The title of the Google 2019 paper was “Quantum supremacy using a programmable superconducting processor”. As the readers may remember, the supremacy claim has largely been refuted by several research groups; see this post, this one, and this one. The calibration process of the Google experiment weakens the claim of a “programmable processor”, and some of our findings from the second paper as well as a few of the findings from the new paper further weaken this claim (see below).

Abstract:In October 2019, Nature published a paper [5] describing an experimental work that was performed at Google. The paper claims to demonstrate quantum (computational) supremacy on a 53-qubit quantum computer. Since then we have been involved in a long-term project to study various statistical aspects of the Google experiment.

In [32] we studied Google’s statistical framework that we found to be very sound and offered some technical improvements. This document describes three main concerns (based on statistical analysis) about the Google 2019 experiment. The first concern is that the data do not agree with Google’s noise model (or any other specific model). The second concern is that a crucial formula for a priori estimation of the fidelity is surprisingly simple and seems to involve an unexpected independence assumption, and yet it gives very accurate predictions. The third concern is about surprising statistical properties of the calibration process

1. There is a large gap between the samples of the Google experiment and the Google noise mode, and any other specific noise model. The gap between the empirical distribution and the model is asymmetric.

2. There are large fluctuations of the empirical behavior which are not understood. Consequently, there is evidence that the distance between the Google noise model and uniform distribution is smaller (when the number of qubits is n > 16) than the distance between the experimental samples and the Google noise model.

3. The empirical behavior of the samples is not stationary.

4. While the empirical distribution is not stationary the XEB fidelity is stable along the samples. Moreover, “high energy events” that lead to abrupt increase in errors that are reported for later experiments with the Sycamore quantum computer cannot be detected in the 2019 quantum supremacy experiment.

5. The predictive power of Formula (77) for the XEB fidelity estimates is statistically surprising: the subsumed independence between components of systems such as quantum computers, which are known to be sensitive to noise and errors caused by interactions with their environment, is striking.

6. The systematic bias of the predictions of Formula (77) for patch circuits seems statistically surprising, and the Google explanation is not convincing.

7. The behavior of the fidelities of the two patches for patch circuits is very different; This appears to be in tension with Formula (77). ( We were not yet provided with the data needed to check this matter.)

8. The success of the experiments fully depends on the very large effect of the calibration adjustments. There are large differences between the effects of the calibration adjustments for different 2-gates, and even for different appearances of the same 2-gate.

9. The calibration adjustments are surprisingly effective, especially given the stated local nature of the calibration. Mathematically speaking, we witness a local optimization process reaching a critical point of a function depending on hundreds of parameters.

Scrutinizing a scientific work necessarily involves punctiliousness and nitpicking, but there are several issues that we find of general interest.

(a) What is the statistical methodology for analyzing samples obtained from noisy quantum computers and for finding appropriate models to describe the empirical data?

(b) We suggest that the statistical independence assumption in a certain predictive model is very surprising. What could be the scientific framework and methodology to study this matter?

(c) We find it surprising that a local optimization process (namely, a process that separately optimizes each variable) of a function of many variables, reaches a critical point. What could be further tools to study this matter?

(d) What are the tools to study if an empirical behavior is non-stationary and perhaps even inherently unpredictable?

(e) What is the appropriate methodology and ethics for scrutinizing major scientific works, and how is it possible to bridge the gap between theoreticians (like us) and experimentalists?

We are in the process of writing a paper on readout errors, gate errors, and Fourier expansion. On the theoretical side, we will study the effect of gate errors on the Fourier expansion, which is of interest in and of itself and could serve as some sanity test for various aspects of the Google 2019 experiment. (The effect of readout errors is well understood – it is essentially the noise operator that I have been studying extensively since the mid-1980s.) On the technical side, this will be our first work where we use simulators for noisy quantum circuits, and we currently use both the Google and the IBM simulators. Then, we may apply some of our statistical tools at other NISQ experiments, and will even try to reproduce, using an IBM quantum computer, a certain random circuit experiment with 6 qubits.

There is a large gap between the samples of the Google quantum supremacy experiments and the Google noise model. In fact, the samples are far away from any noise model we are aware of. There is evidence that the distance between the Google noise model and uniform distribution is smaller (when

the number of qubits is n > 16) than the distance between the experimental samples and the Google noise model. We studied other properties of the empirical distribution like its behavior in different scales, its non-stationary nature, and its Fourier behavior, and there is more to be done mainly for data coming from other NISQ experiments and data from simulators of noisy circuits.

The remarkable predictive power of Formula (77) is statistically surprising: the subsumed independence between components of the quantum computer, is striking. The close agreement of the experimental XEB fidelities between the patch circuits and full circuits shows an unexplained systematic deviation from the predictions of Formula (77). On the other hand, confirmations [13, 23] and replications [33, 35] lend support to the claims in the Google paper. There are various matters that remain to be explored. For example, (i) using simulations of noisy circuits, the magnitude of the difference between the two sides of Formula (77) (Gao et al. [11]) could be estimated, and (ii) the individual values in Formula (77) could be used to study the different XEB fidelities of the two patches in patch circuits.

The calibration process accounts for systematic errors for 2-gates and applies certain adjustments to the definition of the circuits. These adjustments are local, namely the adjustments for a 2-gate involving qubits x and y primarily depend on outcomes for 1- and 2-circuits on these qubits. Some statistical findings regarding the calibration process are: (i) The effects of the calibration is large even for a single 2-gate; (ii) there is a large difference between the effect for different 2-gates and even different appearances of the same 2-gate, and; (iii) the effectiveness of the 2-gate calibrations is remarkable. We note that these findings enhance the tension between the calibration process and Google’s claim for a “programmable quantum computer.” The effectiveness of the calibration process is especially surprising in view of the local nature of the calibration: mathematically speaking, we witness a local optimization process reaching a critical point of a function depending on hundreds of parameters.

**The remarkable effectiveness of the calibration.** The effect of removing the calibration for the kth 2-gate of a circuit for the first Google file (file 0) with n = 12. Note that the same 2-gate occurs periodically along the circuit, indicated by the vertical black dashed lines. Here we remove all ingredients of the calibration: both the 1-gate rotations and 2-gate adjustments.

**The remarkable effectiveness of the calibration II.** The effect of removing the 2-gate adjustments involving the kth 2-gate of a circuit. (The same 2-gate occurs periodically along the circuit.) The 2-gate involving the qubits (3,3) and (3,4) has consistently large effect.

To make matters clear, the calibration is not about tightening the screws in Sycamore; rather, it is about change in the program. We can think about the calibration process as a change in the model that would greatly reduce certain systematic forms of noise. For example, if we discovered that a certain 1-gate that is supposed to apply a 90-degree rotation systematically performs an 80-degree rotation, rather than changing the engineering of the 1-gate, we would change the definition of the circuit.

“We laid the dry facts and findings, and we let the readers make their own interpretation, or rather take note of our concerns and wait for more experimental data from future experiments.”

The title of the Google 2019 paper was “Quantum supremacy using a programmable superconducting processor”. As we mentioned, the supremacy claim has largely (but not fully) been refuted. There are also doubts regarding the claim that the Sycamore 2019 experiment represents a “programmable processor” as the calibration process and other matters weaken this. (This was pointed out in a comment from October 2019 by Craig Gidney from the Google team and also a few months later by a commentator “Till” . Till’s comment led to interesting discussion regarding the nature of the calibration, which was earlier believed by many to represent physical changes in the device.)

Some findings of our papers further weaken the Google claim for “programmable device”. The Google paper describes about 1000 experiments on various circuits but, as it turns out, all these experiments depend on the random choices made for the largest ten circuits, and this fact is also in contrast with the “programmable” claim. (It was quite possible to choose a different random circuit in every case.) A related concern is that improvements of the calibration process were interlaced with the experiment, and that the last minute calibration procedure for the EFGH circuits represented a substantial improvement. We note that while the general principles of the calibration process are publicly available, the precise details are a commercial secret. Of course, concerns regarding the Google calibration process may reflect on other Sycamore experiments.

We wrote two earlier papers:

- Y. Rinott, T. Shoham, and G. Kalai, Statistical Aspects of the Quantum Supremacy Demonstration, Statistical Science (2022)
- G. Kalai, Y. Rinott and T. Shoham, Google’s 2019 “Quantum Supremacy” Claims: Data, Documentation, & Discussion (see this post.)

The question of appropriate methodology, ethics, and culture for scrutinizing major scientific works is related to the replication crisis that we mentioned in an earlier post. There we described our policy regarding data requests. Our experience was overall rather positive. (Things went rather slowly but we were slow as well.) We still did not get the individual terms of Formula (77) (namely, the error rate for individual 1-gates and 2-gates) but the Google team promised to try to push toward getting this information.

(From our paper:) “A few months after the publication of the Google paper we initiated what has become a long-term project to study various statistical aspects of the Google experiment and to scrutinize the Google paper. This is a good place to mention that Google’s quantum supremacy claim appeared to refute Kalai’s theory regarding quantum computation ([15, 16, 18]) and Kalai’s specific prediction that NISQ systems cannot demonstrate `quantum supremacy.’ This fact influenced and may have biased Kalai’s assessment of Google’s quantum supremacy claim. (Recent improved classical algorithms have largely refuted Google’s quantum supremacy claim and therefore the Google results no longer refute Kalai’s theory.)”

For my argument see this post and this one.

Zooming in on the empirical frequency of bitstrings with amplitudes between the median and the 0.55 quantile. The left plot is the empirical occurrences of the bitstrings in Google file 0, n=12. The right plot is based on a simulation with φ = 0.3862. The red line describes the expected number of bitstrings of the Google noise model, and the blue dashed line is 3 standard deviations from the expectation. (We plan to test if these fluctuations are present in samples from IBM quantum computers.)

Comparing the two halves of the Google samples: the black vertical lines are the ℓ1 distance of the occurrences of bitstrings when we partition the samples into two halves according to the sampling order. The histograms give the ℓ1 distances between the occurrences of bitstrings for random partitions of the bitstrings into two halves. Drifts in the fractions of ones. We divided the 500,000 bitstrings into 250 groups of 2,000 bitstrings each, according to the sampling order. For each group calculated the fraction of bitstring having a “1” bit in some place in the bitstring. The Figure shows the fractions of ones in locations 11 and 12 for one circuit (file 0) with n = 12. (The red lines are linear regression fits to the data points.) The trend is consistent along the different circuits and is different for different locations in the bitstrings.

A histogram of differences between the empirical distribution and the values given by the Google noise model, n = 12, Google file 0, φ = 0.3701. What can explain the apparent asymmetry in the gaps between the empirical distribution and the model? Is an explanation at all necessary?

]]>(Click to enlarge.)

- Prologue: “Can we sleep soundly at night?” Meeting Ephraim Halevi (former head of the Israeli Mossad) in 2007.
- Israel and CERN, an evening in honor of Eliezer Rabinovici: The story of how Israel joined CERN is an intriguing story that involves science, academic politics, real politics, money, and diplomacy of various kinds. The evening was quite fascinating with interesting lectures by physicists and diplomats. (Click for the video; some of it was over my head.)
- Ephraim Halevi’s lecture on Israel, CERN, and more. Test your knowledge: did you ever hear about the organizations Pugwash and Global Zero? And do you know who Joseph Rotblat and Frank von Hippel are?
- Reproducibility Crisis zoom conference organized by Sergey Frolov
- Giving quantum talks at the German Israeli quantum academy, and at physics colloquia at PI and Rutgers
- Max the Demon a
**physics comics**by Assa Auerbach and Richard Codor - Epilogue: “Can we sleep soundly at night?” Ephraim Halevi’s 2023 answer.

Two speakers at the evening celebrating Israel’s admission to CERN: Halina Abramowicz and Shikma Bressler

This post is about things around physics where the main topic is an evening at the Israeli Academy of Science honoring my friend Eliezer Rabinovici who is now the president of CERN.

In 2007 we had a special semester at the (Israeli) Institute for Advanced Studies, and one evening the director, Eliezer Rabinovici, hosted the director of the sister IAS at Princeton, Peter Goddard (whose nickname, as we were told, was “God” ) to a party. I was hanging with Imre Barany and a few minutes after we were introduced to Goddard, we saw him again in the crowd:

“How do you enjoy your stay in Israel” I asked Goddard

“I like it” was the answer “but I am an Israeli!”

Looking more carefully at the man we were talking to, I realized my mistake.

“Imre,” I said with enthusiasm “please meet the former head of the Mossad (“Mossad” is the national intelligence agency of Israel) and the current VP of the Hebrew University: Ephraim Halevi! And this is Professor Imre Barany from the Hungarian Academy of Science!” (“actually the question ‘how do you enjoy your stay in Israel` is quite appropriate for a Mossad guy as well,” I thought.)

Imre and I were both very excited, engaged in a small humorous chat with Halevi.

At the end I asked

“So, can we sleep soundly at night?”

“Yes you can,” Halevi answered, “There are people who make sure of it.”

A few weeks ago I met Halevi at the “Israel joining CERN” evening. He gave an interesting lecture that I will mention shortly and I asked him again if we can sleep soundly at night. You can find his answer at the end of this post.

Ephraim Halevi, Petter Goddard, and Imre Barany

The purpose of the evening at the Israeli Academy of Science and Humanities was to honor Eliezer Rabinovici, an old friend of mine who was replaced by Mark Karliner as the representative of Israel in CERN. Rabinovici is still connected to CERN as the president of CERN. The lectures were interesting, although some of them required knowledge of the Israeli physics community or of physics that I don’t have.

Five more speakers at the evening celebrating Israel and CERN: Eliezer Rabinovici, Peter Jenni, Rafi Barak, Giora Mikenberg, and Mark Karliner

Ephraim Halevi served in the Israeli Mossad for many years and was the head of the organization between 1998 and 2002. The lecture’s title was “Track two and track three in the toolbox over the years.” I still don’t understand what “track two” and track three” refer to :). Halevi talked a little about the diplomacy (and his own part) behind Israel joining CERN, and about SESAME, but large parts of Halevi’s lecture were about one of his main activities since he retired from civil service: acting against nuclear weapon proliferation. Halevi mentioned two notable (and noble!) organizations against proliferation: Pugwash (founded in 1957 by Joseph Rotblat) and Global Zero (founded in 2008) and he talked about the very interesting history of these organizations. As it turned out, these types of organizations served as platforms for building informal relations between Israel and other countries (like the Soviet Union). Halevi told two little stories, one from a 2014 Global Zero meeting where he and Uzi Eilam were the Israeli delegates and they had to deal with a detailed program of Frank von Hippel regarding middle-east disarmament. (The picture above are portrays of these three personalities.) To make an exciting short story even shorter, at the end, the program was not adopted. The second story was about a Pugwash meeting in 2015 where Halevi indirectly posed a question to the Iranian Foreign minister. Ephraim Halevi also briefly mentioned the recent war in Ukraine and his hopes that nuclear weapons will not be applied there, and the question regarding the location of next-generation accelerators and his hopes regarding where it will be built. I found Halevi’s (Hebrew) lecture quite exciting and it is very much recommended. I also greatly enjoyed the lecture of another diplomat, Rafi Barak who was the general manager of our Foreign Office.

Eliezer’s lecture (also in Hebrew) told the story of Israel and CERN from his own angle.

Two critical moments (click to enlarge): Israel’s request to join CERN (above); The decision to admit Israel (below); A crucial parameter (in my view): we see in the bottom right picture two women out of ten participants which is rather poor representation of women. (But sadly not surprising for physics/math.)

Shikma Bressler gave a very interesting lecture (click to enlarge the picture) on the role of Israeli scientists for ATLAS detectors.

Sergey Frolov and Vincent Mourik

A few weeks ago I participated in a short zoom workshop organized by Sergey Frolov about the reproduction crisis, and more precisely about the question “Does condensed matter physics need to worry about a replication crisis?”

I think that a central problem is: “What are the appropriate methodology and ethics for scrutinizing major scientific works,” and this problem is relevant to some of my own scientific endeavors over the years. Sergey Frolov and Vincent Mourik themselves put under the microscope several works regarding “Majorana zero modes”, which are important steps toward topological quantum computers.

One issue that speakers elaborated on in the seminar was on getting data from authors, which is something I encountered on several occasions (and overall, had positive experiences). In the last three years, with Yosi Rinot and Tomer Shoham, we have been putting Google’s 2019 “quantum supremacy” paper under the microscope, and naturally, needed some data. Our policy regarding data was: (a) We always “asked” and never “demanded” data; (b) In cases where the answer was negative we did not ask again; (c) Sometimes, when the answer was positive we did send reminders (trying not to be “pushy”) and so we did when we got no answer at all (which is ok).

A few weeks ago I gave a talk at an Israeli-German “Quantum Future academy Workshop” for young students. My lecture was about “Limits of Computations, Noise and Quantum Computers”. At the end, I talked mainly about how quantum computing is related to computational complexity and how quantum computers may give computational advantage and left only little time to discuss my own take on why a noisy quantum computer may not give computational advantage after all.

Alef’s humorous futuristic view of me asking GPT7 whether quantum computers will ever achieve advantage.

I gave two (zoom) physics colloquium talks at Rutgers University and at the Perimeter Institute about my argument against quantum computers, and I felt that both talks went very well with very nice discussions afterwards. Thanks to my hosts Daniel Friedan at Rutgers and Latham Boyle at PI.

The PI talk was in direct competition with the World Cup France vs. Morocco football match, and it was recorded (Video is here). It was very nice to meet face-to-face (in the zoom sense) Lee Smolin, Debbie Leung, Ray Lafflamme, and others.

(Slides: Israeli-German academy; Rutgers; **PI**.)

Max the Demon is a wonderful comics book about thermodynamics written by Assa Auerbach (a famous physicist) and Richard Codor (a famous artist, among the writers of the legendary book Zoo Aretz Zoo)

After the “Israel joining CERN event”, I had a little chat with Halevi. I told him about his answer to my 2007 question and asked him if these days we can also sleep soundly at night. Ephraim Halevi’s response was:

**“I haven’t been sleeping soundly in recent weeks and I estimate that my sleep hours will be even shorter in the coming weeks. The days are gradually getting gloomier and the gates of reconciliation are closing.”**

(This was in February, and I suppose that Halevi referred to the dispute regarding the judicial reforms in Israel. Indeed in later interviews he expressed his objection to these reforms.)

]]>Clarification: we assume Bob’s sequence is fixed, and the randomness is over a probabilistic strategy chosen by Alice.

]]>LLM is the acronym for “large language model” like GPT-3, ChatGPT, GPT-4 etc. Amnon Shashua gave an enlightening clear lecture about the repeated recent breakthroughs for LLM’s and where we stand. Here is the You-Tube link for the lecture (in Hebrew) and I will try to add a link for the slides (which are in English). **Update:** Here are the slides. One aspect of the story that I find mind boggling is that adding to the model the ability to program for the purpose of writing code, led also to improved language abilities! (As a matter of fact, everything about LLMs is mind boggling!)

Two weeks ago Amnon Shasua was awarded the Israel prize, regarded as the state’s highest cultural honor. In his words on behalf of the recipients, Amnon emphasized the need to keep the independence and stature of the Israeli court systems. **Congratulations, Amnon! **

Let me also mention that as part of a project (in its early stages) with Maya Bar-Hillel and Daphna Shahaf, last summer we performed several little experiments with various AI programs and below are some entertaining related Dalle-E2 pictures. Here is another intersting video where Liron Lishinsky-Fisher describes eight AI applications.

AI21 founders: Yoav Shoham, Ori Goshen, and Amnon Shashua. (Roei Shor Photography, source)

**Prompt: Steve is walking from the city hall toward the big fish and Andrew is walking from the big fish toward the city hall.**

**A criminal lawyer at the age of two.**

There are various things to blog about and let me give a quick preview for the plan for the next few posts. The purpose of this post is to give an impression about the hectic mathematical activities around here with special emphasis on combinatorics and early-in-the-week activities. There is so much action around that I feel tired just to write about it. Next post will be around Amnon Shashua’s lecture at Reichman university giving a deep dive on LLMs. Following it I will tell you about developments around physics with special emphasis to the evening at the Israeli Academy celebrating Israel entrance to CERN. Fourth in line is a post about my recent paper with Yosef Rinott and Tomer Shoham on the Google 2019 quantum supremacy experiment (this is our third paper on the subject).

Let’s move on to today’s post which will include more than the usual dose of Hebrew.

A few weeks ago, Avinoam Mann, a dear member of our department in Jerusalem, passed away. Avinoam was a famous group theorist working on many aspects of this theory. I have many warm memories of Avinoam since I was a student when I took with Avinoam two very demanding reading courses, and later as colleagues and friends for many decades. Avinoam was also a poet and here is a poem he wrote.

The seminar takes place on Mondays between 11-13. The 2-hour format allows ample discussions. There were brilliant talks at the HUJI (Hebrew University of Jerusalem) Combinatorics Seminar. The last four were given by Illay Hoshen, Yuval Filmus, Igor Balla, and Nathan Keller. **Ilay Hoshen** spoke about his paper with **Wojtek Samotij** on Simonovits’s theorem for random graphs, and presented a (partial) resolution to a conjecture by DeMarco and Kahn. **Yuval Filmus** talked about a joint work with **Nathan Lindzey**, where the starting point was Fourier expansion for function on the Boolean cube and asked what happens if we study functions on other domains, such as the “slice” or the symmetric group? (very elegant connection with representation theory.) Once it’s on the arxive we will add the link. **Igor Balla** talked about Equiangular lines via matrix projection. This work presents a definite progress on the classical problem of equiangular lines as well as some connections to problems in quantum information theory. **Nathan Keller** talked about his joint work with **Noam Lifshitz, Dor Minzer**, and **Ohad Sheinfeld. **Hypercontractivity for global functions is used for far reaching Erdos-Ko-Rado theorems for permutations. Nati Linial wrote to me about the lecture: **מצויינת, מדוייקת, אינפורמטיבית, א-מחיה** . (which roughly translates to: “Outstanding, Accurate, Informative – Oh the Joy!”). Today **Amir Yehudayoff** will talk about his work with **Dan Carmon** on dual systolic graphs.

This is not the only HUJI combinatorics seminar, on Thursday afternoon we have a joint Jerusalem-Copenhagen combinatorics seminar and at noon we have a joint Jerusalem Copenhagen lunch seminar. I gave a lecture in the lunch seminar a couple of weeks ago about challenges in the combinatorial theory of convex polytopes and spheres beyond the -theorem. Of course, our CS-theory Wednesday seminars have plenty of lectures of combinatorial flavour.

Things in TAU (Tel Aviv University) are not calmer.

TAU combinatorics seminar is on Sundays 10:05-11:05, and last week (April 30) the legendary **Shachar Lovett** gave a talk about his paper with **Alexander Knop**, **Sam McGuire**, and **Weiqiang Yuan **about Structure of monomials of Boolean functions. (Click for the slides.)

The main theorem is the following one. **Theorem:** Let be a Boolean function with . ( is the number of monomials in the presentation of ) Then f can be computed by an AND decision tree of depth .

The proof uses an auxiliary (sharp) result about hitting sets (aka transversals) for monomials. It is not known if the factor in the main theorem is needed. Shachar described exciting connections with the log-rank conjecture and with Frankl’s union-closed conjecture. He also described the analogous (Fourier) question for functions .

After the lecture Shachar told me about some basic details of the recent amazing proof of Kelly and Meka for sharp bound for Roth’s theorems. Shachar promised me that the crucial new ideas giving a Fourier proof for similar bounds for the cup set problem could be presented in four to six hours. (He started with two hours but I flatly disbelieved it.) We also came back to the idea of a polymath project devoted to Frankl’s conjecture.

This Sunday ** Shir Peleg-Priester ** talked about Sylvester-Gallai type theorems for quadratic polynomials. (Joint work with ** Amir Shpilka, Abhibhav Garg, Rafael Oliveira, Akash K Sengupta.)**

Sunday seminar is not the only TAU combinatorics seminar. Two days later, on May 2 **Patrick Morris** gave a special seminar about a robust Corrádi–Hajnal Theorem (joint work with **Peter Allen, Julia Böttcher, Jan Corsten, Ewan Davies, Matthew Jenssen, Barnaby Roberts** and **Jozef Skokan**.) As far as I know there are plans to have special seminars dedicated to both the new ultimate Roth bounds and the Ramsey breakthrough. (Update: just learned too late that the Ramsey seminar already took place on Tuesday.)

Of course, there is also the weekly discrete and computational geometry seminar, and a few weeks ago I gave a talk about “covering problems” which was well accepted and is in line with this year’s theme for the seminar.

There are many other combinatorics seminars around. If you have an urge for combinatorics lectures between the Tel Aviv seminar and the one in Jerusalem, on Sundays at 2 o’clock there is the Bar Ilan weekly Combinatorics Seminar, and on May 7 **Yelena Yuditsky **talked about Conflict-free colouring of subsets (joint work with **Bruno Jartoux, Chaya Keller** and **Shakhar Smorodinsky**.) On Mondays Martin Golumbic runs the seminar: “Monday with Marty and Students of Sunil” devoted to algorithmic graph theory. Tomorrow, **Pradeesha Ashok** talks about: Exact and Parameterised algorithms for Graph Burning (joint work with **Avi Tomar, Shaily Verma, Sayani Das, Lawqueen Kanesh** and **Saket Saurabh**).

Of course, things are just as amazingly intense in other fields of mathematics as well. Last week I attended two great talks by **Shmuel Weinberger**. The first talk gave the answer to the question which groups act without fixed points on some aspherical topological space. Shmuel said that his talk will be structured like a Tarantino’s movie, and at the end he expressed hope that the talk was as entertaining but not as violent. This is based on joint work with **Sylvain Cappell,** and **Min Yan.** The second talk gave (among other things) a lower bound for the number of vertices needed to triangulate -dimensional lens spaces which is the quotient space of an action of on . The proof goes via the notions of -homology and certain invariants of Cheeger and Gromov and it would be really nice to have some simpler proofs. (This is based on an old work with **Stanley Chang**, and a new work with **Geunho Lim**.

Here is a lecture on calculus on extraordinary spaces by Yael Karshon (Hebrew).

Since David Kazhdan moved from Harvard to HUJI he is running four-five semester-long seminars every year on Sundays, and a basic notion seminar on Thursday afternoons. This semester, for example, in one of the seminars, Udi de Shalit presents Wiles’ proof of Fermat’s last theorem (taking Ribet’s part for granted).

Once every decade or so, I serve as a co-teacher in a Kazhdan seminar. In fall 2003 David Kazhdan and I ran a seminar on polytopes, toric varieties, and related combinatorics and algebra. In 2013 David and I felt that it was time to run another such event in 2014, perhaps establishing a tradition for a decennial joint seminar. I announced this coming event in my January 2013 post and wrote: “So next spring, the plan is …[to] devote one of David’s Sunday seminars to computation, quantumness, symplectic geometry, and information.” Alas, David had a terrible car accident and we had to delay the plan to a fall 2019 seminar that Leonid Polterovich, Dorit Aharonov, Guy Kindler and I ran. Also, in fall 2018, Karim Adiprasito gave a Kazhdan seminar on “Positivity in combinatorics and beyond” where Karim presented his proof for the g-conjecture. We are now planning a Kazhdan seminar in fall 2024 around “global hypercontractivity” with Noam Lifshitz, myself and perhaps also Guy Kindler and others. (Kazhdan’s 2023/2024 schedule was fully booked, but come to think of it, since Dor Minzer is in town in fall 2023, maybe we will do something then.)

Noga Alon recently complained that “younger and younger people are celebrating their 60th birthdays”. Indeed, two weeks from now there will be a day-and-a half workshop in Berlin celebrating Günter Ziegler’s birthday and in June there will be a Leonid Polterovich fest in Zurich. Happy birthdays, kids!

A well-known Israeli poet and writer Yonatan Gefen passed away recently and here is a nice song he wrote (performed by Arik Einstein): Yhachol lihyot sheze nigmar (It is possible that it is over.)

]]>**Deep learning 2020**

**Deep learning 2030**

**Deep learning 2040**

The goal of finding *q*-analogs of combinatorial results where, roughly speaking, sets are replaced by subspaces of vector spaces over a field with elements, is common both in enumerative combinatorics and in extremal combinatorics. A recent breakthrough we discussed by Keevash, Sah, and Sawhney was about the existence of -analogs of designs (subspace designs).

I will mention a problem (Question 4) in this direction about incidence matrices, following three questions that have largely been solved.

The incidence matrix of -subsets vs. -subsets of , is a matrix whose rows correspond to -subsets of , whose columns correspond to -subsets of , and the entry equals 1 if , and equals 0 if .

A weighted incidence matrix of -subsets vs. -subsets of , is a matrix whose rows correspond to -subsets of , whose columns correspond to -subsets of , and if and if .

**Question 1:** What is the rank of the incidence matrix of -subsets vs. -subsets of , over a field of characteristic .

This question was beautifully answered by Richard Wilson in 1990 . The problem was posed by Nati Linial and Bruce Rothschild in 1981 and they settled the case . (The answer for , , that motivated the question, had been observed earlier by Perles and by Frankl.) It is unforgivable that I did not present the statement of Wilson’s theorem here on the blog.

**Question 2:** What is the minimum rank, denoted by , of a weighted incidence matrix of -subsets vs. -subsets of $[n]$ over a field of characteristic .

I answered this question in the early 80s (it is related also related to various results presented by other researchers around the same time). The answer is , and remarkably it is independent from the characteristic .

The incidence matrix of -dimensional subspaces vs. -dimensional subspaces of ( has entries if and if .

A weighted incidence matrix of -dimensional subspaces vs. -dimensional subspaces of () has entries if and if .

We pose two additional questions which are the “*q*-analogs” of Questions 1 and 2.

**Question 3:** What is the rank of the incidence matrix over a field of characteristic $p$ (you can simply take the field ).

Frumkin and Yakir settled problem 3 when is not a power of .

The problem I wish to pose (again) here is:

**Question 4: ** What is the minimum rank denoted by of a weighted incidence matrix over a field of characteristic .

In particular, I would like to know if the answer does not depend on and if it agrees with some easy lower bounds (obtained from certain identity submatrices) like in the case of a field with one element (namely, subsets).

**Qoestion 5:** What are the *q*-analogs of trees? (and hypertrees).

The basic idea is first to first find weights so that the incidence matrix of 1-subspace vs 2-subspaces has rank , and then the -trees will correspond to collection of 2-spaces with linearly independent columns. (I don’t expect uniqueness.) This is a somewhat related to a -analog of the notion of symmetric matroids that I studied in the late 80s.

A year ago Ferdinand Ihringer, Motaz Mokatren and I made some very preliminary steps in this direction before moving on (separately) to other projects. It will be nice to come back to it.

**Remark:** There are even greater generalities where problems can be extended from set systems (graphs and hypergraphs) to more general algebraic objects. Those could be related to general primitive permutation groups, to association schemes, and to other objects in algebraic combinatorics.

Let be a normed space. For a set the unit distance graph is the graph whose vertices are points in and two vertices are adjacent if their distance is one.

We can consider the following quantities

1) : The maximum size of a unit distance set in . (In other words, the maximum clique in .)

2) : The number of colors needed for if two points of unit distance are colored with different colors.

3) : The maximum number of colors needed to color points in a set of diameter 1 if every color set has a diameter smaller than 1. (This is the Borsuk number of .)

4) The maximum number of colors needed to color points in a finite set of diameter 1 if two points of unit distance are colored with different colors.

5) The maximum number of points of norm 1 with pairwise distances at least 1.

6) The maximum over all sets with points of pairwise distance at least one, of the minimum degree in the unit distance graph .

7) The maximum over all sets with points of pairwise distance at least one of the chromatic number of the unit distance graph of .

Estimating these seven quantities for Euclidean spaces and for other normed spaces are well-known problems. (See my survey article on problems around Borsuk’s problem.) Alon, Bucić, and Sauermann made a remarkable breakthrough on the first problem of the largest clique in unit distance graphs for arbitrary normed spaces.

Jordan Ellenberg asked: “Does the Alon-Bucic-Sauermann result give you upper bounds for the chromatic number of (the unit distance graph of) with a typical norm? (Or is that already easy for some reason?) “. But I know little about Jordan’s question.

There is much to say about them but I will not discuss these problems here. I will mention a single annoying problem.

**Question 6:** Is there an example of a normed space such that ?

(I am not even sure if for the seventh item it makes a difference to consider finite .)

In the post that I mentioned we also discussed Tomon’s remarkable result on intersection patterns of standard boxes. Here is a loosely related problem. In short, we want to find topological analogs for results on intersection patterns of standard boxes.

Topological Helly-type theorems is an important direction in geometric and topological combinatorics. The idea is to prove Helly type theorems about convex sets in a much wider topological context.

A primary goal of Topological Helly-type theorems is to extend results for nerves of families of convex sets in to the class of -Leray simpilcial complexes. Among the results achieved so far are: The upper bound theorem; Eckhoff’s conjecture; Alon and Kleitman’s (p,q)-theorem; colorful and matroidal Helly’s theorem; topological Amenta’s theorem, and more.

Another goal of Topological Helly-type theorems is to study if results on nerves of standard boxes can be extended to flag -Leray simplicial complexes?

In this direction the immediate goal is to extend Eckhoff’s upper bound theorem. (Item 3 in this post.)

**Question 7.** Conjecture: Let be a Leray flag complex of dimension with $n$ vertices. Then the -vector of obeys Eckhoff’s upper bound theorem for standard boxes.

A closely related question is

**Question 8.** Conjecture: Let be a -Leray flag complex of dimension , then the -vector of is the -vector of a completely balanced -dimensional -Leray complex.

Studying the equality cases of the conjecture is also of interest and the extremal complexes can also be regarded as some sort of ultra-trees.

Roy Meshulam and I worked together on topological Helly type theorems for more than two decades.

]]>Last night, the demonstrations in Israel regarding the “judicial reforms” escalated after prime minister Netanyahu fired the defense minister Gallant who called to stop the legislation. My wife and I were in the midst of enjoying a concert and after it ended we heard loud calls “Bibi fired Gallant” and even “The dictator fired Gallant,” and were surprised by the news and the huge crowds of young people, very angry for a very good reason. And, of course, we joined them. The picture above, ~~provided~~ forwarded from an unknown source by Alon Rosen, is from a major highway in Tel Aviv that was blocked for 9 hours. Whether the picture is genuine or not, it shows the anarchist nature of at least some of the protestors. (We know for sure that some computer scientists were there.) I am not sure if a proof of this bold claim was also provided.

Updates: the picture is authentic see comment and picture by Nir Sochen below; I saw it myself (no proof in the shoulders); Some undocumented reports on street graffiti from protest in Jerusalem the same day; A post about the protestor’s computational complexity worldview and what it means at SO.

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I suppose that most of you have already heard about the first ever aperiodic planar tiling with one type of tiles. It was discovered by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. Amazing!!!

**Update (May 30, 2023):** The same team moved on to another breakthrough! the paper is A Chiral aperiodic monotile. (More details on the Aperiodical.)

Here are excellent blogposts from Complex Projective 4-Space and from the Aperiodical. Update: A beautiful and informative GLL post.

Abstract: A longstanding open problem asks for an aperiodic monotile, also known as an “einstein”: a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the “hat” polykite, can form clusters called “metatiles”, for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical — and hence aperiodic — tilings.

Comments:

The new paper starts with an interesting historical description. From Hilbert’s Entscheidungsproblem, that led to notions of undecidability, Wang’s 1961 work, and Berger’s 1966 undecidability result that showed that some tiles admit only apriodic tilings. (This required 20426 types of tiles.) Penrose’s famous apreiodic tiling from 1978 consists of just two tiles.

There are lovely related results for the hyperbolic plane starting from the 1974 paper by Boroczky, and later papers by Block and Weinberger, Margulis and Mozes, Goodman-Strauss and others.

Browsing the paper, I could not figure out the status of the question “Can we have aperiodic tiling with convex tiles?” (The answer is negative in the plane, see the comment section.)

Half a year ago Rachel Greenfeld and Terry Tao disproved in the paper “A counterexample to the periodic tiling conjecture,“ the *periodic tiling conjecture* (Grünbaum-Shephard and Lagarias-Wang) that asserted that any finite subset of a lattice which tiles that lattice by translations, in fact tiles periodically. (Here is a blog post on Terry’s blog about the result and a Quanta Magazine article.)

Update: Here is a link to a very informative twitter thread by Alex Kontorovich

An interesting street tiling near Dizengoff circle, Tel Aviv

]]>(h/t Michael Simkin and Nati Linial)

Here is a somewhat mysterious announcement for a combinatorics seminar lecture at Cambridge.

Which old problems of Erd**ő**s are we talking about? Here is a picture from the seminar itself.

Stay tuned (or try to test your intuition or guess in the comment section.)

~~Nothing yet here.~~

The paper is now on the arXiv.

Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe

The Ramsey number is the minimum such that every red-blue colouring of the edges of the complete graph on vertices contains a monochromatic copy of . We prove that

for some constant . This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

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