First third of my ICM2018 paper – Three Puzzles on Mathematics, Computation and Games. Corrections and comments welcome

Update: Here is a combined version of all three parts: Three puzzles on mathematics computations and games. Thanks for the remarks and corrections. More corrections and comments welcome.

I have a very strict December 20 deadline (self-imposed, I missed the official one)  for my ICM2018 paper. I plan to talk about three puzzles on mathematics, computation and games, and here is a draft of the first third. Corrections and comments are very welcome! This part is around the simplex algorithm for linear programming.

Preview: The solution by Keller and Lifshitz to several open problems in extremal combinatorics

Peter Frankl (right) and Zoltan Furedi

The news

A new paper by Nathan Keller and Noam Lifshitz settles several open problems in extremal combinatorics for wide range of parameters. Those include the three problems we mention next.

Three central open problems in extremal combinatorics are

The 1975 Erdős-Sós forbidding one intersection problem, asks for the maximal
size of a k-uniform hypergraph that does not contain two edges whose intersection
is of size exactly t−1;

The 1987 Frankl-Füredi special simplex problem asks for the maximal
size of a k-uniform hypergraph that does not contain the following forbidden configuration: d+1 edges $E_1,\dots ,E_{d+1}$ such that there exists a set $S= \{v_1,\dots , v_{d+1}\}$ for which $E_i\cap S = S \backslash \{v_i\}$ for any i and the sets {Ei \ S} are pairwise disjoint.

The 1974 Erdős-Chvátal’s simplex conjecture proposes an answer for the maximal
size of a k-uniform hypergraph that does not contain a d-simplex. Here, a d-simplex is a family of d+1 sets that have empty intersection, such that the intersection
of any d of them is nonempty.

All these questions are related to the Erdős-Ko-Rado theorem (see this post and many others). For $t=1$, two edges whose intersection is of size exactly t−1 are just two disjoint edges and so is a 1-simplex and a special 1-simplex.

The papers by Keller and Lifshitz and by Ellis, Keller, and Lifshitz

The paper by Keller and Lifshitz settles all these problems for a wide range of parameters! A subsequent paper by David Ellis, Nathan Keller, and Noam Lifshitz extends (among various other results) the range of parameters for the  Erdős-Sós problem even further.

Plans for posts

Michel Deza

I have an ambitious plan to devote two or three posts to these developments (but not before January). In the first post I will give some general background on Turan’s problem for hypergraphs and the new new exciting results, Then (perhaps in a second post) I will give little background on two major methods, the Delta-system method initiated by Deza, Erdos and Frankl and used in many earlier papers mainly by Frankl and Furedi, and the Junta method initiated by Friedgut which is used (among other ingredients) in the new paper.  Then I will write about the new results in the last post.

Paul Erdos, Thomas Luczak, Ehud Friedgut, and Svante Janson

Posted in Combinatorics, Open problems, Updates | | 1 Comment

Kazhdan’s Basic Notion Seminar is back!

The “basic notion seminar” is an initiative of David Kazhdan who joined the Hebrew University math department  around 2000. People give series of lectures about basic mathematics (or not so basic at times). Usually, speakers do not talk about their own research and not even always about their field. My first lecture series around 2004 was about computational complexity theory, and another one on extremal combinatorics developed into a series of posts (I,II,III,IV,VI). Since then I talked in the seminar about various other topics like convex polytopes, generalization of planar graphs, and Boolean functions. The seminar did not operate for a few years and we were all very happy to have it back last spring!

Helly-type theorems

I just finished two lectures on Helly-type theorems in the basic notions seminar. which is the topic of my doctoral thesis. (I am interested in Helly type theorems since I was an undergraduate student.) Recently, Helly-type theorems where involved in the study of high dimensional expanders and other topics in high dimensional combinatorics. Alex Lubotzky and Tali Kaufman are running now a special year at the IIAS devoted to high dimensional combinatorics so I thought it would be nice to devote a basic notion seminar to this topic.

My first talk followed quite closely these two posts (I,II). Let me devote this post to the “cascade conjecture” which is a generalization of Tverberg’s theorem.

Tverberg’s theorem (1965): Let $v_1, v_2,\dots, v_m$ be points in $\mathbb R^d$, $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap _{j=1}^rconv (v_i: i \in S_j) \ne \emptyset$.

Given a set $X$ of points  in $\mathbb R^d$, $X= \{ x_1,x_2,\dots ,x_n \}$ we let  $T_r(X)$ be the set of  points in $\mathbb R^d$ which belong to the convex hull of $r$ pairwise disjoint subsets of $X$. (We may allow repetitions among the elements of $X$.) Thus, $T_1(X)$ is just the convex hull of $X$.

Let $\bar t_r(X)=\dim T_r(X)+1$.

Radon’s theorem:  If $\bar t_1(X) < |X|$ then $\bar t_2(X)>0$.

Radon theorem is a simple consequence of the fact that $d+2$ points in an affine space of dimension $d$ are affinely dependent. (Note that $t_1(X)$ is one plus the dimension of the affine span of $X$.)

It seems that the following conjecture requires some “higher linear algebra”

Conjecture 1: If $\bar t_1(X)+ \bar t_2(X) < |X|$ then $\bar t_3(X)>0$.

Conjecture 1 is wide open. It is a special case of the following more general conjecture

Conjecture 2 (The Cascade Conjecture): If $\bar t_1(X)+ \bar t_2(C) +\cdots+\bar t_r(X) < |X|$ then $\bar t_{r+1}(X)>0$.

Another formulation of Conjecture 2 is

The Cascade Conjecture: $\sum_{i=1}^{|X|} \bar t_i(X) \ge |X|$.

Of course, the cascade conjecture implies Tverberg’s theorem since given $(r-1)(d+1)+1$ points in $\mathbb R^d$, we have that $\sum_{i=1}^{r-1} \bar t_i(X) \le (d+1)(r-1) < |X|$, and therefore the conjecture implies that $T_r(X) \ne \emptyset$.

For the 5-point configuration on the left $\bar t_1=3$ and $\bar t_2=2$. For the configuration on the right $\bar t_1=3$ and $\bar t_2=1$. Indeed also $\bar t_3=1$

There are two facts about the cascade conjecture that are separately quite innocuous but combined are mind blogging. The first fact is that the conjecture was made in 1974, namely 43 years ago.  Continue reading

Posted in Combinatorics, Convexity, Open problems | | 4 Comments

I just received an advanced copy of my very first book: “Gina Says: Adventures in the Blogsphere String War” published by Word Scientific. It is a much changed version compared to the Internet version of 8 years ago and it contains beautiful drawings by my daughter Neta Kalai. How exciting!

Here is the World Scientific page, and here is its Amazon page (paperback)  and Amazon page (Kindle).

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Itai Benjamini: Coarse Uniformization and Percolation & A Paper by Itai and me in Honor of Lucio Russo

Here is another video of a smashing short talk by my dear friend Itai Benjamini with beautiful conjectures proposing an important new step in the connection between percolation and conformal geometry.

Here is the link to Itai’s original paper Percolation and coarse conformal uniformization. It turns out that the missing piece for proving the conjecture is a very interesting (innocent looking) theorem of Russo-Seymour-Welsh type.

This conjecture appears also in a paper Itai wrote with me Around two theorems and a lemma by Lucio Russo. Our paper  is part of a special issue of Mathematics and Mechanics of Complex Systems (M&MoCS),  in honor of Lucio Russo, in occasion of his retirementis.  In addition to the conjectural Russo-Seymour-Welsh type theorems, we also present some developments, connections, and problems related to Russo’s lemma and Russo’s 0-1 law.

Lucio Russo

After-Dinner Speech for Alex Lubotzky

An after-dinner speech given on November  9, 2016*, on Alex Lubotzky’s 60th birthday conference: 60 faces to Groups.

As we all realized waking up this morning, today is a historic day*. It is the dinner day for Alex 60th‘s birthday conference, and I am thrilled to say a few words about Alex and to Alex. First let me send my love and my wife Mazi’s love to Alex and Yardena and all the family. A 60th birthday is a somber event, especially so when the birthday boy is younger than you, so I will be serious and try to move you and bring tears to your eyes,  rather than try to be funny and make you laugh. (This is a new experience for me.) I will leave the many funny stories I can tell about Alex to his 70th birthday. To be on the safe side I will write them down soon so not to forget them.

Early days

I don’t know how and when Alex and I first met. Perhaps it was in 1970 or 1971 when Ron Livne (at the time a high school student like me and Alex) invited me to Tel Aviv to their “math club”.   I remember meeting Alex in his home in the late 70s when we were both soldiers and he worked on his thesis and on the beautiful paper merging many different ideas and techniques using the “Golod-Shafarevich” theorem, along with Euler’s theorem for solving a well known problem. The paper appeared in the top journal of mathematics, the “Annals of Mathematics”, a rare achievement.

In 1985 I joined the Hebrew University as a faculty member, and Alex was already a professor there.  When I joined the department I was in awe of the extraordinary group of people, not only in terms of their mathematical achievements, but also when their scholarly nature, their personalities, activities and interests beyond math. This feeling paralyzed me and I hoped it would pass in a couple of months. It didn’t, and even today on the verge of retirement, this is still my feeling and it extends also to the amazing group of people we hired since.

Alex is a great mathematician

I should not tell you how  extraordinarily great a mathematician Alex is, and the mathematicians here can describe Alex’s work better than I can.  But to his family I want to mention a few “buzz words” reflecting some of Alex’s work:  “Ramanujan graphs”, and “powerful groups,” “subgroup growth” and the “Tau property” and the “Sandwich technique” (juggling between Tao and Kazhdan’s T-poperty) , “high dimensional expanders” (I taught with Alex four times a course on this topic. At the end I may understand what it’s about) and many, many more concepts.

Alex is a great friend

both to his close friends and to people who are more distant.  In many cases people enjoy listening to Alex’s life-wisdom (this was already the case when he was in his twenties, even when they were much older) and advice, and confide in him their most intimate experiences and dilemmas.  In these cases, the friendship relations are not quite symmetric, can we characterize what they are? It is not like the relationship you have with a father. Usually you do not want to hear words of wisdom about life from a parent and you certainly don’t want to share your intimate matters with them. It is more like the relation you have with a loving… grandfather. So you can say that Alex’s form of friendship was an early practice for grandfatherhood. Alex practiced being a grandfather all his life and now he can use his skills with his 16 and counting grandchildren. Alex is a great champion of grandchildren and I suppose he is starting to pressure his children to have grandchildren of their own about now.

Alex is an idealist

This is why he and Yardena settled in Efrat, this is why he worked hard to save and advance the “Israel Journal of Mathematics”,  and this is why he went into politics.  Alex joined politics in 95 and the Israeli parliament  after the 99 elections. (A major upset at the time. Commentators and polls predicted victory for one candidate but the other candidate won. This happens from time to time*.)  I remember talking with Hillel about hosting a farewell party for Alex. Hillel was surprised that a mathematical genius like Alex would join the parliament and I thought he would never come back.

Alex’ many virtues and two weaknesses as a leader

Alex had a lot of virtues and two weaknesses as a leader and a politician. Alex was very smart and he could understand the fine details as well as the large picture. He befriended and connected well with other politicians from all parties. In politics and administration just like in mathematics, he could identify the important problems and he went head on to tackle them. He is famous for the Beilin-Lubotzky convention for relations between orthodox and non-orthodox Jews and many other related issues. One of his weaknesses is that he is not malicious and does not appreciate the joy of malice.  As in mathematics, also in politics and in academic administration, he could identify the important problems and he went head on to tackle them. He was also the founder  of the committee for the status of women in the Israeli parliament.  [Here, in the lecture,  Alex corrected me and said that I am exaggerating and he was not the founder. But I made the mistake on purpose to make another point.]  …  Right, Alex was an active member of this committee, not the founder. It was founded 4 years earlier by Yael Dayan (Moshe Dayan’s daughter). This brings me to the second weakness of Alex as a politician. He is too honest. Most politicians and also a few mathematicians will not correct an undeserved credit.

Parents

Alex comes from a family of holocaust survivors and to a large extent regards himself as one. In my impression this accounts primarily for his immense gratitude for what he was blessed with, including the sweet life of a mathematician.  It is also related to a principle he has of never wasting food and never wasting time. We shared apartments quite a few times and he also refused to throw away food. (About Alex’s cooking wait 10 years.) Once I visited him and Yardena when they were renting a house filled with spoiled non kosher food that had been left by the owner and I secretly threw away most of it.  A  legacy from Alex’s father Issar is never to waste time. (I mean time when one is awake, this does not include sleeping, we already heard stories about Alex’s sacred afternoon sleep.) “A time being wasted is like a time being dead,” Alex once told me when he caught me playing a computer game at the computer room at Yale. (I wonder how this statement could be scientifically verified.) Since then I was careful to quickly move to a different screen when Alex entered the computer room.

You can read about the remarkable history of Alex’s  parents in WW2 in  Asael’s soon to be published second book.

Asael

And I want to conclude with mentioning Alex as a father to Asael, and how Alex and Yardena handles Asael’s grave injury in 2006. I remember Mazi and I had dinner with Alex and Yardena and Avi and Edna a day or two before Alex and Yardena got the news that Asael was injured and that  was in critical condition. This was a terrible ordeal and a difficult struggle, mainly for Asael but also for  his parents and the entire family. Asael, now a medical doctor as well as a researcher,  wrote his first book about his experiences, and has recently been translated to English.

Let me send again love from Mazi, myself and my children to Alex, Yardena and the family and wish Alex many more years. I haven’t talked about many joint adventures from the Jordan river to the Amazonas.   I promise to be funny ten years from now.

*At that day November 9, 2016, we woke up in the morning realizing that Donald Trump was the newly-elected President of the United States. Efim Zelmanov mentioned in his after dinner speech that Alex did not share the universal confidence that Trump will not be elected.

Boaz Barak: The different forms of quantum computing skepticism

Boaz Barak  wrote a nice essay which I reblog to you below about skepticism of quantum computers. His essay is centered around a description of four scenarios (or possible “worlds” in the style of Russel Impagliazzo) about the matter. Boaz’ world are called Superiorita (quantum computers are realistic and computational superior), Popscitopia  (Quantum computers are realistic and can solve NP-complete problems), Skepticland  (quantum computers are not realistic), and Classicatopia (quantum algorithms can efficiently be simulated on a classical computers)

Apropos Russel’s universes, Boaz told me about a new cryptography “world” called Obfustopia that can be found (along with some modification of the original worlds) in Boaz’ survey paper. I heard about it and about the related important LWE (learning with errors) problem also from Alon Rosen.

Boaz’ essay also includes Boaz’ own optimistic personal view and also some very brief critique of my skeptical stance. Boaz conclude his personal opinion with:

The bottom line is that, as far as I can tell, Superiorita is the most beautiful and evidence-supported world that is currently on offer.

Boaz’ main argument for his point of view is:

…as far as I can tell, these engineering difficulties are not fundamental barriers and with sufficient hard work and resources the noise can be driven down to as close to zero as needed.

This is indeed the crux of matters and my analysis gives good reasons to think that Boaz is not correct.  The different opinions are described in the pictures below and the crux of matters is “can we cross the green line?”

Common expectations

It is a common belief that by putting more effort for creating qubits the noise level can be pushed down to as  close to zero as we want. Once the noise level is small enough and crosses the green line, quantum error correction allows logical qubits to reduce the noise even further with a small amount of additional effort . Very high quality topological qubits are also expected.

The picture suggested by my analysis

My analysis (see here and here and here) gives good reasons to expect that we will not be able to reach the green line and that all attempts for logical and topological qubits will yield bad quality qubits.

Finer Worlds

One can make a finer division of the superiorita world, and, in my view, the three most relevant worlds are the following:

Quantopia  – The model of quantum circuits is the correct description of local quantum systems in nature and therefore universal quantum computation is possible. Quantum supremacy and robust quantum information are present and perhaps even ubiquitous in the physical world. Quantum systems, quantum information and computation are analogous to classical systems, classical information and computation. Quantum error correction is analogous to classical error correction: an important engineering tool but not the thing that makes the qualitative difference whether scalabale communication or computation is possible. (This represents Boaz’ beliefs expressed in his post and his later comments.)

Superiorita (Quantum noise below the threshold.) Quantum systems are inherently noisy. Time-dependent quantum evolutions necessarily interacts with the environment and are therefore  noisy. The model of noisy quantum circuits is the correct description of local quantum systems in nature. Quantum error-correction  shows that small islands representing noiseless quantum circuits can be created and may also exist in nature. Hence quantum computation is possible. (This is perhaps the most common view among researchers in quantum information.)

Skeptica– (Quantum noise above the threshold.) Quantum systems are inherently noisy. Time-dependent quantum evolutions necessarily interacts with the environment. The model of noisy quantum circuits is the correct description of local quantum systems in nature. The noise level cannot be pushed below the level allowing quantum error-correction and quantum fault-tolerance. Hence quantum computation is not possible. Quantum supremacy is not demonstrated in nature and cannot be demonstrated in the laboratory. (This is were I stand).

There are, of course people who are skeptical about quantum computers from other reasons like the young Boaz who simply did not like physics. There are people who for various reasons are skeptical of quantum mechanics.

Quantopia  suggests that “quantum supremacy” and “robust quantum information” will be present and in fact ubiquitous in the physical world, while under ” superiorita” quantum supremacy represents rare a islands inside a large “decoherence desert” (as Daniel Gottesman referred to it in his beautiful picture portrayed in this post.) The difference between quantopia and  superiorita is relevant to Scott Aaronson’s hope that quantum computers promise  trillion dollars industry via simulations. While additional computing power is always welcome this idea is less promising in the  superiorita scenario where the usefulness of quantum computers to simulation is less clear.

Anyway, without even further ado, here is Boaz’ piece.

Quantum computing is one of the most exciting developments of computer science in the last decades. But this concept is not without its critics, often known as “quantum computing skeptics” or “skeptics” for short. The debate on quantum computing can sometimes confuse the physical and mathematical aspects of this question, and so in this essay I try to clarify those. Following Impagliazzo’s classic essay, I will give names to scenarios or “potential worlds” in which certain physical or mathematical conditions apply.

Potential worlds

Superiorita is the world where it is feasible to build scalable quantum computers, and these computers have exponential advantage over classical computers. That is, in superiorita there is no fundamental physical roadblock to building large quantum computers, and hence the class BQP is a good model of computation that is physically realizable. More precisely, in superioriata the amount of resources (think…

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Bálint Virág: Random matrices for Russ

You can watch now all the videos for Russfest Elegance in Probability. Many great talks! I just watched Bálint Virág‘s lecture “Random matrices for Russ”. Highly recommended.

Test Your Intuition 33: The Great Free Will Poll

Free will is defined (following Wikipedea) as the ability of humans to choose between different possible courses of action unimpeded. But you may take your favorite definition of free will.  Philosophers (and others) have debated the definition of “free will” and the question if humans have free will for over two thousands years.

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The joyous lion of theoretical computer science

(Read here a story about Avi and me related to the above picture.)

Boaz Barak is reporting on Avi Wigderson’s book Mathematics and Computation which is on Avi’s homepage. I read through an earlier version and it is highly recommended! Let me also recommend Avi’s earlier survey on interaction between CS and math.

Avi Wigderson is one of the most prolific and creative theoretical computer scientists (in fact, he is one of the most prolific and creative scientists, period). Over the last several years, Avi had worked hard into distilling his vast knowledge of theoretical computer science and neighboring fields into a book surveying TCS, and in particular computational complexity, and its connections with mathematics and other areas.

I’m happy to announce that he’s just put a draft of this upcoming book on his webpage.

The book contains a high level overview of TCS, starting with the basics of complexity theory, and moving to areas such as circuit complexity, proof complexity, distributed computing, online algorithms, learning, and many more. Along the way there are interludes about the connections of TCS to many mathematical areas.

The book is highly recommended for anyone, but in particular for undergraduate and beginning graduate students that are interested…

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