Jürgen Eckhoff, Ascona 1999
Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first meeting in Oberwolfach in 1982. Here is a description of my mathematical dialogue with Jürgen Eckhoff:
Summary 1) (1980) we found independently two proofs for the same conjecture; 2) (1982) I solved Eckhoff’s Conjecture; 3) Jurgen (1988) solved my conjecture; 4) We made the same conjecture (around 1990) that Andy Frohmader solved in 2007, and finally 5) (Around 2007) We both found (I with Roy Meshulam, and Jürgen with Klaus Peter Nischke) extensions to Amenta’s Helly type theorems that both imply a topological version.
(A 2009 KTH lecture based on this post or vice versa is announced here.)
Let me start from the end:
5. 2007 – Eckhoff and I both find related extensions to Amenta’s theorem.
Nina Amenta proved a remarkable extension of Helly’s theorem. Let be a finite family with the following property:
(a) Every member of is the union of at most r pairwise disjoint compact convex sets.
(b) So is every intersection of members of .
If every r(d+1) members of has a point in common, then all members of have a point in common!
The case r=1 is Helly’s theorem, Grünbaum and Motzkin proposed this theorem as a conjecture and proved the case r=2. David Larman proved the case r=3.
Roy Meshulam and I studied a topological version of the theorem, namely you assume that every member of F is the union of at most r pairwise disjoint contractible compact sets in $R^d$ and that all these sets together form a good cover – every nonempty intersection is either empty or contractible. And we were able to prove it!
Eckhoff and Klaus Peter Nischke looked for a purely combinatorial version of Amenta’s theorem which is given by the old proofs (for r=2,3) but not by Amenta’s proof. An approach towards such a proof was already proposed by Morris in 1968, but it was not clear how to complete Morris’s work. Eckhoff and Nischke were able to do it! And this also implied the topological version for good covers.
The full results of Eckhoff and Nischke and of Roy and me are independent. Roy and I showed that if the nerve of is d-Leray then the nerve of is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of has Helly number d, then the nerve of has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of is d-collapsible then the nerve of F is ((d+1)r-1)-collapsible.
Here, a simplicial comples K is d-Leray if all homology groups vanishes for every and every induced subcomplex L of K.
Roy and I were thinking about a common homological generalization which will include both results but so far could not prove it.
And now let me move to the beginning:
1. 1981 – we give different proofs for the Perles-Katchalski Conjecture
This week we are celebrating in Cambridge MA , and elsewhere in the world, Richard Stanley’s birthday. For the last forty years, Richard has been one of the very few leading mathematicians in the area of combinatorics, and he found deep, profound, and fruitful links between combinatorics and other areas of mathematics. His works enriched and influenced combinatorics as well as other areas of mathematics, and, in my opinion, combinatorics matured greatly as a mathematical discipline thanks to his work.
Correct or incorrect?
(1) Richard drove cross-country at least 8 times (2) In his youth, at a wild party, Richard Stanley found a proof of FLT consisting of a few mathematical symbols. (3) Richard jumped at least once from an airplane (4) Richard is actively interested in the study of consciousness (5) Richard found a mathematical way to divide by zero
Seven Early Papers by Richard Stanley That You Must Read.
Combinatorics and Commutative Algebra
(1) R. P. Stanley, The upper bound conjecture and Cohen-Macaulay rings. Studies in Appl. Math. 54 (1975), no. 2, 135–142. The two seminal papers (1) and (3) (below) showed remarkable and unexpected applications of commutative algebra to combinatorics. In each of these papers a central conjecture in combinatorics was solved in a completely unexpected way which was the basis for a later remarkable theory. Paper (1) is the starting point for the interrelation between commutative algebra and combinatorics of simplicial complexes and their topology. In this work Richard Stanley proved the Motzkin-Klee upper bound conjecture for triangulations of spheres. This conjecture asserts that the maximum number of k-faces for a triangulation of a (d-1)-dimensional sphere with n vertices is attained by the boundary complex of the cyclic d-dimensional polytope with n vertices. Peter McMullen proved this conjecture for simplicial polytopes and Richard Stanley proved it for arbitrary triangulations of spheres. The key point was that a certain ring (the Stanley-Reisner ring) associated with a simplicial polytope has the Cohen-Macaulay property. The connection between combinatorics and commutative algebra is far reaching, and in subsequent works combinatorial problems led to developments in commutative algebra and techniques from the two areas were combined. A more recent important paper by Richard on applications of commutative algebra for the study of face numbers is: R. P. Stanley, Subdivisions and local h-vectors. J. Amer. Math. Soc. 5 (1992), no. 4, 805–851. And here is, a few weeks old important development in this theory: Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums, by Karim A. Adiprasito and Raman Sanyal.
The Cohen-Macaulay property, magic squares and lattice points in polytopes
(2) R. P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43 (1976), no. 3, 511–531. This paper starts with a theorem about enumeration of certain magic squares. Solving a long-standing open problem, Stanley proved that the generating function for the number of k by k integer matrices (k– fixed) with nonnegative entries and row sums and column sums equal to nis rational. This is the starting point of a deep algebraic theory of integral points in polyhedra.
Enters the Hard-Lefshetz Theorem: McMullen’s g-conjecture
(3) R. P. Stanley, The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), no. 3, 236–238. The g-conjecture proposes a complete characterization of face numbers of d-dimensional polytopes. One linear equality that holds among face numbers is, of course, the Euler-Poincaré relation. This relation implies additional [d/2] equalities called the Dehn-Sommerville relations. Peter McMullen proposed an additional system of linear and nonlinear inequalities as a complete characterization of face numbers of polytopes. The sufficiency part of this conjecture was proved by Billera and Lee. Richard Stanley’s brilliant proof for McMullen’s inequalities that established the g-conjecture was based on the Hard Lefschetz Theorem from algebraic topology. Starting from a simplicial polytope P (with rational vertices) we associate to it a toric variety T(P). It turned out that the cohomology ring of this variety is closely related to the Stanley-Reisner ring mentioned above. The Hard Lefschetz Theorem implies an algebraic property of the Stanley-Reisner ring from which McMullen inequalities can be deduced by direct combinatorial reasoning. Richard found a number of other combinatorial applications of the Hard Lefschetz theorem (including the solution of the Erdos-Moser conjecture).
Here is the abstract of Lou Billera’s lecture
LOUIS BILLERA (CORNELL)
Even more intriguing, if rather less plausible…
The title is how Peter McMullen described his own conjectured characterization of the f-vectors of simplicial polytopes in his 1971 lecture notes on the upper bound conjecture written with Geoffrey Shephard. Yet by the end of that decade, the so-called g-conjecture would become the g-theorem, and algebraic combinatorics (as practiced at MIT) would have attracted the attention of mainstream mathematics, almost entirely due to the startling proof given by Richard Stanley.
I will briefly describe some of the events leading to this proof and some of its still developing consequences.
Enumeration is Richard’s true mathematical love. Richard’s monumental books EC1 and EC2 (The picture is of EC1 and a young fan) (4) A baker’s dozen of conjectures concerning plane partitions, in Combinatoire Énumérative (G. Labelle and P. Leroux, eds.), Lecture Notes in Math., no. 1234, Springer-Verlag, Berlin/Heidelberg/New York, 1986, pp. 285-293. 13 beautiful conjectures on counting plane partitions with various forms of symmetry. (5) Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp 100-141. For me this was the best introduction to generating functions, clear and inspiring. The entire MAA 1978 Rota’s blue little volume on combinatorics is great. Buy it!
(6) Supersolvable lattices, Algebra Universalis 2 (1972), 197-217. This paper provides a profound link between group theory and the study of partially ordered sets. It can be seen as a starting point of Stanley’s own work on the Cohen-Macaulay property and it had much influence on later works on combinatorial properties of lattices of subgroups by Quillen and many others, and also on the study of POSETS (=partially ordered sets) arising from arrangements of hyperplanes. The algebraic notion of supersolvable groups is translated to an important combinatorial notion for partially ordered sets. (There is a more detailed paper which I could not find online: R. P. Stanley, Supersolvable semimodular lattices. Mobius algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 80–142. Univ. Waterloo, Waterloo, Ont., 1971.)
Combinatorics and representation theory
(7) On the number of reduced decompositions of elements of Coxeter groups, European J. Combinatorics 5 (1984), 359-372. This paper gives an important result proved using representation theory. It is one of many results by Stanley on connections between enumerative combinatorics, representation theory, and invariant theory. Again, this paper represents an exciting area of research about the connection of enumerative combinatorics and representation theory that I am less familiar with. A very inspiring survey paper is: Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (new series) 1 (1979), 475-511. Here are abstracts of two lectures from the meeting on some recent developments in combinatorial representation theory and symmetric functions.
GRETA PANOVA (UCLA)
The Kronecker coefficients: an unexpected journey
Kronecker coefficients live at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their journey started 75 years ago, they still haven’t found their explicit positive combinatorial formula, and present a major open problem in algebraic combinatorics. Recently, they were given a new role in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the “”P vs NP”” problem.
We will take the Kronecker coefficients to asymptotics land and bound them. As an unexpected consequence of this trip, we find bounds for the difference of consecutive coefficients in the q-binomial coefficients (as polynomial in q), generalizing Sylvester’s unimodality theorem and connecting with results of Richard Stanley. Joint work with Igor Pak.
THOMAS LAM (U MICHIGAN)
Truncations of Stanley symmetric functions and amplituhedron cells
Stanley symmetric functions were invented (by Stanley) with applications to the enumeration of reduced words in the symmetric group in mind. Recently, the “amplituhedron” was introduced in the study of scattering amplitudes in N=4 super Yang Mills. I will talk about a formula for the cohomology class of a (tree) amplituhedron variety as the truncation of an affine Stanley symmetric function.
Two combinatorial applications of the Aleksandrov-Fenchel inequalities;
(7) Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combinatorial Theory (A) 31 (1981), 56-65. In this amazing paper Stanley used inequalities of classical convexity to settle an important conjecture on probability of events in partially ordered sets. A special case of the conjecture was settled earlier by Ron Graham using the FKG inequality. The profound relation between classical convexity inequalities, combinatorial structures, polytopes, and probability theory was further studied by many authors including Stanley himself and there is much more to be done. I see that I ran out of my seven designated slots. Certainly you should read Richard’s combinatorial constructions of polytopes, like Two poset polytopes, Discrete Comput. Geom. 1 (1986), 9-23, and his papers on arrangements. Let me mention a more recent paper of Stanley in this general area: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron (with J. Pitman), Discrete Comput. Geom., 27 (2002), 603-634.
(Mostly from RS’s homepage.)
Chess and Mathematics
An unusual method for proving the Riemann hypothesis.
Richard Stanley’s Mathoverflow question on MAGIC
Richard the Catalan
(From RS’s homepage) Excerpt (27 page PDF file) from EC2 on problems related to Catalan numbers (including 66 combinatorial interpretations of these numbers). Solutions to Catalan number problems from the previous link (23 page PDF file). Catalan addendum (Postscript or PDF) (version of 25 May 2013; 96 pages). An addendum of new problems (and solutions) related to Catalan numbers. Current number of combinatorial interpretations of Cn: 207. The material on Catalan numbers is being collected into a monograph, to be published by Cambridge University Press in late 2014 or early 2015.
My dear friend Itai Benjamini told me that he won’t be able to make it to my Tuesday talk on influence, threshold, and noise, and asked if I already have the slides. So it occurred to me that perhaps I can practice the lecture on you, my readers, not just with the slides (here they are) but also roughly what I plan to say, some additional info, and some pedagogical hesitations. Of course, remarks can be very helpful.
I can also briefly report that there are plenty of exciting things happening around that I would love to report about – hopefully later in my travel-free summer. One more thing: while chatting with Yuval Rabani and Daniel Spielman I realized that there are various exciting things happening in algorithms (and not reported so much in blogs). Much progress has been made on basic questions: TSP, Bin Packing, flows & bipartite matching, market equilibria, and k-servers, to mention a few, and also new directions and methods. I am happy to announce that Yuval kindly agreed to write here an algorithmic column from time to time, and Daniel is considering contributing a guest post as well.
The second AMS-IMU meeting
Since the early 70s, I have been a devoted participants in our annual meetings of the Israeli Mathematical Union (IMU), and this year we will have the second joint meeting with the American Mathematical Society (AMS). Here is the program. There are many exciting lectures. Let me mention that Eran Nevo, this year Erdős’ prize winner, will give a lecture about the g-conjecture. Congratulations, Eran! Among the 22 exciting special sessions there are a few related to combinatorics, and even one organized by me on Wednsday and Thursday.
Contact person: Gil Kalai, firstname.lastname@example.org
TAU, Dan David building, Room 103
|Wed, 10:50-11:30||Van H. Vu (Yale University)||Real roots of random polynomials (abstract)|
|Wed, 11:40-12:20||Oriol Serra (Universitat Politecnica de Catalunya, Barcelona)||Arithmetic Removal Lemmas (abstract)|
|Wed, 12:30-13:10||Tali Kaufman (Bar-Ilan University)||Bounded degree high dimensional expanders (abstract)|
|Wed, 16:00-16:40||Rom Pinchasi (Technion)||On the union of arithmetic progressions (abstract)|
|Wed, 16:50-17:30||Isabella Novik (University of Washington, Seattle)||Face numbers of balanced spheres, manifolds, and pseudomanifolds (abstract)|
|Wed, 17:40-18:20||Edward Scheinerman (Johns Hopkins University, Baltimore)||On Vertex, Edge, and Vertex-Edge Random Graphs (abstract)|
|Thu, 9:20-10:00||Yael Tauman Kalai (MSR, New England)||The Evolution of Proofs in Computer Science (abstract)|
|Thu, 10:10-10:50||Irit Dinur (Weitzman Institute)||Lifting locally consistent solutions to global solutions (abstract)|
|Thu, 11:00-11:40||Benny Sudakov (ETH, Zurich)||The minimum number of nonnegative edges in hypergraphs (abstract)|
And now for my own lecture.
Influence, Threshold, and Noise:
The National Academy of Sciences of Armenia together American University of Armenia are organizing a memorial workshop on extremal combinatorics, cryptography and coding theory dedicated to the 60th anniversary of the mathematician Levon Khachatrian. Professor Khachatrian started his academic career at the Institute of Informatics and Automation of National Academy of Sciences. From 1991 until the end of his short life in 2002 he spent at University of Bielefeld, Germany where Khachatrian’s talent flourished working with Professor Rudolf Ahlswede. Professor Khachatrian’s most remarkable results include solutions of problems dating back over 40 years in extremal combinatorics posed by the world famous mathematician Paul Erdos. These problems had attracted the attention of many top people in combinatorics and number theory who were unsuccessfully in their attempts to solve them. At the workshop in Yerevan we look forward to the participation of invited speakers (1 hour presentations), researchers familiar with Khachatrian’s work, as well as contributed papers in all areas of extremal combinatorics, cryptography and coding theory.
The American University of Armenia (www.aua.am) is proud to host the workshop.
Workshop chair: Gurgen Khachatrian
For any inquiries please send E-mail to: email@example.com
Here is one of the central and oldest problems in combinatorics:
Problem: Can you find a collection S of q-subsets from an n-element set X set so that every r-subset of X is included in precisely λ sets in the collection?
A collection S of this kind are called a design of parameters (n,q,r, λ), a special interest is the case λ=1, and in this case S is called a Steiner system.
For such an S to exist n should be admissible namely should divide for every .
There are only few examples of designs when r>2. It was even boldly conjectured that for every q r and λ if n is sufficiently large than a design of parameters (n,q,r, λ) exists but the known constructions came very very far from this. … until last week. Last week, Peter Keevash gave a twenty minute talk at Oberwolfach where he announced the proof of the bold existence conjecture. Today his preprint the existence of designs, have become available on the arxive.
The existence of designs and Steiner systems is one of the oldest and most important problems in combinatorics.
1837-1853 – The existence of designs and Steiner systems was asked by Plücker(1835), Kirkman (1846) and Steiner (1853).
1972-1975 – For r=2 which was of special interests, Rick Wilson proved their existence for large enough admissible values of n.
1985 -Rödl proved the existence of approximate objects (the property holds for (1-o(1)) r-subsets of X) , thus answering a conjecture by Erdös and Hanani.
1987 – Teirlink proved their existence for infinitely many values of n when r and q are arbitrary and λ is a certain large number depending on q and r but not on n. (His construction also does not have repeated blocks.)
2014 – Keevash’s proved the existence of Steiner systems for all but finitely many admissible values of n for every q and r. He uses a new method referred to as Randomised Algebraic Constructions.
Update: Just 2 weeks before Peter Keevash announced his result I mentioned the problem in my lecture in “Natifest” in a segment of the lecture devoted to the analysis of Nati’s dreams. 35:38-37:09.
Update: Danny Calegary pointed out a bird-eye similarity between Keevash’s strategy and the strategy of the recent Kahn-Markovic proof of the Ehrenpreis conjecture http://arxiv.org/abs/1101.1330 , a strategy used again by Danny and Alden Walker to show that random groups contain fundamental groups of closed surfaces http://arxiv.org/abs/1304.2188 .
Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 -sphere have?
1) Since the number of facets in an n-vertex triangulation of a 3-sphere is at most quadratic in n, an upper bound for the number of triangulations of the 3-sphere with n vertices is . For certain classes of triangulations, Dey removed in 1992 the logarithmic factor in the exponent for the upper bound.
2) Goodman and Pollack showed in 1986 that the number of simplicial 4-polytopes with n vertices is much much smaller . This upper bound applies to simplicial polytopes of every dimension d, and Alon extended it to general polytopes.
3) Before the new paper the world record was the 2004 lower bound by Pfeifle and Ziegler –
4) In 1988 I constructed triangulations of the d-spheres with n vertices. The new construction gives hope to improve it in any odd dimension by replacing [d/2] by [(d+1)/2] (which match up to logn the exponent in the upper bound). [Update (Dec 19) : this has now been achieved by Paco Santos (based on a different construction) and Nevo and Wilson (based on extensions of their 3-D constructions). More detailed to come.]
The conference Poster as designed by Rotem Linial
A conference celebrating Nati Linial’s 60th birthday will take place in Jerusalem December 16-18. Here is the conference’s web-page. To celebrate the event, I will reblog my very early 2008 post “Nati’s influence” which was also the title of my lecture in the workshop celebrating Nati’s 50th birthday.
When do we say that one event causes another? Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places. Now, if causality is not complicated enough, we can ask what is the influence one event has on another one. Michael Ben-Or and Nati Linial wrote a paper in 1985 where they studied the notion of influence in the context of collective coin flipping. The title of the post refers also to Nati’s influence on my work since he got me and Jeff Kahn interested in a conjecture from this paper.
The word “influence” (dating back, according to Merriam-Webster dictionary, to the 14th century) is close to the word “fluid”. The original definition of influence is: “an ethereal fluid held to flow from the stars and to affect the actions of humans.” The modern meaning (according to Wictionary) is: “The power to affect, control or manipulate something or someone.”
Ben-Or and Linial’s definition of influence
Collective coin flipping refers to a situation where n processors or agents wish to agree on a common random bit. Ben-Or and Linial considered very general protocols to reach a single random bit, and also studied the simple case where the collective random bit is described by a Boolean function of n bits, one contributed by every agent. If all agents act appropriately the collective bit will be ‘1’ with probability 1/2. The purpose of collective coin flipping is to create a random bit R which is immune as much as possible against attempts of one or more agents to bias it towards ‘1’ or ‘0’. Continue reading
The Cap Set problem
We presented Meshulam’s bound for the maximum number of elements in a subset A of not containing a triple x,y,x of distinct elements whose sum is 0.
The theorem is analogous to Roth’s theorem for 3-term arithmetic progressions and, in fact, it is a sort of purified analog to Roth’s proof, as some difficulties over the integers are not presented here. There are two ingredients in the proof: One can be referred to as the “Hardy-Littlewood circle method” and the other is the “density increasing” argument.
We first talked about density-increasing method and showed how KKL’s theorem for influence of sets follows from KKL’s theorem for the maximum individual influence. I mentioned what is known about influence of large sets and what is still open. (I will devote to this topic a separate post.)
Then we went over Meshulam’s proof in full details. A good place to see a detailed sketch of the proof is in this post What is difficult about the cap-set problem? on Gowers’ blog.
Let me copy Tim’s sketch over here:
Sketch of proof (from Gowers’s blog).
Next, here is a brief sketch of the Roth/Meshulam argument. I am giving it not so much for the benefit of people who have never seen it before, but because I shall need to refer to it. Recall that the Fourier transform of a function is defined by the formula
where is short for stands for and is short for Now
(Here stands for since there are solutions of ) By the convolution identity and the inversion formula, this is equal to
Now let be the characteristic function of a subset of density Then Therefore, if contains no solutions of (apart from degenerate ones — I’ll ignore that slight qualification for the purposes of this sketch as it makes the argument slightly less neat without affecting its substance) we may deduce that
Now Parseval’s identity tells us that
from which it follows that
Recall that The function is constant on each of the three hyperplanes (here I interpret as an element of ). From this it is easy to show that there is a hyperplane such that for some absolute constant (If you can’t be bothered to do the calculation, the basic point to take away is that if then there is a hyperplane perpendicular to on which has density at least where is an absolute constant. The converse holds too, though you recover the original bound for the Fourier coefficient only up to an absolute constant, so non-trivial Fourier coefficients and density increases on hyperplanes are essentially the same thing in this context.)
Thus, if contains no arithmetic progression of length 3, there is a hyperplane inside which the density of is at least If we iterate this argument times, then we can double the (relative) density of If we iterate it another times, we can double it again, and so on. The number of iterations is at most so by that time there must be an arithmetic progression of length 3. This tells us that we need lose only dimensions, so for the argument to work we need or equivalently
We discussed error-correcting codes. A binary code C is simply a subset of the discrete n-dimensional cube. This is a familiar object but in coding theory we asked different questions about it. A code is linear if it forms a vector space over The minimal distance of a code is the minimum Hamming distance between two distinct elements, and in the case of linear codes it is simply the minimum weight of a non-zero element of the codes. We mentioned codes over larger alphabets, spherical codes and even codes in more general metric spaces. Error-correcting codes are among the most glorious applications of mathematics and their theory is related to many topics in pure mathematics and theoretical computer science.
1) An extremal problem for codes: What is the maximum size of a binary code of length n with minimal distance d. We mentioned the volume (or Hamming) upper bound and the Gilbert-Varshamov lower bound. We concentrated on the case of codes of positive rate.
2) Examples of codes: We mentioned the Hamming code and the Hadamard code and considered some of their basic properties. Then we mentioned the long code which is very important in the study of Hardness of computation.
3) Linearity testing. Linearity testing is closely related to the Hadamard code. We described Blum-Luby-Rubinfeld linearity test and analyzed it. This is very similar to the Fourier theoretic formula and argument we saw last time for the cap problem.
We start to describe Delsartes linear Programming method to be continued next week.
First passage percolation
1) Models of percolation.
We talked about percolation introduced by Broadbent and Hammersley in 1957. The basic model is a model of random subgraphs of a grid in n-dimensional space. (Other graphs were considered later as well.) Here, a grid is a graph whose vertices have integers coordinates and where two vertices are adjacent if their Euclidean distance is one. Every edge of the grid-graph is taken (or is “open” in the percolation jargon) with the same probability p, independently. We mentioned some basic questions – is there an infinite component? How many infinite components are there? What is the probability that the origin belongs to such an infinite component as a function of p?
I mentioned two results: The first is Kesten’s celebrated result that the critical probability for planar percolation is 1/2. The other by Burton and Keane is that in very general situations almost surely there is a unique infinite component or none at all. This was a good point to mention a famous conjecture- The dying percolation conjecture (especially in dimension 3) which asserts that at the critical probability there is no infinite component.
We will come back to this basic model of percolation later in the course, but for now we moved to a related more recent model.
2) First passage percolation
We talked about first passage percolation introduced by Hammersley and Welsh in 1965. Again we consider the infinite graph of a grid and this time we let the length of every edge be 1 with probability 1/2 and 2 with probability 1/2 (independently). These weights describe a random metric on this infinite graph that we wish to understand. We consider two vertices (0,0) and (v,0) (for high dimension the second entry can account for a (d-1) dimensional vectors, but we can restrict our attention to d=2) and we let D(x) be the distance between these two vectors. We explained how D is an integer values function on a discrete cube with Liphshitz constant 1. The question we want to address is : What is the variance of D?
Why do we study the variance, when we do not know exactly the expectation, you may ask? (I remember Lerry Shepp asking this when I talked about it at Bell Labs in the early 90s.) One answer is that we know that the expectation of D is linear, and for the variance we do not know how it behaves. Second, we expect that telling the expectation precisely will depend on the model while the way the variance grows and perhaps D‘s limiting distribution, will be universal (say, for dimension 2). And third, we do not give up on the expectation as well.
Here is what we showed:
1) From the inequality we derived Kesten’s bound var (D) =O(v).
2) We considered the value s so that , and showed by the basic inequality above that the variance of D conditioned on D>s is also bounded by v. This corresponds to exponential tail estimate proved by Kesten.
3) Using hypercontractivity we showed that the variance of D conditioned on D>s is actually bounded above by v/log (1/t) which corresponds to Talagrand’s sub-Gaussian tail-estimate.
4) Almost finally based on a certain very plausible lemma we used hypercontructivity to show that most Fourier coefficients of D are above the log v level, improving the variance upper bound to O(v/log v).
5) Since the plausible lemma is still open (see this MO question) we showed how we can “shortcut” the lemma and prove the upper bound without it.
The major open question
It is an open question to give an upper bound of or even which is the expected answer in dimension two. Michel Ledoux wisely proposes to prove it just for directed percolation in the plane (where all edges are directed up and right) from (0,0) to (v,v) where the edge length is Gaussian or Bernoulli.
Three Further Applications of Discrete Fourier Analysis (without hypercontractivity)
The three next topics will use Fourier but not hypercontractivity. We start by talking about them.
1) The cap-set problem, some perspective and a little more extremal combinatorics
We talked about Roth theorem, the density Hales Jewett theorem, the Erdos-Rado delta-system theorem and conjecture. We mentioned linearity testing.
2) Upper bounds for error-correcting codes
This was a good place to mention (and easily prove) a fundamental property used in both these cases: The Fourier transform of convolutions of two functions f and g is the product of the Fourier transform of f and of g.
3) Social choice and Arrow’s theorem
The Fourier theoretic proof for Arrow’s theorem uses only Parseval’s formula so we are going to start with that.
Fourier-theoretic proof of Arrows theorem and related results.
We talked a little about Condorcet(we will later give a more detailed introduction to social choice). We mentioned Condorcet’s paradox, Condorcet’s Jury Theorem, and the notion of Condorcet winner.
Next we formulated Arrow’s theorem. Lecture 9 was devoted to a Fourier-theoretic proof of Arrow theorem (in the balanced case). You can find it discussed in this blog post by Noam Nisan. Lecture 10 mentioned a few further application of the Fourier method related to Arrow’s theorem, as well as a simple combinatorial proof of Arrow’s theorem in full generality. For the Fourier proof of Arrow’s theorem we showed that a Boolean function with all its non-zero Fourier coefficients on levels 0 and 1 is constant, dictatorship or anti-dictatorship. This time we formulated FKN theorem and showed how it implies a stability version of Arrow’s theorem in the neutral case.