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).

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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This is a (very much) belated update post from the beginning of March (2016).

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; and a blog post on GLL.

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!

**Update (April 17): Outcomes of the poll for the coolest title are in. (See the end of the post)**

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!

Posted in Combinatorics, Computer Science and Optimization, Conferences
Tagged Oded Goldreich
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One of my main research directions in the last decade is quantum information theory and quantum computers. (See this post and this one.) It is therefore a pleasure to report and give many links on the massive efforts carried out these days in these directions and the great enthusiasm and hope these efforts carry.

**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) Continue reading

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:

Posted in Combinatorics, Geometry
Tagged Art Duval, Bennet Goeckner, Carly Klivans, Garsia-Stanley conjecture, Jeremy Martin, Ping Zhang
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The fifteen remarkable individuals in the previous post are all the recipients of the SIGACT Distinguished Service Prize since it was established in 1997. The most striking common feature to all of them is, in my view, that they are all men.

**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.

Posted in Computer Science and Optimization, Women in science
Tagged SIGACT, Women in science
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Test your intuition: What is the most striking common feature to all these fifteen remarkable individuals

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”.**

The Ramsey number *R(s, t)* is defined to be the smallest *n* such that every graph of order *n* contains either a clique of *s* vertices or an independent set of *t* vertices. Understanding the values of *R(s,t)* is among the most important, fruitful, and frustrating questions in combinatorics.

Vigleik Angeltveit and Brendan D. McKay proved that the value of the diagonal Ramsey number * R(5,5) *is at most

The proof of is via computer verification, checking approximately two trillion separate cases! Congratulations Brendan and Vigleik! (I heard about it from Greg Kuperberg.)

The best known lower bound of *42* was established by Exoo in 1989. The previous best upper bound of 49 was proved by Brendan McKay and Stanislaw P. Radziszowski.

By this 4-days old theorem we now have *43 ≤ R(5, 5) ≤ 48*. Brendan and Vigleik write “The actual value of *R(5, 5)* is widely believed to be *43*, because a lot of computer resources have been expended in an unsuccessful attempt to construct a Ramsey *(5,5)*-graph of order *43.” *

In 1995 Brendan McKay and Stanislaw Radziszowski famously proved that **R(4, 5) = 25**. Listing all graphs with 24 vertices which admit a coloring without a clique on 5 vertices or an independent set with 4 vertices is a major part of the current proof.

Here is, more or less, how Greg described the achievement on FB: How many people can you have at a party such that no 5 are all acquainted and no 5 are all strangers? It has been known for nearly 30 years that 42 people is possible, and known for 20 years that 49 is not possible. The momentous news is that 48 is also not possible.

Greg continuation was a little cryptic to me: According to the paper, most mathematicians with any opinion on the matter agree with Douglas Adams that 42 is the answer. (By convention, the question is phrased in terms of the first impossible value, which is now known to be at least 43 and at most 48, and conjectured to be the former.) In response to my confusion Barry Simon added a ring theoretic aspect: Gil Kalai Isn’t the answer to “the ring of what” always: “one ring to rule them all” sort of like what “42” always means.

Posted in Combinatorics, Open problems, Updates
Tagged Brendan D. McKay, Vigleik Angeltveit
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On the occasion of Polymath 12 devoted to the Rota basis conjecture let me remind you about the Alon-Tarsi conjecture and test your intuition concerning a strong form of the conjecture.

The sign of a Latin square is the product of signs of rows (considered as permutations) and the signs of columns. A Latin square is even if its sign is 1, and odd if its sign is -1. It is easy to see that when is odd the numbers of even and odd Latin squares are the same.

**The Alon-Tarsi Conjecture:** When is even the number of even Latin square is different from the number of odd Latin square.

This conjecture was proved when and is prime by Drisko and when and is prime by Glynn. The first open case is .

A stronger conjecture that is supported by known data is that when is even there are actually** more** even Latin squares than odd Latin squares. (This table is taken from the Wolfarm mathworld page on the conjecture.)

**A poll**

Unlike previous “test your intuition” questions the answer is not known.

The Alon-Tarsi conjecture arose in the context of coloring graphs from lists. Alon and Tarsi proved a general theorem regarding coloring graphs when every vertex has a list of colors and the conjecture comes from applying the general theorem to Dinits’ conjecture that can be regarded as a statement about list coloring of the complete bipartite graph . In 1994 Galvin proved the Dinitz conjecture by direct combinatorial proof. See this post. Gian-Carlo Rota and Rosa Huang proved that the Alon-Tarsi conjecture implies the Rota basis conjecture (over ) when is even.

Let be a graph on vertices . Associate to every vertex a variable . Consider the graph polynomial Alon and Tarsi considered the coefficient of the monomial . If this coefficient is non-zero then they showed that for every lists of colors, colors for vertex , there is a legal coloring of the vertices from the lists! Alon and Tarsi went on to describe combinatorially the coefficient as the difference between numbers of even and odd Euler orientations.

Let me mention the paper by Jeannette Janssen The Dinitz problem solved for rectangles that contains a proof based on Alon-Tarsi theorem of a rectangle case of Dinitz conjecture. It is very interesting if the polynomial used by Janssen can be used to prove a “rectangular” (thus a bit weaker) version of the Rota’s basis conjecture. A similar slightly weaker form of the conclusion of Rota’s basis conjecture is conjectured by Ron Aharoni and Eli Berger in much much greater generality.

Posted in Combinatorics, Open problems, Test your intuition
Tagged Alon-Tarsi conjecture, Polymath12
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*This is first of three posts kindly written by Thilo Weinert*

Recently Gil asked me whether I would like to contribute to his blog and I am happy to do so. I enjoy both finite and infinite combinatorics and it seems that these fields drifted apart in recent decades. The latter is a comment not possibly substantiated by personal experience yet rather by my perception of the number and kinds of papers published. I believe that there is no mathematical reason for this separation but that it came about for reasons of the sociology of mathematics—maybe one should create a chair for this subject somewhere sometimes. Anyhow, I would like to write a few words about an area where finite and infinite combinatorics come together, the partition calculus for ordinal numbers.

The Partition Calculus, contemporarily classified as 03E02 under Mathematical Logic/Set Theory was initiated in 1956 with the article *A partition calculus in set theory,* published by Paul Erdos and Richard Rado in the Bulletin of the American Mathematical Society. asserts that for every set of -sized subsets of a set of size there is an of size such that all subsets of of size are in or an of size such that no subset of of size is in . Many mathematicians are concerned with the case where are finite. As in the finite realm the notions of cardinal number and ordinal number coincide, it is there unnecessary to differentiate between these notions of size. Furthermore, via the Well-Ordering-Theorem, the Axiom of Choice implies that for cardinalities the statement is equivalent to where denotes the smallest ordinal number of cardinality . It also implies that there is no such that . Hence, believing the Axiom of Choice, one may limit ones study to the cases in which are order-types and is a natural number. Although there are interesting open questions in contexts where the Axiom of Choice fails and also in contexts where not all of are ordinal numbers, I will for now exclude these from the discussion. That is, what I would like to talk about is the subject of transfinite Ramsey Numbers, is to say that but for no .

First I would like to discuss the lower storeys of the transfinite. It is known that generally for a countable ordinal and a natural number the Ramsey number is countable (If no subscript appears it is understood to be ). When is a finite multiple of or , the number’s calculation is similar in character to the case where is a natural number though it tends to be slightly more involved. For example where is the least number without a digraph on vertices without an independent -tuple and without an induced transitive subtournament on vertices. The are the Tournament Ramsey Numbers which have been investigated slightly more intensely in [1], [2] and exact values are known for . The last paper on these problems was published in 1997 by Jean Larson and William Mitchell in the very first issue of the Annals of Combinatorics. A degree argument yields which gives a good idea about the growth rate of as counterexamples to numbers being finite Ramsey numbers easily carry over. Furthermore they found a digraph on thirteen vertices without a transitive triple or independent quadruple thus establishing .

Recently I started to discuss these problems with Ferdinand Ihringer and Deepak Rajendraprasad. The latter found a digraph with the two aforementioned properties but on fourteen vertices and shortly thereafter they could establish by a triangle-count that there is no such digraph on vertices. So we know now that . Generally these problems should provide a nice playground for people in the business of solving Ramsey-type problems with the help of computers, cf. [3].

The situation is slightly more complicated in the case of finite multiples of as a degree argument only yields a cubic upper bound for which is a number defined such that for all natural numbers and . This is elaborated on in detail in [4].

Recently Jacob Hilton has considered problems of this kind involving additional topological structure alone (cf. [5]) and together with Andr\'{e}s Caicedo (cf. [6]).

In the next post I am going to elaborate on results and open questions

regarding Ramsey numbers for larger countable ordinals.

[1] Kenneth Brooks Reid, Jr. and Ernest Tilden Parker. Disproof of a conjecture of

Erdos and Moser on tournaments. J. Combinatorial Theory, 9 (1970).

[2] Adolfo Sanchez-Flores. On tournaments and their largest transitive subtournaments.

Graphs Combin., 10, 1994.

[3] Stanislaw P. Radziszowski. Small Ramsey numbers. Electron. J. Combin., 1:Dynamic

Survey 1, 30 pp. (electronic), 1994,

[4] Thilo Volker Weinert, Idiosynchromatic poetry. Combinatorica, 34 (2014).

[5] Andres Eduardo Caicedo and Jacob Hilton, Topological Ramsey numbers and countable ordinals.

[6] Jacob Hilton. The topological pigeonhole principle for ordinals.

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