HUJI Thu, 25/10/2018 – 14:30 to 15:30; TAU 29/10/2018 12:15-13:15

**Abstract:** Consider a simplicial complex that allows for an embedding into . How many faces of dimension or higher can it have? How dense can they be? This basic question goes back to Descartes. Using it and other rather fundamental combinatorial problems, I will motivate and introduce a version of Grothendieck’s “standard conjectures” beyond positivity (which will be explored in detail in the Sunday Seminar). All notions used will be explained in the talk (I will make an effort to be very elementary)

Sunday’s seminar refer to a HUJI Kazhdan seminar given by Karim on this topic. (3-5 Ross building’s seminar room.) This is a good opportunity to congratulate Karim Adiprasito and June Huh on receiving the New Horizon Prize, and congratulations also to Vincent Lafforgue who received the breakthrough prize and to all the other winners!

When: Monday Oct.29, 11:00–12:45

Where: Rothberg CS building, room B500, Safra campus, Givat Ram

**Abstract:**

The sharp threshold phenomenon is a central topic of research in the analysis of Boolean functions. Here, one aims to give sufficient conditions for a monotone Boolean function f to satisfy, where , and is the probability that on an input with independent coordinates, each taking the value 1 with probability *p*.

The dense regime, where , is somewhat understood due to seminal works by Bourgain, Friedgut, Hatami, and Kalai. On the other hand, the sparse regime where was out of reach of the available methods. However, the potential power of the sparse regime was suggested by Kahn and Kalai already in 2006.

In this talk we show that if a monotone Boolean function f with satisfies some mild pseudo-randomness conditions then it exhibits a sharp threshold in the interval , with . More specifically, our mild pseudo-randomness hypothesis is that the *p*-biased measure of* f* does not bump up to *Θ(1)* whenever we restrict f to a sub-cube of constant co-dimension, and our conclusion is that we can find such that .

At its core, this theorem stems from a novel hypercontactive theorem for Boolean functions satisfying pseudorandom conditions, which we call `small generalized influences’. This result takes on the role of the usual hypercontractivity theorem, but is significantly more effective in the regime where

We demonstrate the power of our sharp threshold result by reproving the recent breakthrough result of Frankl on the celebrated Erdos matching conjecture, and by proving conjectures of Huang–Loh–Sudakov and Furedi–Jiang for a new wide range of the parameters.

Alef: Koebe’s Karnaf

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**picaboo**

**C***

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I greatly enjoyed all of Monday’s morning lectures. The first very inspiring lecture by my friend and colleague Rafi (Ronald) Coifman was entitled Harmonic analytic geometry in high dimensions – Empirical models (click for the video). Rafi’s research spans across a wide range of areas many of which he himself created and goes from the very applied (e.g., applications of harmonic analysis to pluming, biology, and finance) to the very pure (e.g., applications of wavelets to classical problems in harmonic analysis). The lecture covered a lot of ground, starting with Fourier’s original ideas and his perception that he had discovered the “language of nature” and continuing with wide applications to structural and multi-scale analysis of high dimensional data, and to the possibility, pushing Fourier’s vision one step further, of automatically learning the laws of physics from data.

Toward the third lecture on the history of mathematics by Catherine Goldstein I thought that I could relax and listen to a historical lecture that does not require much mathematical efforts. To my surprise, it was very demanding for me (but fully worth the effort) to follow the mathematics itself. The historical discussion and insights were great. The title of the lecture was Long-term history and ephemeral configurations (click for the video) and it started with a famous quote of Poincaré: Mathematics is the art of giving the same name to different things (Poincaré gave the examples of “groups” and “uniform convergence”.) At the center of the talk was Charles Hermite and the lecture dealt, among other things, with the very interesting question: Is mathematics a natural science? For Hermite the answer was: Yes! Altogether there were a lot of great insights and great lines. (Pictures from these two lectures at the end of the post.)

Knots, three manifolds and instantons

The talk was fantastic, it had great results, the slides were great, the presentation was great, thoughtful, with a lot of food for thought, both for the large audience and (I think) also for experts. A main famous theorem by the speakers is:

**Theorem:** Knots with vanishing Instanton Floer homology and (therefore) also knots with trivial Khovanov’s homology are unknots.

Khovanov’s homology are invariants of groups that refine the famous Jones polynomials and, of course, two problems naturally arise. First, is it the case that the Jones polynomial itself determines unknots? (This is a famous open problem.) And also does Khovanov’s homology or Floer’s homology distinguish different knots? (Maybe the answer for the second bold question is known to be negative…) The lecture had four parts

I was surprised that I had the feeling that I understood everything in the first part. It started with a quick pictorial introduction to what knots are, then looking at the complement of a knot, followed by Dehn’s lemma that was proved by Papakyriakopoulos. (I think but am not sure that Papakyriakopoulos’s proof is still needed for all the stronger results that follow.)

So Papakyriakopoulos’ theorem tells you that the fundamental group of the complement of a non-trivial knot is not Abelian, but could we say something stronger? Peter mentioned that for most, but not all non-trivial small knots the fundamental group maps onto a dihedral group. And the main result is that for all non trivial knots the fundamental group maps onto SO (3). There were two delicate yet important points that were mentioned. The first is that often SO(3) can or should be replaced by its double cover SU(2), and the second is that there is also a crucial condition (that makes the theorem stronger) about the images of “meridians” (small circles around a point on the knot in its complement).

The second part was about ideas, notions and tools needed for the proof of the main theorem. Naturally it was more difficult and for various things I only just pleasantly got a general impression together with some pointers on notions that I should (finally) learn. Connections, flat connections, Chern-Simons functionals, Young-Mills equations and their solutions called “instantons”, and the Floer’s (instanton) homology, … . As you can see from the fourth slide the list of tools that are actually needed for the proof extends even further and Peter and Tomasz also mentioned connections with Ozsváth and Szabo theory of Heegard Floer homology.

Surprisingly, the third part dealt with notions that were somewhat easier for me than those of the second part. The Khovanov homology is a refinement of another famous knot-invariant the Jones polynomial.

I remember hearing a few talks about Khovanov homology in the early 2000. Dror Bar Nathan showed how they appear very naturally and how it is a straight forward matter to compute them (alas, not efficiently). In a different talk some years later David Kazhdan showed how, taking a different point of view, those invariants depend on a sequence of amazing miracles. In any case, the Khovanov homology groups are finite dimensional and the Jones polynomial are just the alternating sums of their dimensions (or Euler characteristics). Like the Jones polynomial themselves there is also some connection (“skein relations“) between the Khovanov homologies of knots when you apply two simple operations on the knots.

The skein relations for Khovanov homology are given in terms of a long exact sequence, and similar relations hold for the Floer homology. Moreover there is some relation (a spectral sequence) between these two exact sequences which shows that when Khovanov homology is not trivial then Floer (instanton) homology is also non trivial and hence from what we already know about Floer homology the knot is not trivial.

and when part IV came I expected that the discussion will be aimed at real experts in the audience and that I could relax and think about other things. However this was not the case. Below the fold I will tell you about the surprising fourth part, and then proceed to talk about various other really interesting things. Statistics tell me that only about a third of the readers read below the fold but this time I truly recommend it.

Next Peter talked about spacial graphs, trivalent graphs to be precise. And when he talked about Tait’s coloring I could see that the aim is to give a new approach for proving the four color theorem. Tait’s colorings are coloring of the edges of trivalent graphs so that the three edges incident to every vertex are colored with the three different colors. (In other words, no two edges that share a vertex have the same color.) A reformulation of the 4CT is that every planar bridgeless trivalent graph has a Tait coloring. In view of the non vanishing theorems of Peter and Tomasz (extended to spacial graphs), for a proof of the four color theorem what is left to be proved is a simple equality:

**Conjecture (Kronheimer and Mrowka):** for (bridgeless) graphs embedded in the plane (hence planar) the dimension of the Floer instanton homology (actually a certain variant called **J**) is equal to the number of Tait colorings.

This is a very nice approach. K&M proved that for spacial graphs embedded in the plane the number of Tait’s coloring is at most the dimension of **J. **This inequality is expected by them to extend to all spacial graphs.

Two remarks on the conjecture:

a) This is an equality, if true it should be provable, no? ( ) Of course the conjecture may well be false. (This conjectural equality implies the 4CT but as far as we know is not a consequence of it, which is also a good sign for the approach.)

b) This equality is proven for bipartite trivalent graphs which reflects a natural test for 4CT approaches. (Can it be proved for bridgeless trivalent Hamiltonian planar graphs?)

c) For more on this part of the lecture see P&T’s papers Tait colorings, and an instanton homology for webs and foams; and Exact triangles for SO(3) instanton homology of webs.

More comments and links:

a) The Kronheimer-Mrowka theorem was used by Greg Kuperberg in his 2012 proof that under GRH, telling if a knot is non-trivial is in NP. An earlier 2002 (completely different) proof was described by Ian Agol. (This is a great result in computational complexity and as far as I can see the two proofs represent a rare meeting point of “two cultures” in 3-dimensional topology.) See this post. Joel Hass, Jeff Lagarias, and Bill Thurston proved that telling if a knot is trivial is in NP.

b) The four color theorem. Both before and after the amazing 1976 proof by Appel and Haken of the 4CT, the four color theorem was surely on the minds of combinatorialists and not only them. Let me come back to the 4CT in some future post but here are some related earlier posts on coloring simple polytopes and triangulations, another on coloring circles and pseudocircles (I, II), an MO question generalization of the 4CT and a GLL post. **Update:** I forgot that one of the answers to my MO question by Ian Agol is actually about Peter and Tomasz’ Instanton homology conjecture.

c) Christos Dimitriou Papakyriakopoulos (papa), was a great mathematical hero. John Milnor wrote a famous poem

**The perfidious lemma of Dehn**

**Was every topologist’s bane **

**‘Til Christos D. Pap- **

**akyriakop-**

**oulos proved it without any strain.**

According to this post, Gina’s cat was named after Papakyriakopoulos. (Gina was the hero of a book that I wrote.)

d) In the first part Tomasz mentioned dihedral representations (pursued by Fox). The fundamental group of most small knots maps onto a dihedral group (with only 22 out of 3000 exceptions). It would be nice to understand it for some model of random knots. (See the paper models of random knots by Chaim Even-Zohar, and references there including to the recent “Pataluma model” of Even-Zohar, Joel Hass, Nati Linial, and Tahl Nowik.)

e) Peter Guthrie Tait (from Tait colorings) conjectured that every planar trivalent bridgeless graph is Hamiltonian which was disproved by Tutte. (This conjecture easily implies the 4CT.) He also made several exciting and important conjectures about knots that were all proved.

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In future posts I will tell you a little about their mathematics. See also these posts about stories and math related to Ricky ( A Discrepancy Problem for Planar Configurations; Many triangulated three-spheres!; Academic Degrees and Sex; GLL: Polls and prediction and P=NP . ) and to Branko ( My Copy of Branko Grünbaum’s Convex Polytopes; The World of Michael Burt: When Architecture, Mathematics, and Art meet; Coloring Simple Polytopes and Triangulations; Budapest, Seattle, New Haven ; How the g-Conjecture Came About; GLL: What are proofs for , anyway.) (Links for blog posts on other blogs are welcome.)

**Branko with Janos Pach**

**The legendary mathematical partnership and friendship of Ricky and Eli Goodman is rare in mathematics.**

**Five generations in Seattle: Branko was the advisor of Micha A Perles (and also of Joram Lindenstrauss, Moshe Rosenfeld, Joseph Zaks and many others). Micha was my advisor (and also of Meir Katchalski, Michael Kallay, Nati Linial Noga Alon and many others). Isabella Novik was my student and here we are with four student of Isabella: Michael Goff, Kurt Luoto, Andy Frohmaderand, and Steven Klee. (From left to right.)**

**Saugata Basu, Ricky Pollack and Marie-Francois Roy’s book Algorithms in real algebraic geometry.**

Here are the (slightly improved) slides.

I made the mistake of trying to improve my slides in the evening before the lecture and in the morning I discovered that the file disappeared (of course, after the lecture it reappeared) and I had to reconstruct it from an earlier version.

And here is the playlist of all plenary lectures:

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Alef’s previous corner: There is still a small gap in the proof.

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**With Peter Sarnak, Stas Smirnov, and Tadashi Tokieda at Sugar Loaf ( Pão de Açúcar), Rio.**

**At Sugar Loaf with Stas and Edward Dunne**

This is my second report from ICM 2018. Some acronyms are used in this post. ICM = International Congress of Mathematicians; IMU = International Mathematical Union, an organization running the ICMs’ GA = General assembly (of the IMU) a body of representatives from the member countries of the IMU.

And a special bonus at the end of the post: Test your intuition question.

Here is a post from the ICM 2018 web-site about my lecture: Kalai plenary ponders possibilities of quantum computing. And here is the link to the ICM 2018 site itself.

The IMU Leelavati Prize is an award for outstanding contribution to public outreach in mathematics. Unlike most prizes “Leelavati” is not a name of a person (a man in most cases) but rather the name of 12th-century Indian mathematical treatise. The story of this year’s laureate, the Turkish mathematicians Ali Nesin and Sevan Nişanyan who built together a mathematical village to educate and advance children, is moving and quite amazing.

A rare standing ovation for Nasin, and Nişanyan

Nesin (left) and Nişanyan

Atiya, Nesin, and Nişanyan

Some members of the Russian delegation after Saint Petersburg was announced as the winner (left), Stas Smirnov and François Loeser shake hands after the announcement. Both François and Stas were stars of my Hyderabad ICM 2010 reports. (I,II).

Mathematicians love sequences – here is the sequence of cities of post WWII ICMs

Cambridge (MA); Amsterdam, Edinburgh, Stockholm, Moscow, Nice, Vancouver, Helsinki, Warsaw, Berkeley, Kyoto, Zurich, Berlin, Beijing, Madrid, Hyderabad, Seoul, Rio,…

What is the law governing these choices? All these countries have good mathematics and a few have superb mathematics. Some of these cities represented democratic countries and some did not, and there are many shades of grey. The hosting countries also differ in terms of economic systems, human rights, foreign policies and various other things. If there is an emerging law for the sequence of cities it is that first, except for extreme cases, it is largely politically blind, and second, that an attempt is made to outreach and make ICMs s truly world wide experience. (The second item also characterizes other important activities of the IMU.)

This time the choice was between two fantastic cities, Paris and Saint Petersburg. The IMU executive committee had chosen Saint Petersburg, but this was challenged (for the first time, as far as I know) and at the end the GA democratically voted 83:63 (4 abstentions) for Saint Petersburg. For the full list of resolutions of the GA see this page. My feeling is that also many mathematicians from Russia and the former Soviet Union who live abroad (many also in Israel) were happy with this opportunity of a “mathematical reunion” in 2022. There was one additional GA resolution that caused some controversy that maybe I will mention in a future post.

Anyway, another exciting moment at the closing ceremony was

Andrei Okounkov and Stas Smirnov came with matching casual clothes and declared that this was because it was time for them to start working. I first met Andrei and Stas in Saint Petersburg in a 2004 conference celebrating Anatoly (Tolya) Vershik’s 70th birthday. (Here is a link to Vershik’s amazing home page.) (Picture: ICM 2018 site)

**The Guardian (yesterday) **

With the landmark decision (Thursday, yesterday) of the Indian supreme court, homosexual sexual activity is now legal in every country that ever hosted the ICM.

A few more ICM 2018 pictures

Answer to trivia question (use your mouse): Homosexuality was nationwide legalized in Brazil in 1831 and in the US in 2003 (172 years later).

Here is when homosexual sexual relations were legalized in some ICM hosted countries and other selected countries. France – 1791; The Netherlands – 1811; Brazil – 1831, Turkey – 1858, Switzerland – 1942; Jordan – 1951; Spain – 1979, United Kingdom – 1982 (Scotland, 1981, England and Wales 1967); Israel – 1988; Russia 1997 (Illegal in practice in Chechnya); China – 1998; U.S. 2003; Korea (both South and North) always legal; India – Apparently 2018, yesterday – August, 6, 2018. (See Omer’s comment about Russia and Wikipedea sources Asia, Europe, Americas, Africa).

]]>**Abstract:** We show that for any finite set *P* of points in the plane and *ϵ>0* there exist points in , for arbitrary small *γ>0*, that pierce every convex set *K* with* |K∩P|≥ϵ|P|*.

This is the first improvement of the bound of that was obtained in 1992 by Alon, Bárány, Füredi and Kleitman for general point sets in the plane.

]]>Here are the slides for my lecture Noise Stability, Noise Sensitivity and the Quantum Computer Puzzle.

There were many national receptions at the ICM (I missed a few but made it to the Abel reception and Japan’s reception). This led to the idea of having a reception by a mathematical discipline. We are thinking of a combinatorial event in ICM2022 as follows:

**Time:** Tuesday evening – the second week of ICM 2022

**Location:** A park in Saint Petersburg (TBA)

**Activity** (tentative): Light food and drinks will be available (to buy). **Music with a DJ and dancing.**

**Invitees** (tentative): The event will be open for all interested ICM2022 participants.

**T-shirts** (very tentative, other suggestions welcome)

background colors: red, blue, white and rainbow

ICM 2018 could be considered as an excellent conference in combinatorics on its own. There were eleven top-of-the-line speakers (one joint with probability) in wide areas of combinatorics who gave very good talks; a separate session of much combinatorial content on theoretical computer science; many lectures in other sessions (alas, a few conflicting) with interest to combinatorialists; and, in addition, many of the plenary talks had a strong combinatorial content or connections. Like for ICM2010 (Post 1 Post 2) I do plan to write more impressions from ICM2018 at a later time. Meanwhile, a few pictures (more, later).

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You can test yourself in the questions below and then look at the Wikipedea article for the answers.

Not since the Mondial’s final match between France and Croatia, has an excitement of this magnitude been witnessed here. On Wednesday August 1, at 14:30 Jerusalem time, mathematics fans will fill bars, restaurants and the beaches, and view on large screens the exciting opening ceremonies of ICM 2018. Well, perhaps not quite like that, but you can all watch the opening ceremony (8:30 Rio Time, August 1, 2018) in this link http://www.icm2018.org/ . (For the actual talks in real time you need to go to Rio, but I was told that videotaped lectures will be available shortly afterwards.) I am very proud that essentially I never knew in advance the identity of prize winners (except twice but I had a good excuse both times).

I asked quite a few people for advice about my talk and Tammy Ziegler mentioned to me that she heard many excellent plenary talks and that Etienne Ghys’ talk stood out among them all. Let me share with you the link to Ghys’ 2006 lecture http://www.icm2006.org/video/ – session four.

My own lecture is scheduled for the last day of the meeting, Thursday August 9, 8:30-9:30.

I dont know the answers to other exciting questions like “will Tim Gowers’s blog from Rio 2018” as he did in ICM 2010 and ICM 2014? Here is, for example, Tim’s beautiful post on Arthur’s plenary lecture. This post mentioned France’s bid for ICM2022, and indeed the choice was between two great cities Paris and Saint Petersburg. Earlier this evening the ICM2018 Facebook posted the following:

This was followed 20 minutes later by this post

This, and other important matters are being decided in the IMU general assembly in Sao Paulo as we speak.

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