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This is the eighth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

Significant progress has been made since the last update; by implementing the “barrier” method to establish zero free regions for H_t by leveraging the extensive existing numerical verification of the Riemann hypothesis (which establishes zero free regions for H_0), we have been able to improve our upper bound on \Lambda from 0.48 to 0.28. Furthermore, there appears to be a bit of further room to improve the bounds further by tweaking the parameters t_0, y_0, X used in the argument (we are currently using t_0=0.2, y_0 = 0.4, X = 5 \times 10^9); the most recent idea is to try to use exponential sum estimates to improve the bounds on the derivative of the approximation to H_t that is used in the barrier method, which currently is the most computationally intensive step of the argument.

Just a quick announcement that Dustin Mixon and Aubrey de Grey have just launched the Polymath16 project over at Dustin’s blog.  The main goal of this project is to simplify the recent proof by Aubrey de Grey that the chromatic number of the unit distance graph of the plane is at least 5, thus making progress on the Hadwiger-Nelson problem.  The current proof is computer assisted (in particular it requires one to control the possible 4-colorings of a certain graph with over a thousand vertices), but one of the aims of the project is to reduce the amount of computer assistance needed to verify the proof; already a number of such reductions have been found.  See also this blog post where the polymath project was proposed, as well as the wiki page for the project.  Non-technical discussion of the project will continue at the proposal blog post.

We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map {f: U \rightarrow V} from an open subset {U} of the complex plane to another open set {V} is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of {f} to be nowhere vanishing. We then say that the two open sets {U,V} are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation {f( z_0 + \Delta z) \approx f(z_0) + f'(z_0) \Delta z + O( |\Delta z|^2)} we see that they almost preserve small circles, indeed for {\varepsilon} small the circle {\{ z: |z-z_0| = \varepsilon\}} will approximately map to {\{ w: |w - f(z_0)| = |f'(z_0)| \varepsilon \}}.

In previous quarters, we proved a fundamental theorem about this concept, the Riemann mapping theorem:

Theorem 1 (Riemann mapping theorem) Let {U} be a simply connected open subset of {{\bf C}} that is not all of {{\bf C}}. Then {U} is conformally equivalent to the unit disk {D(0,1)}.

This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps {f: U \rightarrow D(0,1)} from {U} to {D(0,1)} that map some fixed point {z_0 \in U} to {0}, pick one that maximises the magnitude {|f'(z_0)|} of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If {f(U)} avoids some point in {D(0,1)}, one can compose {f} with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase {|f'(z_0)|} without destroying injectivity; see the previous lecture notes for details. The conformal map {\phi: U \rightarrow D(0,1)} is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points {z_0,z_1} in {U}, and requiring {\phi(z_0)} to be zero and {\phi(z_1)} to be positive real.

It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).

To make the above discussion more precise we need some notation.

Definition 2 (Circle packing) A (finite) circle packing is a finite collection {(C_j)_{j \in J}} of circles {C_j = \{ z \in {\bf C}: |z-z_j| = r_j\}} in the complex numbers indexed by some finite set {J}, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices {\{z_j: j \in J \}} are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)

It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.

The first basic theorem in the subject is the following converse statement:

Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.

Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:

Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.

Exercise 5 Let {G} be a connected planar graph with {n \geq 3} vertices. Show that the following are equivalent:

  • (i) {G} is a maximal planar graph.
  • (ii) {G} has {3n-6} edges.
  • (iii) Every drawing {D} of {G} divides the plane into faces that have three edges each. (This includes one unbounded face.)
  • (iv) At least one drawing {D} of {G} divides the plane into faces that have three edges each.

(Hint: use Euler’s formula {V-E+F=2}, where {F} is the number of faces including the unbounded face.)

Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:

Conjecture 6 (Informal Thurston conjecture) Let {U} be a simply connected domain, with two distinct points {z_0,z_1}. Let {\phi: U \rightarrow D(0,1)} be the conformal map from {U} to {D(0,1)} that maps {z_0} to the origin and {z_1} to a positive real. For any small {\varepsilon>0}, let {{\mathcal C}_\varepsilon} be the portion of the regular hexagonal circle packing by circles of radius {\varepsilon} that are contained in {U}, and let {{\mathcal C}'_\varepsilon} be an circle packing of {D(0,1)} with all “boundary circles” tangent to {D(0,1)}, giving rise to an “approximate map” {\phi_\varepsilon: U_\varepsilon \rightarrow D(0,1)} defined on the subset {U_\varepsilon} of {U} consisting of the circles of {{\mathcal C}_\varepsilon}, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that {\phi_\varepsilon(z_0)} is zero and {\phi_\varepsilon(z_1)} is a positive real. Then {\phi_\varepsilon} converges to {\phi} as {\varepsilon \rightarrow 0}.

A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation {\frac{\partial \phi}{\partial \overline{z}} = 0}, while (sufficiently smooth) quasiconformal maps only obey an inequality {|\frac{\partial \phi}{\partial \overline{z}}| \leq \frac{K-1}{K+1} |\frac{\partial \phi}{\partial z}|}. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map {\phi_\varepsilon} alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).

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I am recording here some notes on a nice problem that Sorin Popa shared with me recently. To motivate the question, we begin with the basic observation that the differentiation operator {Df(x) := \frac{d}{dx} f(x)} and the position operator {Xf(x) := xf(x)} in one dimension formally obey the commutator equation

\displaystyle  [D,X] = 1 \ \ \ \ \ (1)

where {1} is the identity operator and {[D,X] := DX-XD} is the commutator. Among other things, this equation is fundamental in quantum mechanics, leading for instance to the Heisenberg uncertainty principle.

The operators {D,X} are unbounded on spaces such as {L^2({\bf R})}. One can ask whether the commutator equation (1) can be solved using bounded operators {D,X \in B(H)} on a Hilbert space {H} rather than unbounded ones. In the finite dimensional case when {D, X} are just {n \times n} matrices for some {n \geq 1}, the answer is clearly negative, since the left-hand side of (1) has trace zero and the right-hand side does not. What about in infinite dimensions, when the trace is not available? As it turns out, the answer is still negative, as was first worked out by Wintner and Wielandt. A short proof can be given as follows. Suppose for contradiction that we can find bounded operators {D, X} obeying (1). From (1) and an easy induction argument, we obtain the identity

\displaystyle  [D,X^n] = n X^{n-1} \ \ \ \ \ (2)

for all natural numbers {n}. From the triangle inequality, this implies that

\displaystyle  n \| X^{n-1} \|_{op} \leq 2 \|D\|_{op} \| X^n \|_{op}.

Iterating this, we conclude that

\displaystyle  \| X \|_{op} \leq \frac{(2 \|D\|_{op})^{n-1}}{n!} \|X^n \|_{op}

for any {n}. Bounding {\|X^n\|_{op} \leq \|X\|_{op}^n} and then sending {n \rightarrow \infty}, we conclude that {\|X\|_{op}=0}, which clearly contradicts (1). (Note the argument can be generalised without difficulty to the case when {D,X} lie in a Banach algebra, rather than be bounded operators on a Hilbert space.)

It was observed by Popa that there is a quantitative version of this result:

Theorem 1 Let {D, X \in B(H)} such that

\displaystyle  \| [D,X] - I \|_{op} \leq \varepsilon

for some {\varepsilon > 0}. Then we have

\displaystyle  \| X \|_{op} \|D \|_{op} \geq \frac{1}{2} \log \frac{1}{\varepsilon}. \ \ \ \ \ (3)

Proof: By multiplying {D} by a suitable constant and dividing {X} by the same constant, we may normalise {\|D\|_{op}=1/2}. Write {DX - XD = 1 + E} with {\|E\|_{op} \leq \varepsilon}. Then the same induction that established (2) now shows that

\displaystyle  [D,X^n]= n X^{n-1} + X^{n-1} E + X^{n-2} E X + \dots + E X^{n-1}

and hence by the triangle inequality

\displaystyle  n \| X^{n-1} \|_{op} \leq \| X^n \|_{op} + n \varepsilon \|X\|_{op}^{n-1}.

We divide by {n!} and sum to conclude that

\displaystyle  \sum_{n=0}^\infty \frac{\|X^n\|_{op}}{n!} \leq \sum_{n=1}^\infty \frac{\|X^n\|_{op}}{n!} + \varepsilon \exp( \|X\|_{op} )

giving the claim.
\Box

Again, the argument generalises easily to any Banach algebra. Popa then posed the question of whether the quantity {\frac{1}{2} \log \frac{1}{\varepsilon}} can be replaced by any substantially larger function of {\varepsilon}, such as a polynomial in {\frac{1}{\varepsilon}}. As far as I know, the above simple bound has not been substantially improved.

In the opposite direction, one can ask for constructions of operators {X,D} that are not too large in operator norm, such that {[D,X]} is close to the identity. Again, one cannot do this in finite dimensions: {[D,X]} has trace zero, so at least one of its eigenvalues must outside the disk {\{ z: |z-1| < 1\}}, and therefore {\|[D,X]-1\|_{op} \geq 1} for any finite-dimensional {n \times n} matrices {X,D}.

However, it was shown in 1965 by Brown and Pearcy that in infinite dimensions, one can construct operators {D,X} with {[D,X]} arbitrarily close to {1} in operator norm (in fact one can prescribe any operator for {[D,X]} as long as it is not equal to a non-zero multiple of the identity plus a compact operator). In the above paper of Popa, a quantitative version of the argument (based in part on some earlier work of Apostol and Zsido) was given as follows. The first step is to observe the following Hilbert space version of Hilbert’s hotel: in an infinite dimensional Hilbert space {H}, one can locate isometries {u, v \in B(H)} obeying the equation

\displaystyle  uu^* + vv^* = 1, \ \ \ \ \ (4)

where {u^*} denotes the adjoint of {u}. For instance, if {H} has a countable orthonormal basis {e_1, e_2, \dots}, one could set

\displaystyle  u := \sum_{n=1}^\infty e_{2n-1} e_n^*

and

\displaystyle  v := \sum_{n=1}^\infty e_{2n} e_n^*,

where {e_n^*} denotes the linear functional {x \mapsto \langle x, e_n \rangle} on {H}. Observe that (4) is again impossible to satisfy in finite dimension {n}, as the left-hand side must have trace {2n} while the right-hand side has trace {n}.

As {u,v} are isometries, we have

\displaystyle  v^* v = u^* u = 1; \ \ \ \ \ (5)

Multiplying (4) on the left by {v^*} and right by {u}, or on the left by {u^*} and right by {v}, then gives

\displaystyle  v^* u = u^* v = 0. \ \ \ \ \ (6)

From (4), (5) we see in particular that, while we cannot express {1} as a commutator of bounded operators, we can at least express it as the sum of two commutators:

\displaystyle  [u^*, u] + [v^*, v] =1.

We can rewrite this somewhat strangely as

\displaystyle  [\frac{1}{2} u^*, 4u+2v] + [\frac{1}{2} u^* - v^*, -2v] = 2

and hence there exists a bounded operator {a} such that

\displaystyle  [\frac{1}{2} u^*, 4u+2v] = 1+a; \quad [\frac{1}{2} u^* - v^*, -2v] = 1-a.

Moving now to the Banach algebra of {2 \times 2} matrices with entries in {B(H)} (which can be equivalently viewed as {B(H \oplus H)}), a short computation then gives the identity

\displaystyle  \left[ \begin{pmatrix} \frac{1}{2} u^* & 0 \\ a & \frac{1}{2} u^* - v^* \end{pmatrix}, \begin{pmatrix} 4u+2v & 1 \\ 0 & -2v \end{pmatrix} \right] = \begin{pmatrix} 1 & v^* \\ b & 1 \end{pmatrix}

for some bounded operator {b} whose exact form will not be relevant for the argument. Now, by Neumann series (and the fact that {u,v} have unit operator norm), we can find another bounded operator {c} such that

\displaystyle  c + \frac{1}{2} v c u^* = b,

and then another brief computation shows that

\displaystyle  \left[ \begin{pmatrix} \frac{1}{2} u^* & 0 \\ a & \frac{1}{2} u^* - v^* \end{pmatrix}, \begin{pmatrix} 4u+2v & 1 \\ vc & -2v \end{pmatrix} \right] = \begin{pmatrix} 1 & v^* \\ 0 & 1 \end{pmatrix}.

Thus we can express the operator {\begin{pmatrix} 1 & v^* \\ 0 & 1 \end{pmatrix}} as the commutator of two operators of norm {O(1)}. Conjugating by {\begin{pmatrix} \varepsilon^{1/2} & 0 \\ 0 & \varepsilon^{-1/2} \end{pmatrix}} for any {0 < \varepsilon \leq 1}, we may then express {\begin{pmatrix} 1 & \varepsilon v^* \\ 0 & 1 \end{pmatrix}} as the commutator of two operators of norm {O(\varepsilon^{-1})}. This shows that the right-hand side of (3) cannot be replaced with anything that blows up faster than {\varepsilon^{-2}} as {\varepsilon \rightarrow 0}. Can one improve this bound further?

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