which was used for the official T-shirt for Jean-François Le Gall’s birthday conference.

See also this quanta magazine article by Kevin Hartness.

]]>Credit George Csicsery (from the 1993 film “N is a Number”) (source)

(I thank Gady Kozma for telling me about the result.)

An old problem from analysis with a rich history and close ties with combinatorics is now solved!

The paper is: Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littelwood Polynomials exist

**Abstract:** We show that there exist absolute constants Δ > δ > 0 such that, for all *n⩾2*, there exists a polynomial *P* of degree *n*, with ±1 coefficients, such that

for all *z ∈ C* with *|z|=1*. This confirms a conjecture of Littlewood from 1966.

It is still a major open problem to replace by and by (ultra-flat polynomials).

The problem can be traced back to a 1916 paper by Hardy and Littlewood.

Shapiro (1959) and Rudin (1959) showed the existence of such polynomials when you only require for all *z ∈ C* with *|z|=1*.

The new result confirms a conjecture of Littlewood from 1966 and answers a question by Erdős from 1957.

If one allows complex coefficients of absolute value one, ultra flat polynomials exist by a result of Kahane (1980). Bombieri and Bourgain (2009) gave an explicit construction with sharper bounds.

The proof relies strongly on Spencer’s famous result :”Six standard deviation suffices”. (In fact, on a version of Spencer’s result by Lovett and Meka.)

Here is a reminder of Spencer’s theorem. (See this GLL post; We talked about it before and also on related results in Discrepancy’s theory, see this post and this one and also this one on Gowers’s blog.)

Spencer’s “Six standard deviation suffices” theorem: If are linear forms in variables with all , then there exist numbers such that for all .

The determinant thought of as a polynomial of variable is rather flat as far as upper bounds are concerned. (Moreover, if you restrict yourself to matrices where the rows are orthonormal then the determinant is constant.) I will be happy to learn about other rather-flat examples of explicit and algebraically significant polynomials. (Even on sub-varieties.)

Remark: Here is a paper by the same team on Erdős’ covering systems.

]]>The program has sessions on quantum science and technology, computer science and society, computation and the life sciences, and AI and autonomous systems. The keynote speaker is Amnon Shashua who will speak on the promise of machime learning and AI in transforming industries. Other speakers are: Charles Bennett, Adi Stern, Dorit Aharonov, and Thomas Vidick (quantum); Noam Nisan, Omer Reingold, Shafi Goldwasser, and Moshe Vardi (society); Aviv Regev, Ron Shamir, Uri Alon and Leroy Hood (biology), Sarit Kraus, Eva Tardos, Naftali Tishby, and Gal Kaminka (AI), and David Harel (closing remarks).

As for startling connections between the theory of computation and other areas of mathematics let me refer to the very recent post with the big news on the sunflower conjecture.

In the spirit of the coming Israeli elections let me promise to report more on our recent great Oberwolfach conference, on my visit to CERN, and to tell more about the sunflower progress, and about various combinatorics (and more) news I heard about.

**Update: **in partial fulfillment of my promises here are a couple of pictures from CERN and Oberwolfach.

ATLAS@CERN with my son in law Eran Shriker; The view at Oberwolfach.

]]>Richard Ehrenborg with a polyhedron

In the Problem session last Thursday in Oberwolfach, Steve Klee presented a beautiful problem of Richard Ehrenborg regarding the number of spanning trees in bipartite graphs.

Let be a bipartite graph with vertices on one side and vertices on the other side, and with vertex-degrees and .

**Ehrenborg’s problem:** Is it the case that the number of spanning trees of is at most

.

In words, the number of spanning trees is at most the product of the vertex degrees divided by the sizes of the two sides.

For example for the complete bipartite graph the number of the spanning trees is so equality hold. More genarally, equality holds for **Ferrer’s graphs**. Those are graphs where the vertices on the two sides are ordered and if is an edge so are and for and . See this paper of Richard Ehrenborg and Stephanie van Willigenburg.

Steve Klee and Matthiew Stamps found a new point of view for both the weighted and unweighted settings. (See this paper by Klee and Stamps and this paper.)

There is an analogous problem for general graphs where threshold graphs replace Ferrer’s graphs. This is related to a conjecture by Grone and Merris. (There are importat related works by Kelmans.) See this paper by Duval and Reiner and this paper by Martin and Reiner for related results and conjectures on spectrum of Laplacians for general simplicial complexes and shifted simplicial complexes.

I have learned that the Crone-Merris conjecture itself was settled by Hua Bai in 2012. The book Spectra of Graphs By Andries E. Brouwer and Willem H. Haemer has a chapter about the proof. The Duval-Reiner’s high dimensional conjecture is still open. (I thank Vic Reiner for some helpful information.) We also note an interesting 2012 paper Upper bounds for the number of spanning trees of graphs by Bozkurt, Ş. Burcu.

]]>Or Hershkovich reported on FB group “diggings in mathematics” about the solution of the Cartan-Hadamard conjecture by Mohammad Ghomi and Joel Spruck. The conjecture, one of the most famous conjectures in Riemannian geometry, is a far reaching extension of the classic isoperimetric inequality for general Riemannian manifolds.

**Here is the abstract:** We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $latex M^n, n\ge 2$, is bounded below by the volume of the unit sphere in Euclidean space $latex R^n$. This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in M, via Kleiner’s variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the Faber-Krahn and Sobolev inequalities to manifolds of nonpositive curvature.

The proof is quite involved (the paper is 80 pages long) and is based on extending Kleiner’s approch for the 3- and 4- dimensional case. Or Hershkovich raised on FB an even more general question related to isospectral relations based on higher homology.

We also note a 2013 paper by Benoît Kloeckner and Greg Kuperberg The Cartan-Hadamard conjecture and the little prince

A page from Kloeckner-Kuperberg’s paper, and related FB posts by Greg (below).

]]>(Written on my smartphone will expand it when reconnected to my laptop.) (Reconnected) Rather, I will write about it again in a few weeks. Let me mention now that while the old difficult and ingenious improvements stayed in the neighborhood of Erdos and Rado initial upper bound the new result is in the neighborhood of the conjecture! (And is tight for a certain robust version of the problem!)

Let me also mention that we discussed the sunflower conjecture here many times and polymath10 (wiki page, first post, last post) was devoted to the problem.)

**Update:** Let me mention an important progress on the sunflower conjecture from 2018 by Junichiro Fukuyama in his paper Improved Bound on Sets Including No Sunflower with Three Petals. I missed Fukuyama’s paper at the time and I thank Sasha Kostochka and Andrew Thomason for telling me about it now.

A few weeks ago I uploaded to the arXive a new paper with the same title “The argument against quantum computers“. The paper will appear in the volume: Quantum, Probability, Logic: Itamar Pitowsky’s Work and Influence, Springer, Nature (2019), edited by Meir Hemmo and Orly Shenker. A short abstract for the lecture and the paper is:

We give a computational complexity argument against the feasibility of quantum computers. We identify a very low complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explain why it will allow neither good-quality quantum error-correction nor a demonstration of “quantum supremacy.” Some general principles governing the behavior of noisy quantum systems are derived.

The new paper and lecture have the same title as my 2018 interview with Katia Moskvitch at Quanta Magazine (see also this post). Note that Christopher Monroe has recently contributed a very interesting comment to the Quanta article. My paper is dedicated to the memory of Itamar Pitowsky, and for more on Itamar see the post Itamar Pitowsky: Probability in Physics, Where does it Come From? See also this previous post for two other quantum events in Jerusalem: a seminar in the first semester and a winter school on The Mathematics of Quantum Computation on December 15 – December 19, 2019.

**A slide from a lecture by Scott Aaronson where he explains why soap bubble computers cannot solve the NP-complete Steiner-tree problem. Noisy intermediate scale quantum (NISQ) circuits are computationally much more primitive than Scott’s soap bubble computers and this will prevent them from achieving neither “quantum supremacy” nor good quality quantum error correcting codes. **(source for the picture)

**Low-entropy quantum states give probability distributions described by low degree polynomials, and very low-entropy quantum states give chaotic behavior. Higher entropy enables classical information. **

]]>

Paper ; blog post (end of July, 2019)

A collection of equiangular lines is a collection of lines so that the angles between every pair of lines in the same?

Here are two classical questions:

- What is the maximum number of equiangular lines in ?
- Given an angle what is the maximum number of equiangular lines in ? so that the angle between every two lines is

In 2000 Dom de Caen found for the first time an equiangular set of lines in space of size . (The crucial observation that one of the graphs in the Cameron–Seidel association scheme has a certain eigenvalue of large multiplicity. Prior to this construction, the largest sets had sizes of order . In de Caen’s example the lines have angles approaching 90 degrees, and question 2 for a fixed value of led to very different bounds. (For another result by Dom, see this post.)

There were some important progress on this problem by Igor Balla, Felix Dräxler, Peter Keevash, and Benny Sudakov, as featured in this Quanta Magazine article written by Kevin Hartnett. Zilin, Jonathan, Yuan, Shengtong, and Yufei finished off the problem in a clean and crisp manner, in a 10-page paper with a self-contained proof. On the way they proved the following very interesting theorem.

**Theorem:** A bounded degree graph must have sublinear second eigenvalue multiplicity.

Paper, blog post (end of June 2019)

**Abstract**: We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.

“Constant density in all tournaments” means that for some (and hence for all , every tournament with vertices has the same number of copies of . (On the nose!)

This result is related to the famous Sidorenko’s conjecture. Let me copy its description from the paper:

For undirected graphs, conjectures of Sidorenko and Erdős–Simonovits (commonly referred to as Sidorenko’s conjecture) say that for every bipartite graph , the -density in a graph of fixed density is minimized asymptotically by a random graph. Lately the conjecture has been proved for many families of though the conjecture remains open in general. In particular, the case is open.

**Yufei Zhao**

Let me describe briefly three additional posts on Yufei’s blog with two results from 2018 and one result from 2014.

These are two remarkable papers by the same team of researchers. The paper on the number of independent sets in a regular graph settled a famous 2001 conjecture by Jeff Kahn. An earlier breakthrough on the problem was made by Zhao in 2009 when he was an undergraduate student.

Recently I wrote a post Matan Harel, Frank Mousset, and Wojciech Samotij and the “the infamous upper tail” problem describing a major breakthrough on the infamous upper tail problem. Over the years I heard a lot of lectures and private explanation on upper tails and the remarkable related mathematics. I remember lectures by Kim and Vu from the early 2000’s and by Varadhan from ICM 2010 describing (among other things) the fundamental paper by Chatterjee and Varadhan, and later works by DeMarco and Kahn, and by Chatterjee and Dembo and Lubetzky and Zhao and others. But when I wrote the post I realized my knowledge is too sparse for giving a thorough description, and I decided not to wait and write a short post. This post by Yufei describes some of the history very nicely as well as the major papers by Eyal and Yufei from 2012 and 2014.

]]>Emmy Noether (left) Grete Hermann (right)

“In the fall of 2019 Dorit Aharonov, Gil Kalai, Guy Kindler and Leonid Polterovich intend to run a new one semester course (as a Kazhdan seminar) attempting to connect questions about noise and complexity in quantum computation, with ideas and methods about “classical-quantum correspondence”.that are well studied in symplectic geometry. The course will be highly research-oriented, and will attempt to teach the basics in both areas, and define and clarify the interesting research questions related to the connections between these areas, with the hope that this will lead to interesting insights. The course is oriented to grad students (and faculty), with reasonable background in mathematics, and with interest in the connections between mathematical and computational aspects of quantum mechanics. (See below for a full description.)”

The course will by on Sundays 14:00-16:00 in Ross building.

See also the post Symplectic Geometry, Quantization, and Quantum Noise from January 2013. (The seminar was initially planned to 2014 but some bumps in the road delayed it to 2019.)

The Mathematics of Quantum Computation

The 4th Advanced School in Computer Science and Engineering

Event date: December 15 – December 19, 2019

“We will be organizing a math-oriented quantum computation school in the IIAS at the Hebrew university. No prior knowledge on quantum will be assumed. The school will introduce TCS and math students and faculty, who are interested in the more mathematical side of the area, to the beautiful and fascinating mathematical open questions in quantum computation, starting from scratch. We hope to reach a point where participants gain initial tools and basic perspective to start working in this area. (See below for a full description.)

Organizers: Dorit Aharonov, Zvika Brakerski, Or Sattath, Amnon Ta-Shma,

**Main (confirmed) Speakers: **Sergey Bravyi, Matthias Christandl, Sandy Irani, Avishay Tal, Thomas Vidick, (1-2 additional speakers may be added later).

**Additional (confirmed) lectures will be given by: **Dorit Aharonov, Zvika Brakerski, and/ Or Sattath. (1-2 additional speakers may be added later).”

The Isreali Institute of Advanced Study hosted already a 2014 school about quantum information as part of its legendary physics series of schools, and also hosted QSTART in 2013.

In the fall of 2019 Aharonov, Kalai, Kindler and Polterovich intend to run a new

one semester course (as a Kazhdan seminar) attempting to connect questions about noise and complexity in quantum computation, with ideas and methods about “classical-quantum correspondence”.that are well studied in symplectic geometry. The course will be

highly research-oriented, and will attempt to teach the basics in both areas, and define and clarify the interesting research questions related to the connections between these areas, with the hope that this will lead to interesting insights.

The course is oriented to grad students (and faculty), with reasonable background in mathematics, and with interest in the connections between mathematical and computational aspects of quantum mechanics. Students who attend it will be awarded two N”Z after passing an exam. The goal of the course is to initiate and lead to new connections between the seemingly unrelated areas of quantum computation and symplectic geometry.

The topics will include:

– Introduction to quantum computation, quantum universality, quantum algorithms

and quantum computational complexity classes such as BQP and Quantum NP (QMA)

– quantum measurement and quantum noise explained using the standard quantum computational model.

– questions about quantum error correction and quantum noise – fault tolerance,

quantum error correcting codes, and the breakdown of robustness when the locality of

the noise does not hold.

– quantum measurement/quantum information (noise and speed limit) having classical counterparts, studied from the symplectic geometry perspective.

– Konsevich theorem and quantization,

– towards a the semi classical approximation of quantum computers.

Examples of questions we would like to initiate research on are:

1) what would be a semi classical model of quantum computation, and what would be

its computational power?

2) what is a good notion of complexity in a symplextic geometry computational model?

3) What can we learn from basic symplectic geometry results (such as non squeezing)

about the limitations on quantum computation in the semi classical limit?

4) Can noise in quantum computation be related in any way with the semi classical limit

of quantum computing systems?

5) can we learn anything about the possible noise models in quantum computers,

using our knowledge from symplectic geometry?

Hope to see you in the course!

On 15-19 December 2019, we will be organizing a math-oriented quantum computation school in the IIAS at the Hebrew university. No prior knowledge on quantum will be assumed. The school will introduce TCS and math students and faculty, who are interested in the more mathematical side of the area, to the beautiful and fascinating mathematical open questions in quantum computation, starting from scratch. We hope to reach a point where participants gain initial tools and basic perspective to start working in this area.

To achieve this, we will have several mini-courses, each of two or three hours, about central topics in the area. These will include quantum algorithms, quantum error correction, quantum supremacy, delegation and verification, interactive proofs, cryptography, and Hamiltonian complexity. We will emphasize concepts, open questions, and links to mathematics. We will have daily TA sessions with hands-on exercises, to allow for a serious process of learning.

There will be two rounds of registration. The first deadline is 23rd of August. If there is room, there will be another deadline sometime in October; please follow this page for further announcements.

Hope to see you this coming December!

]]>The cover of Avi Wigderson’s book “Mathematics and computation” as was first exposed to the public in Avi’s Knuth Prize videotaped lecture. (I had trouble with 3 of the words: What is EGDE L WONK 0? what is GCAAG?GTAACTC ? TACGTTC ? I only figured out the first.)

Avi Wigderson’s book “Mathematics and computation, a theory revolutionizing technology and science” is about to be published by Princeton University Press. The link is to a free copy of the book which will always be available on Avi’s homepage. (See also this re-blogged post here of Boaz Barak.)

One main theme of the book is the rich connection between the theory of computing and other areas (in fact, most areas) of mathematics. See also this self contained survey (based on Chapter 13 of the book) by Avi Interactions of Computational Complexity Theory and Mathematics, which in 27 pages overviews relations to number theory, Geometry, Operator Theory, Metric Geometry, Group Theory, Statistical Physics, Analysis and Probability, Lattice Theory and Invariant Theory. Of course, Avi himself is among the main heroes in finding many paths between mathematics and the theory of computing over the last four decades.

Another theme of the book and of several talks by Avi is that the theory of computing has revolutionary connections with many fields of science and technology. Again, this theme is present in the entire book and is emphasized in Chapter 20, which also appeared as a self contained paper “On the nature of the Theory of Computation (ToC).” Let me quote one sentence from Avi’s book that I propose for discussion. (For the complete quote see the end of the post.)

## The intrinsic study of computation transcends human-made artifacts, and its expanding connections and interactions with all sciences, integrating computational modeling, algorithms, and complexity into theories of nature and society, marks a new scientific revolution!

Of course, similar ideas were also expressed by several other prominent scientists, and let me mention Bernard Chazelle’s essay: The Algorithm: Idiom of Modern Science. (Feel free to give further examples and links in the comment section.)

I propose to discuss in the comment section the idea that the theory of computing offers a scientific revolution. Very nice cases to examine are the computational study of randomness and connections to statistics, connections with economy and connections with biology. Comments on the relations between the theory of computation and other areas of mathematics are also very welcome.

Avi’s concluding slide compared these three great theories of human understanding.

(Previous attempts of open discussions were made in the following posts on this blog: 10 Milestones in the History of Mathematics according to Nati and Me; Why is mathematics possible? (and a follow up post); When it rains it pours; Is it Legitimate/Ethical for Google to close Google+?; An Open Discussion and Polls: Around Roth’s Theorem; Is Mathematics a Science?)

Avi promotes the idea of the central place of the theory of computing in his talks and writings

And at the same time he is also humorously skeptical about it. (And mainly emphasizes that his far reaching claim requires careful discussion and ample evidence.)

The Theory of Computation is as revolutionary, fundamental and beautiful as major theories of mathematics, physics, biology, economics… that are regularly hailed as such. Its impact has been similarly staggering. The mysteries still baffling ToC are as challenging as those left open in other fields. And quite uniquely, the theory of computation is central to most other sciences. In creating the theoretical foundations of computing systems ToC has already played, and continues to play a major part in one of the greatest scientific and technological revolutions in human history. But the intrinsic study of computation transcends man-made artifacts. ToC has already established itself as an important mathematical discipline, with growing connections to nearly all mathematical areas. And its expanding connections and interactions with all sciences, naturally integrating computational modeling, algorithms and complexity into theories of nature and society, marks the beginning of another scientific revolution!

- Avi’s talk Scientific revolutions, ToC and PCP at the Tel Aviv PCP meeting and an interview of Avi by Alon Rosen.
- A talk by Avi on the Stepanov method
- The recent works on polytopes arising from moment maps and related optimization problems and algorithmic aspects. Avi’s Knuth prize videotaped lecture; Avi’s lecture Complexity, Optimization and Math (or, Can we prove that P != NP by gradient descent?) in the recent conference honoring Bill Cook. (I plan to come back to this fascinating topic.)
- An essay by Oded Goldreich and Avi Wigderson (essentially from 1996) “The theory of computing – a scientific perspective.”

The volume of comments in the first decade of this blog was modest. I recently read, however, wordpress’s advice on how to reply to blog comments like a pro.

And finally, EGDE L WONK 0 is 0-knowledge.

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