Here is the abstract of a recent paper by Andrew Suk. (I heard about it from a Facebook post by Yufei Zhao. I added a link to the original Erdős Szekeres’s paper.)

Let *ES(n)* be the smallest integer such that any set of *ES(n)* points in the plane in general position contains *n* points in convex position. In their seminal 1935 paper, Erdős and Szekeres showed that

In 1960, they showed that

,

and conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of magnitude has ever been made on the upper bound over the last 81 years. In this paper, we nearly settle the Erdős-Szekeres conjecture by showing that

.

This is amazing! The proof uses a 2002 “positive-fraction” version of the Erdős-Szekeres theorem by Pór and Valtr.

Among the many beautiful results extending, applying, or inspired by the Erdős Szekeres theorem let me mention an impressive recent body of works on the number of points in which guarantee *n* points in *cyclic position. *A good place to read about it is the paper by Bárány, Matoušek and Pór Curves in intersecting every hyperplane at most *d+1* times, where references to earlier papers by Conlon, Eliàš, Fox, Matoušek, Pach, Roldán-Pensado, Safernová, Sudakov, Suk, and others.

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My paper “the quantum computer puzzle” has just appeared in the May 2016 issue of Notices of the AMS. Here are the beautiful drawings for the paper (representing the “optimistic view” and the “pessimistic view”) by my daughter Neta.

And the summary of my view

Understanding quantum computers in the presence of noise requires consideration of behavior at different scales. In the small scale, standard models of noise from the mid-90s are suitable, and quantum evolutions and states described by them manifest a very low-level computational power. This small-scale behavior has far-reaching consequences for the behavior of noisy quantum systems at larger scales. On the one hand, it does not allow reaching the starting points for quantum fault tolerance and quantum supremacy, making them both impossible at all scales. On the other hand, it leads to novel implicit ways for modeling noise at larger scales and to various predictions on the behavior of noisy quantum systems.

The nice thing is that my point of view is expected to be tested in various experimental efforts to demonstrate quantum computational supremacy in the next few years.

Updates (April, 24 2016): Here is an expanded version of the paper, with references, additional predictions and discussion. Here is a related post on GLL.

The plan for polymath10: I hope to come back to it soon, report on some computer experimentation and, of course, further comments on post 4 are most welcome. I hope to be able to report on some computer experimentation regarding the various conjectures and ideas. I am planning to launch a fifth post in May. Overall, I consider** one year** as a good time span for the project. Post 4 of Polymath11 is still active on Gowers’s blog, and I think that a fifth post is also in planning.

Here on the blog, I plan a mathematical post about my visit to Yale on February. The visit have led to Stefan Steinerberger’s beautiful post on Ulam sequences. There are also newer interesting things, from our combinatorics seminar at HUJI, and from the third Simons’ conference on the analysis of Boolean functions (I hoped Ryan will blog about the conference). In celebration of the recent breakthrough on sphere packing in dimensions 8 and 24 I also plan to write more on sphere packing.

**Happy Passover!**

Pictures with Avi Wigderson at Nogafest and with Alex Lubotzky at Yale.

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**Quick TYI. See if you can guess the title and speaker for a lecture described by “where the mathematics of Cauchy, Fourier, Sobolev, Gelfand and Bourgain meet. (Answer at the end of the post.)”**

Joel Spencer’s 70th birthday conference is coming up on April. Here is the website

Joel Hass’ 60th birthday conference is coming up in May at UC Berkeley. Here is the website.

A conference celebrating Jean Bourgain’s work is coming up in May at Princeton. Here is the conference page.

**Answer: ** Speaker: Haim Brezis; Title**: ***Old-new perspectives on the winding number; *

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Two days from now on Wednesday April 20 there will be a splendid open day at the math department. Do not miss it!!

For online registration form and more information click here or on the picture!

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The Ulam sequence

is defined by starting with 1,2 and then repeatedly adding the smallest integer that is (1) larger than the last element and (2) can be written as the sum of two distinct earlier terms in a unique way. It was introduced by Stanislaw Ulam in a 1962 paper (`On some mathematical problems connected with patterns of growth of figures’) where he vaguely describes this as a one-dimensional object related to the growth of patterns. He also remarks (in a later 1964 paper) that `simple questions that come to mind about the properties of a sequence of integers thus obtained are notoriously hard to answer.’ The main question seems to have been whether the sequence has asymptotic density 0 (numerical experiments suggests it to be roughly 0.07) but no rigorous results of any kind have been proven so far.

A much stranger phenomenon seems to be hiding underneath (and one is tempted to speculate whether Ulam knew about it). A standard approach in additive cominatorics is to associate to the first elements of a sequence a

function

and work with properties of . If we do this with the elements of the Ulam sequence and plot the real part of the function, we get a most curious picture with a peak around

Such spikes are generally not too mysterious: if we take the squares we can observe a comparable peak at for the simple reason that squares are (mod 4). However, here things seem to be very different: numerically, the Ulam sequence does seem to be equidistributed in every residue class. Due to periodicity, the function only sees the set of numbers

and it makes sense to look at the distribution of that sequence on the torus for that special value . A plot of the first 10 million terms reveals a very strange distribution function.

The distribution function seems to be compactly supported (among the first 10 million terms only the four elements give rise to elements on the torus that lie outside .) The same phenomenon seems to happen for some other initial conditions (for example, 2,3 instead of 1,2) as well and the arising distribution functions seem to vary greatly.

**Question 1: What is causing this?**

**Question 2: Are there other `natural’ sequences of integers with that property?**

See also Stefan’s paper A Hidden Signal in the Ulam sequence .

**Update: **See also** **Daniel Ross’ ** **subsequent study of Ulam’s sequence, presented in Daniel’s sort of public ongoing “research log”. (“It includes a summary at the top of the most interesting observations to date, which usually lags a couple weeks behind the most current stuff.”)

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Test your intuition: is this probability bounded away from 0? tends to 0 like ? Quicker? Slower? Is there a nice formula?

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**Maryna Viazovska**

**Maryna Viazovska** has solved the densest packing problem in dimension eight! Subsequently, **Maryna Viazovska** with **Henry Cohn**, **Steve Miller**, **Abhinav Kumar**, and **Danilo Radchenko** solved the densest packing problem in 24 dimensions!

Here are the links to the papers:

Maryna Viazovska, The sphere packing problem in dimension 8

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska,

The sphere packing problem in dimension 24

(I thank Steve Miller and Peter Sarnak for telling me about it.)

**Additional sources:** An article by Frank Morgan in The Huffington Post; A blog post by John Baez on the n-Category Cafe; An article by Erica Klarreich on Quanta Magazine; A blog post in Mathbya girl;

**Kepler, Gauss, Hales, Cohn and Kumar.** A central mathematical problem is to find the densest sphere packing in . The case is known as the Kepler conjecture. Gauss solved it for lattice packings, and Thomas Hales proved it for general packing using a massive use of computations. Cohn and Kumar settled the lattice case for dimensions 8 and 24.

**Conway and Sloane. **The “bible” regarding sphere packing is the classic book by two major player of the theory John Conway and Neil Sloane.

**Hales and Fejes Toth.** Announced in 1998 and published a few years later, Hales’ proof relies on some early work of Laszlo Fejes Toth. Since a full verification would require developing much of the whole project from scratch, Hales himself led a team of researchers to find a formal proof which was published in 2015.

**Lie and Leech.** Lower bounds for higher dimensions. For some dimensions, special lattices of Lie type give surprisingly dense lattice packings. The Leech lattice gives a remarkably dense packing in dimension 24.

**Minkowski,…, Ball, Vance and Venkatesh, ** For asymptotically large dimensions a probabilistic method by Minkowski gives the best known lower bound up to small (but exciting) improvements. It gives a packing of density $2 \cdot 2^{-n}$. Here is a slide from a lecture by Henry Cohn on the state of the art for the asymptotic question. (And here is the link to the slides of the full lecture.)

**Delsartes, Kabatiansky and Levenshtein.** Upper bounds via linear programming. Delsartes’ linear programming method (that can be seen as a Fourier/spectral attack with special features,) had led to important results towards general upper bounds by Kabatiansky and Levenshtein.

**Cohn and Elkies** developed related spectral methods applying directly to sphere packing, which allow to improve the upper bounds in dimensions 4–31 and give strikingly good results in dimensions 8 and 24. Cohn and Kumar used these linear programming methods to settle the densest lattice problem in dimensions 8 and 24 and to give extremely good numerical upper bounds for the non-lattice case.

This is the starting point for Viazovska’s breakthrough.

**Related problems/issues to keep in mind:** The densest packing problem in other dimensions and when the dimension tends to infinity; Kissing numbers and spherical codes; Upper bounds for error correcting codes; packing in other symmetric spaces; packing covering and tiling in combinatorics and geometry.

The little I can tell you is that for a solution one needs to identify certain functions to plug in to the spectral machine. And Maryna’s starting point was some familiar extraordinary elliptic functions and modular forms. More details on the comment section are most welcome. (Update:) John Baez wrote on the n-Category Cafe some elementary comments on the proofs: E8 is the best.

A key ingredient for the result in dimension 24 is the earlier numerical rationality conjectures by Cohn and Miller. Those now appear in the preprint: Henry Cohn, Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24 .

**Congratulations to Maryna, Abhinav, Danilo, Henry, and Steve!**

I remember a decade ago that Steve Miller explained to me some developments, ideas, and dreams regarding two problems. One was the sphere packing problem in dimensions 8 and 24 that he now took part in solving, and the other was the irrationality questions regarding zeta functions at odd integers (and maybe also the Euler constant.) Time to move to the second problem, Steve

(And a trivia question: name a player in both these stories. As usual if you answer in the comment section please give a zero-knowledge answer demonstrating that you know the solution without revealing it.)

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Because of polymath10, I did not discuss over here other things. Let me mention two super major developments that I am sure you all know about. One is Laci Babai’s quasi-polynomial algorithm for Graph isomorphism. (This is a good time to mention that my wife’s mother’s maiden name is Babai.) You can read about it here (and the next three posts) and here. Another is the solution by Jean Bourgain, Ciprian Demeter, Larry Guth of Vinogradov’s main conjecture. You can read about it here and here.

We denoted by the largest cardinality of a family of -sets without a sunflower of size , and the largest cardinality in a family of -subsets of with the property , namely without a sunflower of size with head of size at most . Adding the subscript for refer to balanced families rather than to arbitrary family. The “Erdos-Ko-Rado regime” is when and is arbitrary or when and is arbitrary. (In this regime the property is preserved under shifting.)

Ferdinand gave interesting upper bounds and lower bounds for in terms of .

For the upper bound see this writeup by Ferdinand. For more details see this comment and the following ones. It will be interesting to find sharp lower and upper bounds.

Domotor raised in this comment and the following ones some ideas for using algebraic methods for the sunflower conjecture. This have led to an interesting discussion (following these comments) between Domotor and Ferdinand on connection between Sunflowers and Ramsey numbers (mainly ).

So that this post will not interrupt this interesting discussion let me quote the beginning of Domotor’s comment posted minutes ago:

“So let us denote the size of the largest -uniform family without (different) sets whose pairwise intersection has the same size by . As we have seen in the earlier discussion, trivially and (where in the last Ramsey notation we forbid in a -coloring of the complete graph). Mysteriously, for all three functions we have similar lower and upper bounds. In this comment I propose to try proving for some .”

Shachar Lovett proposed in this comment how to use the topological Tverberg theorem for attacking a certain important special case of the conjecture.

We started with the observation that every family of -sets contains a balanced subfamily of size . (BTW for graphs there is a sharper result by Noga Alon and it would be interesting to extend it to hypergraphs!) This gives that

Phillip Gibbs showed in this comment that if the sunflower conjecture is true then

It will be interesting to describe precisely what (1+o(1)) stands for, and improve it as much as possible. All left to be done for a counterexample to the sunflower conjecture is to complement Phillip construction by another one showing an exponential gap between the balanced and general case. This is certainly a tempting direction.

More drastic reductions then the very basic one are also possible, for example: for families of -subsets without a sunflower of size 3 it is enough to consider families with elements colored by colors such that every set is colored by consecutive colors and two sets sharing elements are not using the same color-intervals. (This implies “high girth” of some sort.) We can ask if this type of reductions is useful for proving the conjecture, if one can show (like Phillip) that assuming the conjecture, the gap between the bounds for this version is not too large compared to the general version, and hope that this can be part of a construction going in the negative direction.

In quite a few comments to post 3, I tried to further develop the homological ideas (and to mention some low hanging fruits and related questions). We did reach a major obstacle sending us back to the drawing board. I see some possible way around the difficulty and I will now describe it. For this purpose I will review the homological ideas in somewhat more general and very elementary context. (We only briefly mention the connection with cycles/homology but it is not needed).

Let be a generic matrix. The -th compound matrix is the matrix of by minors. Namely, , where .

Given two -unform hypergraphs we say that and are **weakly isomorphic** if the minor of whose rows and columns correspond to sets in and respectively is non-singular. (It is fairly easy to see that if and are isomorphic then they are weakly isomorphic. This relies on the condition that is generic.) We will say that **dominates** if and is full rank. Thus, and are weakly isomorphic if each dominates the other.

Let be the set of -subsets of such that . The connection with our notions of acyclicity is as follows: is acyclic (i.e. ) iff is dominated by . In greater generality, iff is dominated by . (In particular, iff is dominated by , and iff is dominated by $D[m,m]$.) **Remark:** Here and later in the post with since we deal only with cycles for the top dimension, so there are no “boundaries” to mod out.

A family of -subsets of is **shifted** if whenever and is obtained by replacing an element by a smaller element then . It can be shown that two shifted families of the same size are weakly isomorphic only if they are equal! We can use our compound matrix to describe an operation (in fact, several operations) called shifting which associated to a family a shifted family . Suppose that . is the lexicographically smallest family of sets which is weakly isomorphic to . In more details: we first consider a total ordering on -subsets of . Then we greedily build a hypergraph which is dominated by . When we reach sets we obtain a hypergraph weakly isomorphic to .

Now, if the total ordering is the lexicographic order on then we denote and call the “algebraic shifting” of . In this case, it can be shown that the resulting family is shifted. Also of special importance is the case that is *the reverse lexicographic order*.

For sets of integers of size , the lexicographic order refers to the lexicographic order when we ordered the elements of every set from increasingly, and the reverse lexicographic order is the lexicographic order when we order the elements decreasingly.

From 12<13<14<15<…<21<22<…<31<32<… we get

and from 21<31<32<41<42<43<51<52<… we get

We mention again some connection with acyclicity and homology: is acyclic if all sets in contains ‘1’. is -acyclic iff all sets in in intersect . is -acyclic iff all sets in contains . For general , -acyclicity is not expressed by those two versions of algebraic shifting, however, is -cyclic if .

The following properties are preserved under algebraic shifting:

(1) Every two members of has at least elements in common.

(2) There are no pairwise disjoint sets in .

For balanced families we know even more:

(3) If is balanced and every two members of has at least elements in common then all sets in contains .

and

(4) If is balanced and there are no pairwise disjoint sets in , then every set in intersects .

As it turns out we have much stronger statements:

The following properties are preserved under reverse-lexicographic algebraic shifting:

(5) Every two members of has at least elements in common.

(6) There are no pairwise disjoint sets in .

(I dont know if (3) and (4) extends as well. It certainly worth the effort to check it.)

Our ultimate conjecture remains the same:

**Main Conjecture:** If has no sunflower of size then it is -acyclic. (I.e., )

For balanced family we conjectured that and for the non-balance we had a larger value .

Our homological approach mainly for the balanced case was to use the local homological condition from (2) (or(4)) (with some additional homological condition yet to be proved – this was Conjecture A) to conclude (This was Conjecture B) that is -acyclic. However, Conjecture B turned out to be false. Our new idea is to replace the local conditions on homology obtained from (2), (4) by those given by (6) which are apparently quite stronger.

We also tried to explore how to prove the main conjecture via an even stronger statement for “combinatorial cycles”. This works for the balanced case in the Erdos-Ko-Rado-regime (leading to some interesting consequences). But we did not manage to extend it beyond this regime. Domotor shoot down some half-baked attempts towards such a goal.

What we propose now is to use the following theorem instead of Conjecture A and to greatly modify Conjecture B accordingly. (For general families; being balanced is not used.)

**Theorem:** Let is a family of -sets without a sunflower of size . Then

(*) For every family of -sets which is the link of a set of size (including the case , Every set in intersect .

**Conjecture B** (greatly modified): For a family of -sets satisfying (*) we have

(**) .

This is the new plan! Below are a couple of comments.

We can, I think, translate the condition on reverse lexicographic shifting also to some conditions on homology, given in this comment, but the connection is still a little dubious and need some further checking and explanation. (Specifically, it relies on a comment I make in my paper on algebraic shifting that if is a family of -subsets of and is its complement, then the shifting of is related to the reverse lexicographic shifting of as follows: takes the complement of and apply the involution sending element to .)

If is the simplicial complex spanned by our family and is the simplicial complex spanned by the complement of we want to replace conditions of the form

(*)

by the stronger condition

(**)

For example, for a graph with vertices and edges gives all edges containing ‘1’ (which is equivalent to (*) for ) if and only if is a tree. But requiring it for implies (I think) that be a star and is equivalent (I think) to , where is the complement of .

We plan to run computer experimentation to test some of these ideas on small cases.

Let me mention that there is a variant of compound matrix, weak equivalence and of algebraic shifting (and hence also of the various homology groups we considered,) when we use symmetric products instead of exterior products. The theories based on those variants are very similar, and the advantage of the notions based on symmetric powers is that they are closer to studied notions of commutative algebra. (But I think they will be harder to compute as determinants are replaced by permanents.)

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j) **Polymath11 (?)** Tim Gowers’s proposed a polymath project on Frankl’s conjecture. If it will get off the ground we will have (with polymath10) two projects running in parallel which is very nice. (In the comments Jon Awbrey gave a links for a first in a series posts also on Frankl’s conjecture, with the catchy title, Frankl my dear.)

a) NogaFest started a few days ago. It is a wondeful meeting! My lecture entitled “polymath” refers to the older meaning of the word, so appropriate to describe Noga. (I was not aware that the word has a meaning until recently). I talked, among other things, about **polymath10**. I prepared the talk a week ahead and presented our Conjectures A and B (from polymath10 last post) hoping that perhaps I could add some positive information toward them. Well, just after my presentation was ready, I realized that Conjecture B is false. Here are the slides.

Two quotes from the lecture. First about the birthday boy: the idea of the polymath was expressed by Leon Battista Alberti (1404–1472), in the statement, most suitable to Noga **“****a man (who) can do all things if he will”.**** **Second, about polymath projects (by Gowers): “a large collaboration in which no single person has to work all that hard.”

b) Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals (perhaps mainly to see what researchers regard as central major projects,) in applied mathematics, and various areas of geometry, algebra, analysis and logic.

c) A very nice polymath proposal by Dinesh Thakur was posted by Terry Tao on the polymath blog. The task was to explain some numerically observed identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two. David Speyer managed to prove Thakur’s observed identities! Here is the draft of the paper.

d) This reminds me that some years ago David Speyer solved a question that interested me for decades and was presented here on the blog and later on MathOverflow about systems of skew lines in three dimensional vector spaces over division rings (and especially the Quaternions).

e) Related to **polymath3**, let me mention that Michael Todd proved a small but very elegant improvement of the upper bounds by Kleitman and me from 1992. (The new bound is . The first improvement, I think, in two decades!

f) I have a very nice thing to tell you about **polymath4!** Shafi Goldwasser abstract for Nogafest talked about a new notion of randomized algorithms: A randomized algorithm to achieve a certain task (for example to find a perfect matching in a graph,) which is guaranteed to reach the same answer with high probability! Such an algorithm is called pseudo-deterministic. It is both an amazing concept, and it is quite amazing that it was not introduced before. The polymath4 question was to find deterministically a prime with n digits and A new challenge (that Shafi asks about) is to find a pseudo-deterministic efficient algorithm. Namely, a randomized algorithm which will find an n-digit prime, but with high probability the same one! (I would guess that it is still hopeless.)

g) And Terry Tao gave a beautiful lecture on Erdos discrepancy problem (the topic of **polymath5**). I understood a little better the argument (which is similar to Roth density increase argument for 3 term AP,) that allows Tao to use the logarithmically-averaged Chowla inequalities.

h) The old conjecture that centrally symmetric convex sets have nonnegative correlation w.r.t. the Gaussian distribution was proved! Let me refer you to the paper Royen’s proof of the Gaussian correlation inequality for a simple exposition of a proof by Thomas Royen, and more information on the solutions and solvers.

i) The Nogafest participants are invited to a Jazz night at Gilly’s

The third Polymath10 post is active. I hope to post a new polymath10 post in about 1-2 weeks. I hope also to return to various amazing things I am hearing on Nogafest and other places (and also on my ownfest some months ago).

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Let [] be the maximum size of a family of -sets [balanced family] with no sunflower of size . Let and .

Phillip gave a beautiful construction showing that if is finite than . This is very interesting and I found it surprising. (Earlier we only knew that .) Of course, if you can find another construction which starts with balanced families (even only those constructed by Philip) and end with exponentially larger general families (you can even enlarge k a little,) this will provide a counterexample to the Erdos-Rado conjecture.

In the previous post and some comments (like this comment and the following ones; and this comment) I continued to meditate about my homological approach. For we want to show that for balanced families, a (2m,m)-cycle will contain a sunflower with head of size smaller than m. (This will now give .) Juggeling between the homological notion of (b,c)-cycles and a combinatorial one can be useful.

Tim observed that if all pairwise intersections between sets in a sunflower free family have the same size then this leads to exponential upper bounds. This can be a starting point for an argument where “same size” is replaced by a weaker condition, or to ideas about how to construct a counterexample.

We also want to understand the expected number of sets in the sunflower-free process when we consider k-subsets fron [n]. (We would also like to understand the expected size of the union of sets in the resulting family.) This is interesting both for the general case and the balanced case. Simulations by Philip (e.g. here) and by Gonzalo (e.g. here) were presented. (I am still confused about the outcomes of the simulations, maybe we shoud run more simulations.)

In the later comments to the previous thread (starting here) that I did not digest yet, Philip offered some ideas on new constructions of various types.

Avi Wigderson asked: Is it enough to prove the Erdos-Rado conjecture for ? ? etc? Dömötör asked the following Kneser-type coloring question:

How many colors to we need to color all k-tuples of an n element set avoiding monochromatic 3-sunflowers?

Shachar mentioned in comments to post I (starting here) an exciting special case (related to matrix multiplication.)

It can be a good time to look at the classic papers by Abott, Hanson, and Sauer and , Spencer and Kostochka (for the general case; for sunflower of size three). Here is again the link for Kostochka’s survey. There are many other papers (even recent) about sunflowers and many aspect of the problem and related problems that we did not talk about.

Here are the six pages of Joel’s paper.

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