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.

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^*


\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?

The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces {M = M^n} that locally look like open subsets of a Euclidean space {{\bf R}^n}, and which can be equipped with an atlas {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}} of coordinate charts {\phi_\alpha: U_\alpha \rightarrow V_\alpha} from open subsets {U_\alpha} covering {M} to open subsets {V_\alpha} in {{\bf R}^n}, which are homeomorphisms; in particular, the transition maps {\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )} defined by {\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}} are all continuous. (It is also common to impose the requirement that the manifold {M} be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.

In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is {{\bf C}^n} rather than {{\bf R}^n}, and the transition maps {\tau_{\alpha\beta}} are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of {{\bf C}} – are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:

Definition 1 (Riemann surface) If {M} is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}} of homeomorphisms from open subsets {(U_\alpha)_{\alpha \in A}} of {M} that cover {M} to open subsets {V_\alpha} of the complex numbers {{\bf C}}, such that the transition maps {\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )} defined by {\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}} are all holomorphic. Here {A} is an arbitrary index set. Two atlases {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}, {(\phi'_\beta: U'_\beta \rightarrow V'_\beta)_{\beta \in B}} on {M} are said to be equivalent if their union is also an atlas, thus the transition maps {\phi'_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U'_\beta) \rightarrow \phi'_\beta(U_\alpha \cap U'_\beta)} and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space {M} equipped with an equivalence class of one-dimensional complex atlases.

A map {f: M \rightarrow M'} from one Riemann surface {M} to another {M'} is holomorphic if the maps {\phi'_\beta \circ f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap f^{-1}(U'_\beta)) \rightarrow {\bf C}} are holomorphic for any charts {\phi_\alpha: U_\alpha \rightarrow V_\alpha}, {\phi'_\beta: U'_\beta \rightarrow V'_\beta} of an atlas of {M} and {M'} respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).

Here are some basic examples of Riemann surfaces.

Example 2 (Quotients of {{\bf C}}) The complex numbers {{\bf C}} clearly form a Riemann surface (using the identity map {\phi: {\bf C} \rightarrow {\bf C}} as the single chart for an atlas). Of course, maps {f: {\bf C} \rightarrow {\bf C}} that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup {\Lambda} of {{\bf C}}, the quotient {{\bf C}/\Lambda} is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood {U} of the origin in {{\bf C}} and take the atlas {(\phi_\alpha: U_\alpha \rightarrow U)_{\alpha \in {\bf C}/\Lambda}} where {U_\alpha := \alpha+U} and {\phi_\alpha(\alpha+z) := z} for all {z \in U}. In particular, given any non-real complex number {\omega}, the complex torus {{\bf C} / \langle 1, \omega \rangle} formed by quotienting {{\bf C}} by the lattice {\langle 1, \omega \rangle := \{ n + m \omega: n,m \in {\bf Z}\}} is a Riemann surface.

Example 3 Any open connected subset {U} of {{\bf C}} is a Riemann surface. By the Riemann mapping theorem, all simply connected open {U \subset {\bf C}}, other than {{\bf C}} itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).

Example 4 (Riemann sphere) The Riemann sphere {{\bf C} \cup \{\infty\}}, as a topological manifold, is the one-point compactification of {{\bf C}}. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets {U_1 := {\bf C}} and {U_2 := {\bf C} \cup \{\infty\} \backslash \{0\}}, and give these two open sets the charts {\phi_1: U_1 \rightarrow {\bf C}} and {\phi_2: U_2 \rightarrow {\bf C}} defined by {\phi_1(z) := z} for {z \in {\bf C}}, {\phi_2(z) := 1/z} for {z \in {\bf C} \backslash \{0\}}, and {\phi_2(\infty) := 0}. This is a complex atlas since the {1/z} is holomorphic on {{\bf C} \backslash \{0\}}.

An alternate way of viewing the Riemann sphere is as the projective line {\mathbf{CP}^1}. Topologically, this is the punctured complex plane {{\bf C}^2 \backslash \{(0,0)\}} quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes {[z,w] := \{ (\lambda z, \lambda w): \lambda \in {\bf C} \backslash \{0\}\}} with the usual quotient topology. One can cover this space by two open sets {U_1 := \{ [z,1]: z \in {\bf C} \}} and {U_2: \{ [1,w]: w \in {\bf C} \}} and give these two open sets the charts {\phi: U_1 \rightarrow {\bf C}} and {\phi_2: U_2 \rightarrow {\bf C}} defined by {\phi_1([z,1]) := z} for {z \in {\bf C}}, {\phi_2([1,w]) := w}. This is a complex atlas, basically because {[z,1] = [1,1/z]} for {z \in {\bf C} \backslash \{0\}} and {1/z} is holomorphic on {{\bf C} \backslash \{0\}}.

Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.

Example 6 (Smooth algebraic plane curves) Let {P(z_1,z_2,z_3)} be a complex polynomial in three variables which is homogeneous of some degree {d \geq 1}, thus

\displaystyle P( \lambda z_1, \lambda z_2, \lambda z_3) = \lambda^d P( z_1, z_2, z_3). \ \ \ \ \ (1)


Define the complex projective plane {\mathbf{CP}^2} to be the punctured space {{\bf C}^3 \backslash \{0\}} quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes {[z_1,z_2,z_3] := \{ (\lambda z_1, \lambda z_2, \lambda z_3)\}}. Inside this plane we can define the (projective, degree {d}) algebraic curve

\displaystyle Z(P) := \{ [z_1,z_2,z_3] \in \mathbf{CP}^2: P(z_1,z_2,z_3) = 0 \};

this is well defined thanks to (1). It is easy to verify that {Z(P)} is a closed subset of {\mathbf{CP}^2} and hence compact; it is non-empty thanks to the fundamental theorem of algebra.

Suppose that {P} is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple {(z_1,z_2,z_3) \in {\bf C}^3 \backslash \{(0,0,0)\}} such that the four numbers {P(z_1,z_2,z_3), \frac{\partial}{\partial z_j} P(z_1,z_2,z_3)} simultaneously vanish for {j=1,2,3}. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity

\displaystyle \sum_{j=1}^3 z_j \frac{\partial}{\partial z_j} P(z_1,z_2,z_3) = d P(z_1,z_2,z_3)

that arises from differentiating (1) in {\lambda}. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial {z_1^2 z_3 - z_2^3} is irreducible but singular (there is a “cusp” singularity at {[0,0,1]}). With this hypothesis, we call the curve {Z(P)} smooth.

Now suppose {[z_1,z_2,z_3]} is a point in {Z(P)}; without loss of generality we may take {z_3} non-zero, and then we can normalise {z_3=1}. Now one can think of {P(z_1,z_2,1)} as an inhomogeneous polynomial in just two variables {z_1,z_2}, and by nondegeneracy we see that the gradient {(\frac{\partial}{\partial z_1} P(z_1,z_2,1), \frac{\partial}{\partial z_2} P(z_1,z_2,1))} is non-zero whenever {P(z_1,z_2,1)=0}. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve

\displaystyle Z(P)_{aff} := \{ (z_1,z_2) \in {\bf C}^2: P(z_1,z_2,1) = 0 \}

is a Riemann surface in a neighbourhood of {(z_1,z_2,1)}; we leave this as an exercise. This can be used to give a coordinate chart for {Z(P)} in a neighbourhood of {[z_1,z_2,z_3]} when {z_3 \neq 0}. Similarly when {z_1,z_2} is non-zero. This can be shown to give an atlas on {Z(P)}, which (assuming the connectedness claim that we will prove later) gives {Z(P)} the structure of a Riemann surface.

Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.

Exercise 8

  • (i) Show that all (irreducible plane projective) algebraic curves of degree {1} are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as {z_3}.)
  • (ii) Show that all (irreducible plane projective) algebraic curves of degree {2} are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as {z_1^2+z_2^2+z_3^2} or {z_2 z_3 - z_1^2}.)

Exercise 9 If {a,b} are complex numbers, show that the projective cubic curve

\displaystyle \{ [z_1, z_2, z_3]: z_2^2 z_3 = z_1^3 + a z_1 z_3^2 + b z_3^3 \}

is nonsingular if and only if the discriminant {-16 (4a^3 + 27b^2)} is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)

A recurring theme in mathematics is that an object {X} is often best studied by understanding spaces of “good” functions on {X}. In complex analysis, there are two basic types of good functions:

Definition 10 Let {X} be a Riemann surface. A holomorphic function on {X} is a holomorphic map from {X} to {{\bf C}}; the space of all such functions will be denoted {{\mathcal O}(X)}. A meromorphic function on {X} is a holomorphic map from {X} to the Riemann sphere {{\bf C} \cup \{\infty\}}, that is not identically equal to {\infty}; the space of all such functions will be denoted {M(X)}.

One can also define holomorphicity and meromorphicity in terms of charts: a function {f: X \rightarrow {\bf C}} is holomorphic if and only if, for any chart {\phi_\alpha: U_\alpha \rightarrow {\bf C}}, the map {f \circ \phi^{-1}_\alpha: \phi_\alpha(U_\alpha) \rightarrow {\bf C}} is holomorphic in the usual complex analysis sense; similarly, a function {f: X \rightarrow {\bf C} \cup \{\infty\}} is meromorphic if and only if the preimage {f^{-1}(\{\infty\})} is discrete (otherwise, by analytic continuation and the connectedness of {X}, {f} will be identically equal to {\infty}) and for any chart {\phi_\alpha: U_\alpha \rightarrow X}, the map {f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha) \rightarrow {\bf C} \cup \{\infty\}} becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space {{\mathcal O}(X)} of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space {M(X)} of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order {k}, or a pole of order {k}, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on {X} this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic {1}-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on {X} agree on a non-empty open set, then they agree everywhere.

On the complex numbers {{\bf C}}, there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).

It turns out, however, that the situation changes dramatically when the Riemann surface {X} is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:

Lemma 11 Let {f \in \mathcal{O}(X)} be a holomorphic function on a compact Riemann surface {X}. Then {f} is constant.

This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)

Proof: As {f} is continuous and {X} is compact, {|f(z_0)|} must attain a maximum at some point {z_0 \in X}. Working in a chart around {z_0} and applying the maximum principle, we conclude that {f} is constant in a neighbourhood of {z_0}, and hence is constant everywhere by analytic continuation. \Box

This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic {1}-forms (also known as Abelian differentials).

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This is the seventh “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.

The most recent news is that we appear to have completed the verification that {H_t(x+iy)} is free of zeroes when {t=0.4} and {y \geq 0.4}, which implies that {\Lambda \leq 0.48}. For very large {x} (for instance when the quantity {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor} is at least {300}) this can be done analytically; for medium values of {x} (say when {N} is between {11} and {300}) this can be done by numerically evaluating a fast approximation {A^{eff} + B^{eff}} to {H_t} and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of {x}, in part due to some new, and significantly faster, numerical ways to evaluate {H_t} in this range.

One obvious thing to do now is to experiment with lowering the parameters {t} and {y} and see what happens. However there are two other potential ways to bound {\Lambda} which may also be numerically feasible. One approach is based on trying to exclude zeroes of {H_t(x+iy)=0} in a region of the form {0 \leq t \leq t_0}, {X \leq x \leq X+1} and {y \geq y_0} for some moderately large {X} (this acts as a “barrier” to prevent zeroes from flowing into the region {\{ 0 \leq x \leq X, y \geq y_0 \}} at time {t_0}, assuming that they were not already there at time {0}). This require significantly less numerical verification in the {x} aspect, but more numerical verification in the {t} aspect, so it is not yet clear whether this is a net win.

Another, rather different approach, is to study the evolution of statistics such as {S(t) = \sum_{H_t(x+iy)=0: x,y>0} y e^{-x/X}} over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times {t_0>0}. However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for {H_t}, which at present is only available asymptotically (or at time {t=0}). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.

Participants are also welcome to add any further summaries of the situation in the comments below.

This is the sixth “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.

The last two threads have been focused primarily on the test problem of showing that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}. We have been able to prove this for most regimes of {x}, or equivalently for most regimes of the natural number parameter {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor}. In many of these regimes, a certain explicit approximation {A^{eff}+B^{eff}} to {H_t} was used, together with a non-zero normalising factor {B^{eff}_0}; see the wiki for definitions. The explicit upper bound

\displaystyle  |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3

has been proven for certain explicit expressions {E_1, E_2, E_3} (see here) depending on {x}. In particular, if {x} satisfies the inequality

\displaystyle  |\frac{A^{eff}+B^{eff}}{B^{eff}_0}| > \frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}

then {H_t(x+iy)} is non-vanishing thanks to the triangle inequality. (In principle we have an even more accurate approximation {A^{eff}+B^{eff}-C^{eff}} available, but it is looking like we will not need it for this test problem at least.)

We have explicit upper bounds on {\frac{E_1}{|B^{eff}_0|}}, {\frac{E_2}{|B^{eff}_0|}}, {\frac{E_3}{|B^{eff}_0|}}; see this wiki page for details. They are tabulated in the range {3 \leq N \leq 2000} here. For {N \geq 2000}, the upper bound {\frac{E_3^*}{|B^{eff}_0|}} for {\frac{E_3}{|B^{eff}_0|}} is monotone decreasing, and is in particular bounded by {1.53 \times 10^{-5}}, while {\frac{E_2}{|B^{eff}_0|}} and {\frac{E_1}{|B^{eff}_0|}} are known to be bounded by {2.9 \times 10^{-7}} and {2.8 \times 10^{-8}} respectively (see here).

Meanwhile, the quantity {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} can be lower bounded by

\displaystyle  |\sum_{n=1}^N \frac{b_n}{n^s}| - |\sum_{n=1}^N \frac{a_n}{n^s}|

for certain explicit coefficients {a_n,b_n} and an explicit complex number {s = \sigma + i\tau}. Using the triangle inequality to lower bound this by

\displaystyle  |b_1| - \sum_{n=2}^N \frac{|b_n|}{n^\sigma} - \sum_{n=1}^N \frac{|a_n|}{n^\sigma}

we can obtain a lower bound of {0.18} for {N \geq 2000}, which settles the test problem in this regime. One can get more efficient lower bounds by multiplying both Dirichlet series by a suitable Euler product mollifier; we have found {\prod_{p \leq P} (1 - \frac{b_p}{p^s})} for {P=2,3,5,7} to be good choices to get a variety of further lower bounds depending only on {N}, see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range {300 \leq N \leq 2000}.

In the range {11 \leq N \leq 300}, we have been able to obtain a suitable lower bound {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}| \geq c} (where {c} exceeds the upper bound for {\frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}}) by numerically evaluating {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} at a mesh of points for each choice of {N}, with the mesh spacing being adaptive and determined by {c} and an upper bound for the derivative of {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|}; the data is available here.

This leaves the final range {N \leq 10} (roughly corresponding to {x \leq 1600}). Here we can numerically evaluate {H_t(x+iy)} to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on {H'_t(x+iy)}. It seems that we can get reasonable estimates using some contour shifting from the original definition of {H_t} (see here). We are close to finishing off this remaining region and thus solving the toy problem.

Beyond this, we need to figure out how to show that {H_t(x+iy) \neq 0} for {y > 0.4} as well. General theory lets one do this for {y \geq \sqrt{1-2t} = 0.447\dots}, leaving the region {0.4 < y < 0.448}. The analytic theory that handles {N \geq 2000} and {300 \leq N \leq 2000} should also handle this region; for {N \leq 300} presumably the argument principle will become relevant.

The full argument also needs to be streamlined and organised; right now it sprawls over many wiki pages and github code files. (A very preliminary writeup attempt has begun here). We should also see if there is much hope of extending the methods to push much beyond the bound of {\Lambda \leq 0.48} that we would get from the above calculations. This would also be a good time to start discussing whether to move to the writing phase of the project, or whether there are still fruitful research directions for the project to explore.

Participants are also welcome to add any further summaries of the situation in the comments below.

Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA.  This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).

246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series.  The initial topics I have in mind to cover are

As usual, I will be posting lecture notes on this blog as the course progresses.

[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]

This is the fifth “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.

We have almost finished off the test problem of showing that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}. We have two useful approximations for {H_t}, which we have denoted {A^{eff}+B^{eff}} and {A^{eff}+B^{eff}-C^{eff}}, and a normalising quantity {B^{eff}_0} that is asymptotically equal to the above expressions; see the wiki page for definitions. In practice, the {A^{eff}+B^{eff}} approximation seems to be accurate within about one or two significant figures, whilst the {A^{eff}+B^{eff}-C^{eff}} approximation is accurate to about three or four. We have an effective upper bound

\displaystyle  |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3^*

where the expressions {E_1,E_2,E_3^*} are quite small in practice ({E_3^*} is typically about two orders of magnitude smaller than the main term {B^{eff}_0} once {x} is moderately large, and the error terms {E_1,E_2} are even smaller). See this page for details. In principle we could also obtain an effective upper bound for {|H_t - (A^{eff} + B^{eff} - C^{eff})|} (the {E_3^*} term would be replaced by something smaller).

The ratio {\frac{A^{eff}+B^{eff}}{B^{eff}_0}} takes the form of a difference {\sum_{n=1}^N \frac{b_n}{n^s} - e^{i\theta} \sum_{n=1}^N \frac{a_n}{n^s}} of two Dirichlet series, where {e^{i\theta}} is a phase whose value is explicit but perhaps not terribly important, and the coefficients {b_n, a_n} are explicit and relatively simple ({b_n} is {\exp( \frac{t}{4} \log^2 n)}, and {a_n} is approximately {(n/N)^y b_n}). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product {\prod_{p \leq P} (1 - \frac{b_p}{p^s})} to cancel much of the initial oscillation; also one can take advantage of the fact that the {b_n} are real and the {a_n} are (approximately) real. See this page for details. The upshot is that we seem to be getting good lower bounds for the size of this difference of Dirichlet series starting from about {x \geq 5 \times 10^5} or so. The error terms {E_1,E_2,E_3^*} are already quite small by this stage, so we should soon be able to rigorously keep {H_t} from vanishing at this point. We also have a scheme for lower bounding the difference of Dirichlet series below this range, though it is not clear at present how far we can continue this before the error terms {E_1,E_2,E_3^*} become unmanageable. For very small {x} we may have to explore some faster ways to compute the expression {H_t}, which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions {E_1,E_2} by something more manageable. For instance, right now these quantities depend on the continuous variable {x}; it would be preferable to have a quantity that depends only on the parameter {N = \lfloor \sqrt{ \frac{x}{4\pi} + \frac{t}{16} }\rfloor}, as this could be computed numerically for all {x} in the remaining range of interest quite quickly.

As before, any other mathematical discussion related to the project is also welcome here, for instance any summaries of previous discussion that was not covered in this post.

This is the fourth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing Progress will be summarised at this Polymath wiki page.

We are getting closer to finishing off the following test problem: can one show that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}, {x \geq 0}? This would morally show that {\Lambda \leq 0.48}. A wiki page for this problem has now been created here. We have obtained a number of approximations {A+B, A'+B', A^{eff}+B^{eff}, A^{toy}+B^{toy}} to {H_t} (see wiki page), though numeric evidence indicates that the approximations are all very close to each other. (Many of these approximations come with a correction term {C}, but thus far it seems that we may be able to avoid having to use this refinement to the approximations.) The effective approximation {A^{eff} + B^{eff}} also comes with an effective error bound

\displaystyle |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3

for some explicit (but somewhat messy) error terms {E_1,E_2,E_3}: see this wiki page for details. The original approximations {A+B, A'+B'} can be considered deprecated at this point in favour of the (slightly more complicated) approximation {A^{eff}+B^{eff}}; the approximation {A^{toy}+B^{toy}} is a simplified version of {A^{eff}+B^{eff}} which is not quite as accurate but might be useful for testing purposes.

It is convenient to normalise everything by an explicit non-zero factor {B^{eff}_0}. Asymptotically, {(A^{eff} + B^{eff}) / B^{eff}_0} converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range {10^5 \leq x \leq 10^6}, and we seem to be able to keep it (or at least the toy counterpart {(A^{toy} + B^{toy}) / B^{toy}_0}) away from zero starting from about {x \geq 4 \times 10^6} (here it seems that there is a useful trick of multiplying by Euler-type factors like {1 - \frac{1}{2^{1-s}}} to cancel off some of the oscillation). Also, the bounds on the error {(H_t - A^{eff} - B^{eff}) / B^{eff}_0} seem to be of size about 0.1 or better in these ranges also. So we seem to be on track to be able to rigorously eliminate zeroes starting from about {x \geq 10^5} or so. We have not discussed too much what to do with the small values of {x}; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute {H_t}.

In addition to this main direction of inquiry, there have been additional discussions on the dynamics of zeroes, and some numerical investigations of the behaviour Lehmer pairs under heat flow. Contributors are welcome to summarise any findings from these discussions from previous threads (or on any other related topic, e.g. improvements in the code) in the comments below.