece of piano improvisation. The Hebrew title of his page is “Music will … Continue reading

Posted in Games, Music, Number theory | Tagged | Leave a comment

To cheer you up in difficult times 5: A New Elementary Proof of the Prime Number Theorem by Florian K. Richter

Here is a piece of news that will certainly cheer you up: Florian Richter found A new elementary proof of the prime number theorem. (I thank Tami Ziegler for telling me about the new result.) From left to right: Atle Selberg, … Continue reading

Posted in Number theory, Updates | Tagged , , | 12 Comments

Hoi Nguyen and Melanie Wood: Remarkable Formulas for the Probability that Projections of Lattices are Surjective

Following a lecture by Hoi Nguyen at Oberwolfach, I would like to tell you a little about the paper: Random integral matrices: universality of surjectivity and the cokernel by Hoi Nguyen and Melanie Wood. Two background questions: Hoi started with … Continue reading

Posted in Algebra, Combinatorics, Number theory, Probability | Tagged , | 6 Comments

The largest clique in the Paley Graph: unexpected significant progress and surprising connections.

The result on Paley Graphs by Hanson and Petridis On May 2019, Brandon Hanson and Giorgis Petridis posed a paper on the arXive: Refined Estimates Concerning Sumsets Contained in the Roots of Unity. The abstract was almost as short as … Continue reading

Posted in Combinatorics, Number theory | Tagged , , , , , | 3 Comments