Category Archives: Open problems

More Reasons for Small Influence


Readers of the big-league ToC blogs have already heard about the breakthrough paper An average-case depth hierarchy theorem for Boolean circuits by Benjamin Rossman, Rocco Servedio, and Li-Yang Tan. Here are blog reports on Computational complexity, on the Shtetl Optimized, and of Godel Lost letter and P=NP. Let me mention one of the applications: refuting a 1999 conjecture by Benjamini, Schramm and me.

Update: Li-Yang Tang explained matters in an excellent comment below. Starting with: “In brief, we believe that an average-case depth hierarchy theorem rules out the possibility of a converse to Hastad-Boppana-LMN when viewed as a statement about the total influence of *constant*-depth circuits. However, while the Inf(f) <= (O(\log(S)))^{d-1} bound is often applied in the setting where d is constant,  it in fact holds for all values of d. It would interesting to explore the implications of our result in regimes where d is allowed to be super-constant.

Let me add that the bounded depth case is an important case (that I referred to here), that there might be some issues failing the conjecture for non-constant depth “for the wrong reasons”,  and that I see good prospect that RST’s work and techniques will refute BKS conjecture in full also for non-bounded depth.

Update:  Rossman, Servedio, and Tan refuted some important variations of our conjecture, while other variations remain open. My description was not so accurate and in hindsight I could also explained the background and motivation better. So rather than keep updating this post, I will write a new one in a few weeks. 


Theorem: If f is described by a bounded depth circuit of size s and depth d then I(f) the total influence of f,  is at most (\log s)^{d-1}.

The total influence I(f) of f is defined as follows: for an input x write I(x) for the number of neighbors y of x with f(y) \ne f(x). I(f) = \mathbb EI(x).

The history of this result as I  remember it is that: it is based on a crucial way on  Hastad Switching lemma going back to Hastad 1986 thesis, and for monotone functions one can use an even earlier 1984 result by Boppana.  It was first proved  (with exponent “d”) in 1993 by Linial-Mansour and  Nisan, as  a consequence of their theorem on the decay of Fourier coefficients for AC0 functions, (also based on the switching lemma). With the correct exponent d-1 it is derived from the switching lemma in a short clean argument in a 97 paper by Ravi Boppana; and finally it was extended to a sharpening of LMN result about the spectral decay by Hastad (2001).

Sipser MIT photo

Mike Sipser

Inverse Boppana-Hastad-LMN

Conjecture: (Benjanmini, Kala, and Schramm, 1999): Every Boolean function  f  is “close” to some depth-d size s circuit with (\log s)^{d-1} not much larger than I(f).

Of course, the exponent (d-1) is strongest possible but replacing it with some constant times d  is also of interest. (Also the monotone case already capture much interest.)

As we will see the conjecture is false even if the exponent d-1 is replaced by a constant times d. I do not know what is the optimal function u(d) if any for which you can replace the exponent d-1 by u(d).

Update: Following some comments by Boaz Barak I am not sure that indeed the new examples and results regarding them leads to disproof of our conjecture. The remarkable part of RST’s paper is that the RST example cannot be approximated by a circuit of smaller depth – even by one. (This has various important applications.) In order to disprove our conjecture one need to show that the influence of the example is smaller than what Boppana’s inequality ((log size)^{depth-1} ) gives. This is not proved in the paper (but it may be true). 

The RST’s result does say that if the influence is (say) logn (where n is the number of variables,) and the function depends on a small number of variables then it need not be correlated with a function in AC0.  

Anyway I will keep you posted.  

in 2007 O’Donnell and Wimmer showed that our inverse conjecture is false as stated. They took a Boolean function which is a tribe function on half the variables and “anti-tribes” on the rest. This still left the possibility that the exponent d-1 could be replaced by d or that “close” could be replaced by a weaker conclusion on substantial correlation.

Rossman,  Servedio, and a genuinely new reason for small influence!Their example, named after Mike Sipser,  is based on the AND-OR tree – a Boolean formula with alternating AND and OR levels and carefully designed parameters.  The crucial part is to show that you cannot approximate this function by lower depth circuits. The theorem proved by RST  is amazingly strong and does not allow reducing the depth even by one! The novel technique for proving it of random projections is very exciting.

It is still possible (I think) that such inverse theorems hold when the individual influences of all variables is below polylog(n)/n where n is the number of variables. Let me pose it as a conjecture:

Conjecture: Every Boolean function  f  with n variables and individual influences below polylog (n)/n is close to a function g in AC0   of size s depth d where (log s)^d is polylog (n).

And here is a post on TCSexchange with a question about “monotone  vs positive” for the class P. Similar questions for AC0 and TC0 were asked in this post.

Coloring Simple Polytopes and Triangulations


Edge-coloring of simple polytopes

One of the equivalent formulation of the four-color theorem asserts that:

Theorem (4CT) : Every cubic bridgeless planar graph is 3-edge colorable

So we can color the edges by three colors such that every two edges sharing a vertex are colored by different colors.

Abby Thompson asked the following question:

Question: Suppose that G is the graph of a simple d-polytope with n vertices. Suppose also that n is even (this is automatic if d is odd). Can we always properly color the edges of G with d colors?

Vising theorem reminded

Vising’s theorem asserts that a graph with maximum degree D can be edge-colored by D+1 colors. This is one of the most fundamental theorems in graph theory. (One of my ambitions for the blog is to interactively teach the proof based on a guided way toward a proof, based on Diestel’s book, that I tried in a graph theory course some years ago.) Class-one graphs are those graphs with edge chromatic number equal to the maximum degree. Those graphs that required one more color are called class-two graphs.

Moving to triangulations

Thompson asked also a more general question:

Question: Let G be a dual graph of a triangulation of the (d-1)-dimensional sphere. Suppose that G has an even number of vertices.  Is G d-edge colorable?

Grunbaum’s question and counterexample

Branko Grunbaum proposed a beautiful generalization for the 4CT: He conjectured that the dual graph of a triangulation of every two-dimensional manifold is 3-edge colorable. This conjecture was refuted in 2009 by Martin Kochol.

Triangulations in higher dimensions

A third question, even more general, posed by Thompson is: Let G be a dual graph of a triangulation of a (d-1)-dimensional manifold, d ≥ 4. Suppose that G has an even number of vertices.  Is G d-edge colorable?

Hamiltonian cycles

Coloring graph is notoriously difficult but finding a Hamiltonian cycle is even more difficult.

Tait’s conjecture and Barnette’s conjectures

Peter Tait conjectured in 1884 that every 3-connected cubic planar graph is Hamiltonian. His conjecture was disproved by William Tutte in 1946. A cubic Hamiltonian graph must be of class I and therefore Tait’s conjecture implies the 4CT. David Barnette proposed two ways to save Tait’s conjecture: one for adding the condition that all faces have an even number of edges or, equivalently that the graph is bipartite, and another, by moving up in the dimension.

Barnette’s conjecture I: Planar 3-connected cubic bipartite graphs are Hamiltonian.

Barnette’s conjecture II: Graphs of simple d-polytopes d ≥ 4 are Hamiltonian.

Barnette’s hamiltonicity conjecture in high dimension does not imply a positive answer to Thompson’s quaestion. We can still ask for the following common strengthening:  does the graph of a simple d-polytope, d 4, with an even number of vertices contain [d/2] edge-disjoint Hamiltonian cycles?

There are few more things to mention: Peter Tait made also three beautiful conjectures about knots. They were all proved, but it took a century more or less. When we move to high dimensions there are other notions of coloring and other generalizations of “Hamiltonian cycles.” You can Test Your Imagination and try to think about such notions!

Update (Dec 7): Following rupeixu’s comment I asked a question over: generalizations-of-the-four-color-theorem.

Jim Geelen, Bert Gerards, and Geoff Whittle Solved Rota’s Conjecture on Matroids


Gian Carlo Rota

Rota’s conjecture

I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor.  The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors).  Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements.  The notices paper gives an excellent self-contained introduction to the conjecture.

This is a project that started in 1999 and it will probably take a couple more years to complete writing the proof. It relies on ideas from the Robertson-Seymour forbidden minor theorem for graphs. Congratulations to Jim, Bert, and Geoff!

Well, looking around I saw that this was announced in August 22’s post in a very nice group blog devoted by matroids- Matroid Union, with contributions by Dillon Mayhew, Stefan van Zwam, Peter Nelson, and Irene Pivotto. August 22? you may ask, yes! August 22, 2013. I missed the news by almost a year. It was reported also here and here  and here, and here, and here, and here!

This is a good opportunity to mention two additional conjectures by Gian-Carlo Rota. But let me ask you, dear readers, before that a little question.

Rota’s unimodality conjecture and June Huh’s work

Rota’s unimodality conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. This implies that the coefficients are unimodal. A special case of the conjecture is an earlier famous conjecture (by Read) asserting that the coefficients of the chromatic polynomial of a graph are unimodal (and log-concave). This conjecture about matroids was made also around the same time by Heron and Welsh.

June Huh proved Reads’ unimodality conjecture for graphs and the more general Heron-Rota-Welsh conjecture for representable matroids for characteristic 0. Later Huh and  Eric Katz proved the case of  representable matroids for arbitrary characteristics. I already mentioned these startling results earlier and we may come back to them later.

Huh’s path to mathematics was quite amazing. He wanted to be a science-writer and accomponied Hironaka on whom he planned to write. Hironaka introduced him to mathematics in general and to algebraic geometry and this led June to study mathematics and a few years later to use deep connections between algebraic geometry and combinatorics to prove the conjecture.

Rota’s basis conjecture

Rota’s basis conjecture from the late 80’s appears to remain wide open. The problem first appeared in print in a paper by Rosa Huang and Rota. Here is a post about it also from “the matroid union.” It is the first problem in Rota’s article entitled “Ten Mathematics problems I will never solve“. Having access only to page one of the paper I can only guess what the other nine problems might be.


Rota’s portrait by Fan Chung Graham



My Mathematical Dialogue with Jürgen Eckhoff

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

Nina Amenta proved a remarkable extension of Helly’s theorem. Let \cal F be a finite family with the following property:

(a) Every member of \cal F is the union of at most r pairwise disjoint compact convex sets.

(b) So is every intersection of members of \cal F.

(c) |{\cal F}| > r(d+1).

If every r(d+1) members of \cal F has a point in common, then all members of \cal F 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

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 \cal G is d-Leray then the nerve of \cal F is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of \cal G has Helly number d, then the nerve of \cal F has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of \cal G 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 H_i(L) vanishes for every i \ge d 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

Continue reading

Navier-Stokes Fluid Computers


Smart fluid

Terry Tao posted a very intriguing post on the Navier-Stokes equation, based on a recently uploaded paper Finite time blowup for an averaged three-dimensional Navier-Stokes equation.

The paper proved a remarkable negative answer for the regularity conjecture for a certain variants of the NS equations, namely (or, perhaps, more precisely) the main theorem demonstrates finite time blowup for an averaged Navier-Stokes equation. (This already suffices to show that certain approaches for a positive answer to the real problem are not viable.) The introduction ends with the following words.

“This suggests an ambitious (but not obviously impossible) program (in both senses of
the word) to achieve the same e ffect for the true Navier-Stokes equations, thus obtaining a negative answer to Conjecture 1.1 (the regularity conjecture for 3D NS equation)… Somewhat analogously to how a quantum computer can be constructed from the laws of quantum mechanics [Here Tao links to Benio ff’s 1982 paper: “Quantum mechanical Hamiltonian models of Turing machines,”], or a Turing machine can be constructed from cellular automata such as “Conway’s Game of Life” , one could hope to design logic gates entirely out of ideal fluid (perhaps by using suitably shaped vortex sheets to simulate the various types of physical materials one would use in a mechanical computer). If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid that, when it runs, behaves qualitatively like the blowup solution used to establish Theorem 1.4.[The paper’s main theorem] Note that such replicators, as well as the related concept of a universal constructor, have been built within cellular automata such as the “Game of Life.”

Once enough logic gates of ideal fluid are constructed, it seems that the main difficulties in executing the above program are of a “software engineering” nature, and would be in principle achievable, even if the details could be extremely complicated in practice. The main mathematical difficulty in executing this “fluid computing” program would thus be to arrive at (and rigorously certify) a design for logical gates of inviscid fluid that has some good noise tolerance properties. In this regard, ideas from quantum computing (which faces a unitarity constraint somewhat analogous to the energy conservation constraint for ideal fluids, albeit with the key di fference of having a linear evolution rather than a nonlinear one) may prove to be useful. (Emphasis mine.)

Interesting idea!

And what Tao does go well beyond an idea, he essentially implement this program for a close relative of the NS equation!  I am not sure if universal computing is established for these systems but the proofs of the finite-time blow up theorem certainly uses some computational-looking gadget, and also as Terry explains some form of fault-tolerance.

Somewhat related ideas (unsupported by any results, of course,) appeared in the seventh post “Quantum repetition” of my debate with Aram Harrow on quantum computing.  (See, e.g., this remark, and this one, and this one.) The thread also contains interesting links, e.g. to Andy Yao’s paper “Classical physics and the Curch-Turing Thesis.”  In addition to the interesting question:

Does the NS-equation in three-dimension supports universal (classical) computation,

we can also ask what about two-dimensions?

Can NS-equations in two dimension be approximated in any scale by bounded depth circuits?

One general question suggested there was the following: “It can be of interest (and perhaps harder compared to the quantum case) to try to describe classical evolutions that do not enable/hide fault tolerance and (long) computation.”

Another interesting comment by Arie Israel is: “I was surprised to learn that experimental fluid mechanics people had thought of this analogy before. Apparently the key name is ‘Fluidics’ and those ideas date back at least to the sixties.”


Update: Here is the first paragraph from a nice article by  Erica Klarreich entitled A Fluid New Path in Grand Math Challenge on this development in Quanta Magazine:

In Dr. Seuss’s book “The Cat in the Hat Comes Back,” the Cat makes a stain he can’t clean up, so he calls upon the help of Little Cat A, a smaller, perfect replica of the Cat who has been hiding under the Cat’s hat. Little Cat A then calls forth Little Cat B, an even smaller replica hidden under Little Cat A’s hat. Each cat in turn lifts his hat to reveal a smaller cat who possesses all the energy and good cheer of the original Cat, just crammed into a tinier package. Finally, Little Cat Z, who is too small to see, unleashes a VOOM like a giant explosion of energy, and the stain disappears.

And here is a follow up post on Tao’s blog (and a few more II, III), and a post on Shtetl Optimized.

The flip side

Update (June 14): It is worth noting that while the purpose of Tao’s program is to show finite-time blow up of the 3D Navier Stokes equations (as is often the case) these lines of ideas can potentially be useful also toward a positive solution of the regularity conjectures. Specifically, one can try to show that 3D Navier-Stokes equations do not support universal classical computation and even more specifically do not support classical fault-tolerance and error correction. Also here some analogy with quantum computation can be useful: It is expected that for adiabatic processes computation requires “spectral gap” and that gapped evolutions with local Hamiltonians support only bounded depth computation. Something analogous may apply to NS equations in bounded dimensions.

There are many caveats, of course,  the quantum results are not proved for D>1, NS equations are non-linear which weakens the analogy, and showing that the evolution does not support computation does not imply, as far as we know, regularity.

Three more remarks: 1) On the technical level an important relevant technical tool for the results on gapped systems with local Hamiltonians is the Lieb-Robinson inequality. (See, e.g. this review paper.)  2) for classical evolutions a repetition mechanism (or the “majority function”) seems crucial for robust computation and it will be interesting specifically to test of 3D Navier-stokes support it; 3) If computation is not possible beyond bounded depth this fact may lead to additional conserved quantities for NS, beyond the classical ones. (One more, June 28): It looks to me that the crucial question is if NS equations only support bounded computation or not. So this distinction captures places where circuit complexity gives clear mathematical distinctions.

Amazing: Peter Keevash Constructed General Steiner Systems and Designs


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 {{q-i} \choose {r-i}} should divide \lambda {{n-i} \choose {r-i}} for every 1 \le i \le r-1.

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.

Brief history

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: Some other blog post on this achievement: Van Vu Jordan Ellenberg, The aperiodical . A related post from Cameron’s blog Subsets and partitions.

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 , a strategy used again by Danny and Alden Walker to show that random groups contain fundamental groups of closed surfaces .

Many triangulated three-spheres!

The news

Eran Nevo and Stedman Wilson have constructed \exp (K n^2) 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?

Quick remarks:

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 \exp(n^2 \log n). 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 \exp (O(n\log n)). 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 – \exp (Kn^{5/4}).

4) In 1988 I constructed \exp (K n^{[d/2]}) 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.]