The aim of the Ostrowski Foundation is to promote the mathematical sciences. Every second year it provides a prize for recent outstanding achievements in pure mathematics and in the foundations of numerical mathematics.

The prize has been awarded every two years since 1989. The most recent winners are Ben Green and Terence Tao in 2005; Oded Schramm in 2007; Sorin Popa in 2009; Ib Madsen, David Preiss, and Kannan Soundararajan in 2011; Yitang Zhang in 2013; and Peter Scholze in 2015.

The jury invites nominations for candidates for the 2017 Ostrowski Prize. Nominations should include a CV of the candidate, a letter of nomination, and 2-3 letters of reference. Nominations should be sent to the Chair of the jury for 2017, Gil Kalai (Hebrew University of Jerusalem, Israel), kalai@math.huji.ac.il by June 30, 2017.

(Because of some technical difficulties in advertising the deadline was extended from the original deadline of May 15, 2017.)

Links: Ostrowski Prize (Webpage) Ostrowski Prize (Wikipedea)

Please share this information.

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On June 18-23 2017 we will celebrate in Budapest the 70th birthday of Imre Bárány. Here is the webpage of the conference. For the occasion I wrote a short paper with problems in discrete geometry, mainly around Helly’s and Tverberg’s theorem. This represents

one of the areas on which Imre Bárány had immense impact. Here is the paper: Problems for Imre Bárány’s birthday.

(Our 1999 picture was taken by Emo Welzl.)

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Very nice mathematical videos!

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May 4 2:30-3:30; May 7 11:00-13:00; May 10 10:30-12:00

See the event webpage for titles and abstracts (or click on the picture below).

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I spent six weeks in February (2016) in New Haven. It was very nice to get back to Yale after more than two years. Here is a picture from the spectacular Yale’s art museum.

A few years ago I wrote about amazing developments in low dimensional topology. Ian Agol proved the Thurston’s virtual Haken conjecture. The proof relies on earlier major works by Dani Wise and others (see this post) and an important ingredient was Micha Sageev work on cubical complexes arising from 3-manifolds. I remember thinking that as a mathematician in another field it is unrealistic for me to want to understand these developments in any detail (which is not an obstacle for writing about them), but that one day I want to understand Sageev’s construction. At Yale, Ian Agol gave three lectures on his proof, mentioned a combinatorial form of Sageev construction, and gave me some references. A few days after landing Dani Wise talked at our seminar and he explained to me at lunch how it goes in details. (BTW, a few days before landing Micha Sageev gave a colloquium at HUJI that I missed.) I have some notes, and it is not so difficult so stay tuned for some details in a future post!. (**Update:** I will need some refreshing for writing about it. But I can assure you that Micha Sageev’s construction is a beautiful combinatorial construction that we can understand and perhaps use.)

Hee Oh talked at Yale about various analogs of the prime number theorem (PNT) and in some cases even of weak forms of the Riemann hypothesis for hyperbolic manifolds and for rational maps. This is based on a recent exciting paper by Oh and Dale Winter. Some results for hyperbolic manifolds were known before but moving to dynamics of rational maps is completely new and it follows a “dictionary” between hyperbolic manifolds and rational maps offered by Dennis Sullivan. It was nice to see in Oh’s lecture unexpected connections between the mathematical objects studied by three Yale mathematicians, Mandelbrot, Margulis, and Minsky. Are these results related to the real Riemann Hypothesis? I don’t know.

Going well over my head I want to tell you about somethings I learned from two lectures. It is about Weil’s conjectures for curves including the “Riemann hypothesis for curves over finite fields.” One lecture is by Peter Sarnak and the other by Ravi Vakil. Both lectures were given twice (perhaps a little differently) in Jerusalem and at Yale few weeks apart. Peter Sarnak mentioned in a talk the very concrete proof by Stepanov to the Rieman hypothesis for curves over finite fields. The proof uses some sort of the polynomial method, and it is this concrete proof that is useful for a recent work on Markoff triples by Bourgain, Gamburd and Sarnak. Ravi Vakhil mentioned a very very abstract form of the conjecture or, more precisely, of the “rationality” part of it proved in 1960 by Bernard Dwork (yes, Cynthia’s father!). It started from an extremely abstract version offered by Kapranov in 2000 to which Larsen and Lunts found a counterexample. A certain weaker form of Kapranov’s conjecture that Ravi discussed might still be correct. (A related paper to Ravi’s talk is Discriminants in the Grothendieck ring by Ravi Vakil and Melanie Matchett Wood.) These very abstract forms of the conjectues are also extremely appealing and it is especially appealing to see the wide spectrum from the very concrete to the very abstract. It is also nice that both the polynomial method and the quest for rationality (of power series) are present also in combinatorics.

Alex Lubotzky and I gave a 6-weeks course on high dimensional expanders. This was the fourth time we gave a course on the subject and quite a lot have happened since we first taught a similar course some years ago so it was quite interesting to get back to the subject. I certainly plan to devote a few posts to HD-expanders and Ramanujan complexes at some point in the future.

**Update:** There will be a special semester on high-dimensional combinatorics and the Israeli Institute for Advanced Studies in Jerusalem in the academic year 2017/2018.

At the Simons Center in NYC Rafał Latała gave a beautiful lecture on the solution by Thomas Royen of the Gaussian correlation conjecture. Here is a review paper by Rafał Latała and Dariusz Matlak.

**Update:** Here is an article about this proof in Quanta magazine.

I also gave three lectures at Yale about polymath projects (at that time both Polymath10 and Polymath11 were active), and a special welcome lecture to fresh Ph. D. potential students POLYMATH and more – Mathematics over the Internet (click for the presentation) about polymath projects and mainly Polymath5, MathOverflow, mathematical aspects of Angry Birds and why they should all choose Yale.

One of repeated rather unpleasant dreams I had over the years (less so in the last decade) was that the Israeli army discovered that I still owe some months of service, and I find myself confusingly and inconveniently back in uniform. A few weeks (+ one year) ago I had a new variant of that dream appropriately scaled to my current age: MIT discovered that I still owe some months of my postdoctoral service. This was much more pleasant!

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Oded Goldreich’s 60 birthday meeting, April 19-20 at the Weitzmann Institute promises to be a great event. Here is the webpage of the event. Many cool talks with cool titles. To celebrate the birthday we run a poll for the coolest title among eight selected titles. Please participate!

The coolest titles are:

First place: 31% of votes (22/71)

Shafi Goldwasser, Wednesday, April 19, 14:20-14:50

Second place: 20% (14/71)

Boaz Barak, Wednesday, April 19, 14:50-15:30

Third place: 17% (12/71)

Avi Wigderson, Thursday, April 20, 10:00-10:40

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**Updates:** I will try to update this post by adding new items at the end (and some important older items that I forgot to include). Update 1, April-May 2017, posted May 24, 2017. Update 2, June-July 2017, posted July 19; revised July 27, 2017.

European Commission will launch €1 billion quantum technologies flagship; The full quantum manifesto;

A general article about quantum technologies and computing in the Economist.

“Nature” article Quantum computers ready to leap out of the lab in 2017. It is mainly on Google, Microsoft, and Monroe’s lab mentioning also IBM, Rigetti and Quantum Circuits (a new startup by the Yale group). A little debate between John Martinis and Robert Schoelkopf about the target of “quantum supremacy.” Regarding topological quantum computing Leo Kouwenhoven declares that “2017 is the year of braiding,”

How to say “Quantum computers” in other languages? Hebrew: מחשבים קוונטים pronounced: maghshevim kvantim; Spanish: Computadoras cuánticas (google translate)*. *

Please contribute! For earlier efforts in this spirit see “more or less in various languages” and “When it rains it pours” in various languages. Of course, ordinary comments are welcome as well.

The Race to Sell True Quantum Computers Begins Before They Really Exist, (Wired) Mainly on Google and IBM. IBM Inches Ahead of Google in Race for Quantum Computing Power, MIT Technology Review. Commercialize *quantum* technologies in five years (Nature)

IBM: IBM’s quantum cloud computer goes commercial (Nature) IBM moves from a 5-qubit quantum computer to 50-qubit commercial quantum computer in the near future. Also IBM’s first commercial quantum computer paves way to overhaul of molecular simulations (chemistry world); IBM Building First Universal Quantum Computers for Business and Science (IBM Press Release); and a PC World article.

Google: Researchers Report Milestone in Developing Quantum Computer (NYT); Quantum computing is poised to transform our lives. Meet the man leading Google’s charge (Wired, and interview with John Martinis)

Microsoft: Microsoft Makes Bet Quantum Computing Is Next Breakthrough (NYT) ; Inside Microsoft’s quest for a topological quantum computer (Nature) Microsoft doubles down on quantum computing bet. Microsoft approach is based on topological quantum computing.

Others: Scientists are close to building a quantum computer that can beat a conventional one, Science, Chris Monroe and the Startup ionQ, and various other groups and methods. Quantum Computing on Cusp, EE Times, Yale’s group and Quantum Circuits. Quantum hanky-panky (Seth Lloyd, Edge); Its much bigger than it looks ( ;

NIST: National Institute of Standards and Technology, quantum divisions. Super *quantum* simulator ‘entangles’ hundreds of ions (Science daily) *Quantum computers* may have higher ‘speed limits’ than thought (Phys.org.) A thought experiment by Stephen Jordan.

D-wave. D-wave is building large scale quantum computers with (rather noisy) superconducting qubits, and implement specific optimization algorithms. *D-Wave* quantum computers: The smart person’s guide . Quantum computer learns to ‘see’ trees Science Magazine. An article in Quanta Magazine.

List of companies involved in quantum computers. A few webpages: 1Qbit ; D-wave ; Quantum circuits (Yale group) ; Rigetti ; Monroe’s blog; Station Q (Microsoft); Google; IBM-Q;

The key for large scale universal quantum computers is quantum error-correcting codes.

IBM to develop hardware to wipe out errors in quantum computing (New Scientists)

Error fix for long-lived qubits brings quantum computers nearer (New Scientist)

Physicists show that real-time *error correction* in *quantum* … (Phys.Org, reporting on research in South Africa)

New *Yale*-developed device lengthens the life of quantum information (Yale news)

A convincing demonstration of computational complexity supremacy is expected by researchers in the near future. Main approaches are using quantum circuits with around 50 qubits or via demonstration of BosonSampling with 20-30 bosons.

( NextBigFuture. )

Revealed: Google’s plan for quantum computer supremacy (New Scientist); How Widely Should We Draw The Circle? (Scott Aaronson on imminent demonstration of quantum supremacy); News release from Bristol U. ; Proto *quantum* computer inspired by Victorians gets a speed boost About BosonSampling. (New Scientist.); Massive Disruption Is Coming With Quantum Computing, Singularity Hub

Forging a Qubit to Rule Them All (Quanta Magazine); Inside the Knotty World of ‘Anyon’ Particles (Quanta Magazine); Why Insights of Nobel Physicists Could Revolutionize 21st-Century (Scientific Computing): Via Majorana fermions: Physicists In China Detect The *Majorana Fermion* (Asian Scientist Magazine); The discovery of *Majorana fermion* (EurekAlert);

Quantum computer makes first high-energy physics simulation, Nature, about the Innsbruck group.

A general article: The trouble with quantum computing by Ben Skuse, Engineering and Technology. The author expresses overall an optimistic bottom line, but explains the difficulties and also brings my point of view.

Here is the ending paragraph:

Much like outcomes in the quantum world itself, the future of quantum computing involves a number of unknowns, none of which we can predict. When and whether quantum supremacy and universal quantum computers can be achieved is currently up in the air, but the work of Chow, Lucas, Montanaro and even Kalai is contributing to tilting the odds in its favour. And each ‘step closer’ is only that. The road to quantum computing stretches off into the distance and may have no end.

Quantum computers will allow breaking most of current cryptosystems. This leads to much effort for “post-quantum cryptography,” namely developing cryptographic methods immune to quantum attacks.

HACKING, CRYPTOGRAPHY, AND THE COUNTDOWN TO QUANTUM COMPUTING (New Yorker)

Post-Quantum Cryptography: NIST’s Plan for the Future (U.S. National Institute of Standards and Technology (NIST)) *NIST* requests ideas for crypto that can survive *quantum* computers (The Register)

An article regarding RSA conference. (bank info security.); An article in Quanta Magazine

EXCLUSIVE – Mathematically provable communication security and other applications of Quantum Key Distribution (OpenGovAsia); Developing a quantum key system to make mobile transactions safer (New Atlas); China’s 2000-km Quantum Link Is Almost Complete (IEEE Spectrum); SK Telecom, Nokia team on quantum cryptography (Telecom Asia); Chinese Physicists Achieve Record-Breaking Quantum Cryptography Breakthrough (Hecked).

Atomic clocks: *Physicists* Find That as Clocks Get More Precise, Time Gets More, ScienceAlert; JILA atomic clock mimics long-sought synthetic magnetic state; Time crystals: *Time Crystals* Are Real, But That Doesn’t Mean Time Is Crystallized (Forbes); Entanglement and theoretical physics: The quantum source of space-time (Nature); The equation that could transform physics: Researchers say ER=EPR … (Daily mail). Quantum biology: Can *Quantum Physics* Explain Consciousness? (The Atlantic) Are we ready for quantum biology?, New Scientist; A New Spin on the Quantum Brain, Quanta Magazine; quantum effects in biological systems (a conference at IIAS).

Metaculus: When will the first 49+ qubit universal quantum computer be constructed? Posted March 2017, 42 predictions, median answer February 19, 2020.

U.S. report : ADVANCING QUANTUM INFORMATION SCIENCE: NATIONAL CHALLENGES AND OPPORTUNITIES

Europe’s leap into the *quantum* computing arms race (Financial Times)

U.K. The future is quantum: solution to the world’s critical problems; Quantum computers will solve problems that conventional computers never could. (Financial times)

China: Chinese satellite is one giant step for the quantum internet (Nature); 5 reasons why *China* will rule tech, 2017 edition (computerworld. quantum computers is one of the items). *China’s* push to become a tech superpower triggers alarms abroad (Financial times, quantum computing emphasized ); *China* can help drive global progress in *quantum computing* (ZD net)

Singapore: CEO conversations: *Quantum computing*, leadership, and … (Enterprise Innovation).

Australia: Sydney University’s David Reilly part of Microsoft’s billion-dollar push, The Sydney Morning Heral; Quantum computing is the next space race says Telstras Hugh Bradlow (Financial Review, Australia).

Canada: Canada leads in the race to create *Quantum* Valley, The Globe and Mail; Canada’s PM on quantum computing.

Israel: The future is here: the quantum computer revolution (in Hebrew); Calcalist article (Hebrew); Israeli scientists discovery towards QC (in Hebrew Haaretz) ; Quantum computing breakthrough: Israeli scientists invent cannon for entangled photon clusters (International Business Times).

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Links to further items (mainly from the media) on quantum technology: April May 2017

Supercomputer Simulation Offers Peek at the Future of *Quantum Computers*. MIT Technology review.

QUANTUM COMPUTING IS GOING COMMERCIAL WITH THE POTENTIAL TO DISRUPT EVERYTHING (Newsweek)

“Indistinguishable Photons” Could Unleash *Quantum Computing* (Inverse)

*Quantum computing* the new frontier for banking (Australia) “Commonwealth Bank’s announcement that it has developed a quantum computer simulator takes the bank a step closer to its ambition of building a small, 10-qubit (quantum bits) machine by 2020.”

Microsoft’s cool *quantum computing* plan embraces cryogenic memory (PCworld)

Finland: Refrigerator for *quantum computers* discovered Aalto University (EurekAlert), Nanofridge could keep *quantum computers* cool enough to calculate New Scientist. China: World’s First *Quantum Computer* Is Here (Wall street Pit) China Has Built The World’s 1st *Quantum Computer*, And It’s 24000 (India Times)

Google’s New Chip Is a Stepping Stone to *Quantum Computing* (MIT Technology Review)

Microsoft’s cool *quantum computing* plan embraces cryogenic memory (PC World)

Artificial Intelligence (AI) and *Quantum Computing* (Nanalyze) (Various cool stuff) Is Bitcoin at Risk as Google and IBM Aim for 50-Qubit *Quantum* … (Nigeria Today)

China team achieve 5-bosons 5-modes BosonSampling. High-efficiency multiphoton boson sampling : (Nature photonics); Blogpost on the Shtetl. However, UK team indicates over 50 photons needed for Boson Sampling quantum supremacy. (The next Big Future).

Quanta: The quantum thermodynamics revolution; Why Quantum Computers Might Not Break cryptography;

IBM: Moving from 5 qubits to 16/17 qubits: IBM reveals prototype of its first commercial quantum computer processor (New Atlas); *IBM* makes a leap in *quantum computing* power; *IBM’s quantum computer* gets powerful upgrade (Reuters); and ComputerWord New Zealand; The Inquirer; IBM: Αποκαλυπτήρια πρωτότυπου κβαντικού επεξεργαστή για εμπορική χρήση; MIT Technology Review;

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(Sources: Google search, articles on quanta magazine, Facebook posts/ and blogposts by QI/QC groups, John Preskill, Sabine Hossenfelder , Scott Aaronson and others.)

General: *Quantum Computing* Might Be Here Sooner Than You Think, Bloomberg; Scientific American;

The race between ion trapped and superconducting for “supremacy”: Split decision in first-ever *quantum computer* faceoff (Science) Quantum computers compete for “supremacy” (Scientific American)

Google: Google on track for *quantum computer* breakthrough by end of 2017 (New Scientist) Big Think ; Daily mail ;

Topological: New method could enable more stable and scalable quantum computing, physicists report Phys. Org; Purdue joins Microsoft in five-year mission to create topological quantum computer. Yahoo News

51 qubits via cold atoms: World’s Most Advanced Quantum Computer Created By A Team Of US And Russian Scientists (Fossbytes). The arxived paper.

IonQ plans to bring a general-purpose quantum computer to the market by the end of 2018. IonQ Raises $20M Series B Round Led By NEA, GV To Advance Quantum Computing For Commercial Applications ; Venture Firms Back Startup with Novel Twist on Quantum Computing (Wall Street Journal).

The European large company Atos joins the quantum race. *Atos* Launches 40-Qubit *Quantum Computing* Simulator;

A Science article; Media coverage: Physicists Just Generated a Particle That Acts as Its Own Antiparticle (Science Alert); Latest Quantum Breakthrough: Majorana Fermions Discovered

Teleportation: Teleportation: Photon particles today, humans tomorrow? (BBC, the article is quite reasonable.) (Time )

LIGO

The recent doubts regarding LIGO are not related to quantum computers but are related to correlated noise which is also an issue for quantum computers. Was It All Just Noise? Independent Analysis Casts Doubt On LIGO’s Detections, Forbes. A very nice post by Sabine Hossenfelder; Strange Noise in Gravitational-Wave Data Sparks Debate. Quanta Magazine;

A skeptical look: Delivering on a quantum promise by Karl Svozil physicsword.com. Svozil writes “While many of the quantum manifesto’s short- and medium-term goals appear feasible, some of the long-term goals might not be achievable even in principle.”

This is quite the opposite to my own skeptical view which is that some crucial short-term goals, primarily creating the necessary quantum error correcting codes, are infeasible. (The tragedy of skepticism – skeptics rarely agree with each other, are skeptical to views of other skeptics and also to their own views.)

Quantum machine learning: A Hybrid of Quantum Computing and Machine Learning Is Spawning New Ventures, IEEE Spectrum (Scot Aaronson is skeptical.)

New breakthrough discovery—every quantum particle travels backwards (Phys.Org)

Optical (classical) computers: Light-Powered *Computers* Brighten AI’s Future Scientific American

How *quantum* trickery can scramble cause and effect (Nature)

Harry Buhrman and Yfke Dulek give ‘Paradiso lecture’ on artificial intelligence and quantum computers CWI (I met Harry at “IlanFest)

Blind and private quantum computers: Private quantum computation: an introduction to blind quantum computing and related protocols, Joe Fitzsimons, Nature

After 2000 years, a Jewish quantum computer? : Jewish news: Cutting-edge tech brings super-secure quantum Israel a step closer

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Art Duval, Bennet Goeckner, Carly Klivans, and Jeremy Martin found a counter example to the Garsia-Stanley partitioning conjecture for Cohen-Macaulay complexes. (We mentioned the conjecture here.) Congratulations Art, Bennet, Carly and Jeremy! Art, Carly, and Jeremy also wrote an article on the the Partitionability Conjecture in the Notices of the AMS. (Here is an article about it in the Brown Daily Herald.) Let me tell you about the conjecture and related questions. Here are the last few sentences of the paper.

Even though statements like the Partitionability Conjecture can seem too beautiful to be false, we should remember to keep our minds open about the mathematical unknown—the reality might be quite different, with its own unexpected beauty.

I will start with an analogy.

Let *P(n)* be the family of all subsets of *{1,2,…, n}. *A family of subsets of *{1,2,…, n} is an antichain *if no set in the family contains another set in the family*. *Sperner’s theorem (see this post) asserts that for every antichain of subsets of we have .

So we have a combinatorial notion: **an antichain**, and we have a numerical consequence: **Sperner’s theorem**. Our aim is to give a structure theorem which implies the numerical consequence. A saturated symmetric chain is a chain of sets of sizes , for some .

**Symmetric chain decomposition:** *P(n)* can be partitioned into saturated symmetric chains

The partition of *P(n)* into saturated symmetric chain gives you a combinatorial structure that implies the “numerical” content of Sperner’s theorem. The Garsia-Stanley’s conjecture has a somewhat similar flavor.

Let me note that there are stronger numerical results about antichains such as the LYM inequality (see the same post) and an interesting question (I think) is:

Question: Are there decomposition theorems for *P(n)* that support stronger numerical results about antichains, such as the LYM inequality?

The general framework of our discussion is described in the following diagram:

We will consider classes of objects (usually simplicial complexes) described by algebraic or topological properties (TOP) and a smaller class defined by a combinatorial notion (RIGHT). The topological/algebraic properties has some interesting numerical consequences (LEFT). Our task is to find general decomposition theorems that are implied by the topological notions and suffice to imply the numerical consequences.

The algebraic topological notion: **acyclic simplicial complex**.

The combinatorial notion: **collapsible simplicial complexes**.

Numerical relations: (1) Euler-poincare relation, (2) Morse inequalities, (3) a certain system of non linear inequalities.

The decomposition theorem was proved by Stanley.

Fixing a field of coefficients, a simplicial complex is acyclic if all its (reduced) homology groups vanish. A simplicial complex is *collapsible* if you can reduce it to the void complex by repeated deletion of pairs of faces* (F,G)* where *G* is a facet (maximal face) *F* is contained in *G* and is not contained in any other facet and . Note that a collapsing of gives a perfect matching among the faces of (We discussed collapsible and acyclic complexes in this post, and gave Bing’s example for non-collapsible contractible complex is this follow up post.)

Let be the number of -faces of , and let . The Euler-Poincare relation asserts that , Morse inequalities (a baby version) assert that all the s are nonnegative, and (I showed that) they also satisfy some non linear relations of Kruskal-Katona nature. In the paper A combinatorial decomposition of acyclic simplicial complexes, Stanley proved that if *K* is acyclic, there is a perfect matching in the set of faces of which resembles the matching obtained by collapsing, Namely (1) , , and (2) the ‘s form a simplicial complex. The existence of this perfect matching implies the numerical consequences for acyclic complexes. (But it *does not* imply that the complex is acyclic.) Earlier, I proved that a perfect matching with the first property exists. This implies the linear (Morse) inequalities.

Zeeman conjectured that if a 2-dimensional space is contractible then is collapsible. ( is the unit interval. This innocent-looking conjecture is widely believed to be false (perhaps even outrageous) because it implies both the Poincare conjecture (now theorem) and also the Andrews-Curtis conjecture.

The algebraic topological notion: **simplicial complex with prescribed Betti-numbers**.

The combinatorial notion: **Discrete** **Morse theory**. (This is an important notion developed by Robin Forman in the mid 90s.)

Numerical relations (Bjorner and me, late 80s): (1) Euler-poincare relation, (2) Morse inequalities, (3) a certain system of non linear inequalities.

The decomposition theorem was proved by Art Duval in the paper A Combinatorial Decomposition of Simplicial Complexes.

In this case Duval proved that the set of faces in the complex admits a matching (leaving isolated faces that correspond to the Betti numbers) which implies the numerical consequences found by Bjorner and me. It resembles the structure of discrete Morse theory.

The algebraic topological notion: **Cohen-Macaulay simplicial complex**.

The combinatorial notion: **shellable simplicial complexes**.

Numerical relations: (1) The *h*-numbers are non negative , (3) a certain non linear inequalities (The *h*-vector form an *M*-vector).

A decomposition conjecture: the the Garsia Stanley conjecture (now disproved). (A weaker form was proved by Duval and Zhang.)

Richard Stanley in 1979 and Adriano Garsia in 1980 conjectured that the faces of every -dimensional Cohen-Macaulay simplicial complex can partitioned into intervals [] where is a facet (top dimensional) for every . The paper by Duval, Geockner, Klivans, and Martin, A non-partitionable Cohen-Macaulay simplicial complex, presents a counter example. A weaker form of the conjecture was proved by Duval and Zhang.

We discussed face rings, and the Cohen-Macauly property in various previous posts like the one by Eran Nevo on on the commutative algebra connection to the g-conjecture, and the happy birthday post for Richard Stanley. A -dimensional simplicial complex is Cohen-Macaulay (with respect to a field of coefficients,) if when and for every face latex when . This property is equivalent (by a deep theorem of Riesner) to the Cohen-Macaulay property for the face ring of . The ring theoretic definition of Cohan-Macaulayness can be seen as an algebraic form (on the level of vector spaces) of the combinatorial decomposition conjectured by Garsia and Stanley.

The Cohen-Macauly property implies that a certain linear combinations of face numbers called the *h*-numbers are non-negative, and in addition satisfies certain nonlinear relations originated in the 1927 work of Macaulay. For complexes that admit the partitioning conjectured by Garsia and Stanley the* h*-numbers count the number of intervals in the partition according to the dimensions of the s. (This implies the nonnegativity of the *h*-numbers.)

A pure -dimensional simplicial complex is *shellable* if it can partitioned into intervals [] where is a facet (top dimensional) for every , and the union of the first intervals is a simplicial complex for every . A shellable simplicial complex is Cohen Macaulay (a direct proof of the ring-theoretic statement was was given by Kind and Kleinschmidt in 1979.) Shelling of a simplicial complex can be seen as a special form of decomposition. It is a partitioned into intervals [] where is a facet (top dimensional) for every with the additional property that the union of the first intervals in the partition is a simplicial complex for every .

Cohen-Macauly complexes are not necessarily shellable, since shellability implies stronger homotopical properties. But even these stronger homotopical properties do not suffice, Marry Ellen Rudin found an example of non-shellable simplicial ball. (See also here.)

The algebraic topological notion: **General simplicial complex.** (With prescribed shifted complex, if you wish).

The combinatorial notion:** non-pure shellability**. (This notion introduced by Michelle Wachs is also sometimes called **michellability**.)

**A decomposition conjecture** (by me) decomposition of the shifted complex into intervals can be mimicked on the original complex. (More general than the Garsia-Stanley conjecture and thus false.) This decomposition implies the system of linear equaltions satisfied by simplicial complexes with prescribed algebraic shifting. (See, this paper on algebraic shifting, and this post on algebraic shifting.)

**A decomposition theorem (by Duval and Zhang): **decomposition of the shifted complex into intervals can be mimicked by a decomposition of the original complex into certain trees in the paper Iterated Homology and Decompositions of Simplicial Complexes.

Boolean trees

1) Find a version of Duval-Zhang decomposition theorem for Cohen-Macauly complexes that supports also the *non-linear *inequalities for the -vector. (Like the decomposition theorems of Stanley and of Duval.) Extend to the general case of Duval-Zhang theorem!

2) Is there an analog for discrete Morse theory (using either Boolean trees or full Boolean intervals) for triangulated manifolds (and more generally Buchsbaum compexes) for the usual homology and for various classes of iterated homology?

3) For the class of triangulated spheres is there a tree-decomposition a la Duval and Zhang that supports the non negativity of *g*-numbers or the full *g*-conjecture?

4) Can these partitioning theorems be extended to more general classes of regular CW complexes?

And let me end with two more sentences from the ending paragraph of the Notices paper.

The story of the Partitionability Conjecture has many facets. Shellability, partitionability, constructibility, and the Cohen-Macaulay property come from different but overlapping areas of mathematics: combinatorics, commutative algebra, topology, and discrete geometry. The hierarchy of these structural properties turned out to be more complicated than we had anticipated, just as Rudin’s and Ziegler’s examples demonstrate that even the simplest spaces can have intricate combinatorics.

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**Update: **As Lance Fortnow the chair of the selection committee commented below the deadline for this year nomination is April 2, 2017. (Nominations should be sent to Lance to fortnow@cc.gatech.edu and if you need more time nicely ask Lance). Lance wrote that the selection committee welcomes diverse applications.

Here is the description of the prize given in the prize webpage.

The Theory community benefits in many ways from the dedicated service, above and beyond the call of duty, of many of its members. Among other contributions, the field’s members underpin the operation of conferences, journals, prizes, funding agencies, and other community activities, help ensure funding for the field, and promote the recognition of the field by external communities. The SIGACT Distinguished Service Prize is intended to recognize and promote their contributions, as well as to raise awareness of the need for and importance of such service, for the health of our community.

The prize is given annually to an individual who has made substantial service contributions to the Theoretical Computer Science community. (From 2002 to 2012, the award was given biennially.)

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László Babai; Avi Wigderson; Lance Fortnow; Lane Hemaspaandra; Sampath Kannan; Hal Gabow; Richard Karp; Tom Leighton; Rockford J. Ross; Alan Selman; Michael Langston; S. Rao Kosaraju; Fred S. Roberts; Ian Parberry; David S. Johnson

**A poll: Please ignore the “answers” and enter your own answer under “other”.**

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