This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution

$\displaystyle \partial_t P(t,z) = \partial_{zz} P(t,z)$

on the zeroes of a time-dependent family of polynomials ${z \mapsto P(t,z)}$, with a particular focus on the case when the polynomials ${z \mapsto P(t,z)}$ had real zeroes. Here (inspired by some discussions I had during a recent conference on the Riemann hypothesis in Bristol) we record the analogous theory in which the polynomials instead have zeroes on a circle ${\{ z: |z| = \sqrt{q} \}}$, with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when ${P}$ is the numerator of the zeta function of a curve.

More precisely, let ${g}$ be a natural number. We will say that a polynomial

$\displaystyle P(z) = \sum_{j=0}^{2g} a_j z^j$

of degree ${2g}$ (so that ${a_{2g} \neq 0}$) obeys the functional equation if the ${a_j}$ are all real and

$\displaystyle a_j = q^{g-j} a_{2g-j}$

for all ${j=0,\dots,2g}$, thus

$\displaystyle P(\overline{z}) = \overline{P(z)}$

and

$\displaystyle P(q/z) = q^g z^{-2g} P(z)$

for all non-zero ${z}$. This means that the ${2g}$ zeroes ${\alpha_1,\dots,\alpha_{2g}}$ of ${P(z)}$ (counting multiplicity) lie in ${{\bf C} \backslash \{0\}}$ and are symmetric with respect to complex conjugation ${z \mapsto \overline{z}}$ and inversion ${z \mapsto q/z}$ across the circle ${\{ |z| = \sqrt{q}\}}$. We say that this polynomial obeys the Riemann hypothesis if all of its zeroes actually lie on the circle ${\{ z = \sqrt{q}\}}$. For instance, in the ${g=1}$ case, the polynomial ${z^2 - a_1 z + q}$ obeys the Riemann hypothesis if and only if ${|a_1| \leq 2\sqrt{q}}$.

Such polynomials arise in number theory as follows: if ${C}$ is a projective curve of genus ${g}$ over a finite field ${\mathbf{F}_q}$, then, as famously proven by Weil, the associated local zeta function ${\zeta_{C,q}(z)}$ (as defined for instance in this previous blog post) is known to take the form

$\displaystyle \zeta_{C,q}(z) = \frac{P(z)}{(1-z)(1-qz)}$

where ${P}$ is a degree ${2g}$ polynomial obeying both the functional equation and the Riemann hypothesis. In the case that ${C}$ is an elliptic curve, then ${g=1}$ and ${P}$ takes the form ${P(z) = z^2 - a_1 z + q}$, where ${a_1}$ is the number of ${{\bf F}_q}$-points of ${C}$ minus ${q+1}$. The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

$\displaystyle P(z) := \det( 1 - \sqrt{q} F ) \ \ \ \ \ (1)$

of ${2g \times 2g}$ matrices ${F}$ in the compact symplectic group ${Sp(g)}$. These polynomials obey both the functional equation and the Riemann hypothesis. The Sato-Tate conjecture (in higher genus) asserts, roughly speaking, that “typical” polyomials ${P}$ arising from the number theoretic situation above are distributed like the rescaled characteristic polynomials (1), where ${F}$ is drawn uniformly from ${Sp(g)}$ with Haar measure.

Given a polynomial ${z \mapsto P(0,z)}$ of degree ${2g}$ with coefficients

$\displaystyle P(0,z) = \sum_{j=0}^{2g} a_j(0) z^j,$

we can evolve it in time by the formula

$\displaystyle P(t,z) = \sum_{j=0}^{2g} \exp( t(j-g)^2 ) a_j(0) z^j,$

thus ${a_j(t) = \exp(t(j-g)) a_j(0)}$ for ${t \in {\bf R}}$. Informally, as one increases ${t}$, this evolution accentuates the effect of the extreme monomials, particularly, ${z^0}$ and ${z^{2g}}$ at the expense of the intermediate monomials such as ${z^g}$, and conversely as one decreases ${t}$. This family of polynomials obeys the heat-type equation

$\displaystyle \partial_t P(t,z) = (z \partial_z - g)^2 P(t,z). \ \ \ \ \ (2)$

In view of the results of Marcus, Spielman, and Srivastava, it is also very likely that one can interpret this flow in terms of expected characteristic polynomials involving conjugation over the compact symplectic group ${Sp(n)}$, and should also be tied to some sort of “${\beta=\infty}$” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if ${z \mapsto P(0,z)}$ obeys the functional equation, then so does ${z \mapsto P(t,z)}$ for any other time ${t}$. Now we investigate the evolution of the zeroes. Suppose at some time ${t_0}$ that the zeroes ${\alpha_1(t_0),\dots,\alpha_{2g}(t_0)}$ of ${z \mapsto P(t_0,z)}$ are distinct, then

$\displaystyle P(t_0,z) = a_{2g}(0) \exp( t_0g^2 ) \prod_{j=1}^{2g} (z - \alpha_j(t_0) ).$

From the inverse function theorem we see that for times ${t}$ sufficiently close to ${t_0}$, the zeroes ${\alpha_1(t),\dots,\alpha_{2g}(t)}$ of ${z \mapsto P(t,z)}$ continue to be distinct (and vary smoothly in ${t}$), with

$\displaystyle P(t,z) = a_{2g}(0) \exp( t g^2 ) \prod_{j=1}^{2g} (z - \alpha_j(t) ).$

Differentiating this at any ${z}$ not equal to any of the ${\alpha_j(t)}$, we obtain

$\displaystyle \partial_t P(t,z) = P(t,z) ( g^2 - \sum_{j=1}^{2g} \frac{\alpha'_j(t)}{z - \alpha_j(t)})$

and

$\displaystyle \partial_z P(t,z) = P(t,z) ( \sum_{j=1}^{2g} \frac{1}{z - \alpha_j(t)})$

and

$\displaystyle \partial_{zz} P(t,z) = P(t,z) ( \sum_{1 \leq j,k \leq 2g: j \neq k} \frac{1}{(z - \alpha_j(t))(z - \alpha_k(t))}).$

Inserting these formulae into (2) (expanding ${(z \partial_z - g)^2}$ as ${z^2 \partial_{zz} - (2g-1) z \partial_z + g^2}$) and canceling some terms, we conclude that

$\displaystyle - \sum_{j=1}^{2g} \frac{\alpha'_j(t)}{z - \alpha_j(t)} = z^2 \sum_{1 \leq j,k \leq 2g: j \neq k} \frac{1}{(z - \alpha_j(t))(z - \alpha_k(t))}$

$\displaystyle - (2g-1) z \sum_{j=1}^{2g} \frac{1}{z - \alpha_j(t)}$

for ${t}$ sufficiently close to ${t_0}$, and ${z}$ not equal to ${\alpha_1(t),\dots,\alpha_{2g}(t)}$. Extracting the residue at ${z = \alpha_j(t)}$, we conclude that

$\displaystyle - \alpha'_j(t) = 2 \alpha_j(t)^2 \sum_{1 \leq k \leq 2g: k \neq j} \frac{1}{\alpha_j(t) - \alpha_k(t)} - (2g-1) \alpha_j(t)$

which we can rearrange as

$\displaystyle \frac{\alpha'_j(t)}{\alpha_j(t)} = - \sum_{1 \leq k \leq 2g: k \neq j} \frac{\alpha_j(t)+\alpha_k(t)}{\alpha_j(t)-\alpha_k(t)}.$

If we make the change of variables ${\alpha_j(t) = \sqrt{q} e^{i\theta_j(t)}}$ (noting that one can make ${\theta_j}$ depend smoothly on ${t}$ for ${t}$ sufficiently close to ${t_0}$), this becomes

$\displaystyle \partial_t \theta_j(t) = \sum_{1 \leq k \leq 2g: k \neq j} \cot \frac{\theta_j(t) - \theta_k(t)}{2}. \ \ \ \ \ (3)$

Intuitively, this equation asserts that the phases ${\theta_j}$ repel each other if they are real (and attract each other if their difference is imaginary). If ${z \mapsto P(t_0,z)}$ obeys the Riemann hypothesis, then the ${\theta_j}$ are all real at time ${t_0}$, then the Picard uniqueness theorem (applied to ${\theta_j(t)}$ and its complex conjugate) then shows that the ${\theta_j}$ are also real for ${t}$ sufficiently close to ${t_0}$. If we then define the entropy functional

$\displaystyle H(\theta_1,\dots,\theta_{2g}) := \sum_{1 \leq j < k \leq 2g} \log \frac{1}{|\sin \frac{\theta_j-\theta_k}{2}| }$

then the above equation becomes a gradient flow

$\displaystyle \partial_t \theta_j(t) = - 2 \frac{\partial H}{\partial \theta_j}( \theta_1(t),\dots,\theta_{2g}(t) )$

which implies in particular that ${H(\theta_1(t),\dots,\theta_{2g}(t))}$ is non-increasing in time. This shows that as one evolves time forward from ${t_0}$, there is a uniform lower bound on the separation between the phases ${\theta_1(t),\dots,\theta_{2g}(t)}$, and hence the equation can be solved indefinitely; in particular, ${z \mapsto P(t,z)}$ obeys the Riemann hypothesis for all ${t > t_0}$ if it does so at time ${t_0}$. Our argument here assumed that the zeroes of ${z \mapsto P(t_0,z)}$ were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial ${z \mapsto P(0,z)}$ obeying the functional equation, the rescaled polynomials ${z \mapsto e^{-g^2 t} P(t,z)}$ converge locally uniformly to ${a_{2g}(0) (z^{2g} + q^g)}$ as ${t \rightarrow +\infty}$. By Rouche’s theorem, we conclude that the zeroes of ${z \mapsto P(t,z)}$ converge to the equally spaced points ${\{ e^{2\pi i(j+1/2)/2g}: j=1,\dots,2g\}}$ on the circle ${\{ |z| = \sqrt{q}\}}$. Together with the symmetry properties of the zeroes, this implies in particular that ${z \mapsto P(t,z)}$ obeys the Riemann hypothesis for all sufficiently large positive ${t}$. In the opposite direction, when ${t \rightarrow -\infty}$, the polynomials ${z \mapsto P(t,z)}$ converge locally uniformly to ${a_g(0) z^g}$, so if ${a_g(0) \neq 0}$, ${g}$ of the zeroes converge to the origin and the other ${g}$ converge to infinity. In particular, ${z \mapsto P(t,z)}$ fails the Riemann hypothesis for sufficiently large negative ${t}$. Thus (if ${a_g(0) \neq 0}$), there must exist a real number ${\Lambda}$, which we call the de Bruijn-Newman constant of the original polynomial ${z \mapsto P(0,z)}$, such that ${z \mapsto P(t,z)}$ obeys the Riemann hypothesis for ${t \geq \Lambda}$ and fails the Riemann hypothesis for ${t < \Lambda}$. The situation is a bit more complicated if ${a_g(0)}$ vanishes; if ${k}$ is the first natural number such that ${a_{g+k}(0)}$ (or equivalently, ${a_{g-j}(0)}$) does not vanish, then by the above arguments one finds in the limit ${t \rightarrow -\infty}$ that ${g-k}$ of the zeroes go to the origin, ${g-k}$ go to infinity, and the remaining ${2k}$ zeroes converge to the equally spaced points ${\{ e^{2\pi i(j+1/2)/2k}: j=1,\dots,2k\}}$. In this case the de Bruijn-Newman constant remains finite except in the degenerate case ${k=g}$, in which case ${\Lambda = -\infty}$.

For instance, consider the case when ${g=1}$ and ${P(0,z) = z^2 - a_1 z + q}$ for some real ${a_1}$ with ${|a_1| \leq 2\sqrt{q}}$. Then the quadratic polynomial

$\displaystyle P(t,z) = e^t z^2 - a_1 z + e^t q$

has zeroes

$\displaystyle \frac{a_1 \pm \sqrt{a_1^2 - 4 e^{2t} q}}{2e^t}$

and one easily checks that these zeroes lie on the circle ${\{ |z|=\sqrt{q}\}}$ when ${t \geq \log \frac{|a_1|}{2\sqrt{q}}}$, and are on the real axis otherwise. Thus in this case we have ${\Lambda = \log \frac{|a_1|}{2\sqrt{q}}}$ (with ${\Lambda=-\infty}$ if ${a_1=0}$). Note how as ${t}$ increases to ${+\infty}$, the zeroes repel each other and eventually converge to ${\pm i \sqrt{q}}$, while as ${t}$ decreases to ${-\infty}$, the zeroes collide and then separate on the real axis, with one zero going to the origin and the other to infinity.

The arguments in my paper with Brad Rodgers (discussed in this previous post) indicate that for a “typical” polynomial ${P}$ of degree ${g}$ that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to ${1/g}$, basically because the average spacing is ${1/g}$ and hence by (3) the typical velocity of the zeroes should be comparable to ${g}$, and the diameter of the unit circle is comparable to ${1}$, thus requiring time comparable to ${1/g}$ to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant ${\Lambda}$ should typically take on values comparable to ${-1/g}$ (since typically one would not expect the initial configuration of zeroes to be close to evenly spaced). I have not attempted to formalise or prove this claim, but presumably one could do some numerics (perhaps using some of the examples of ${P}$ given previously) to explore this further.

I have just uploaded to the arXiv my paper “Commutators close to the identity“, submitted to the Journal of Operator Theory. This paper resulted from some progress I made on the problem discussed in this previous post. Recall in that post the following result of Popa: if ${D,X \in B(H)}$ are bounded operators on a Hilbert space ${H}$ whose commutator ${[D,X] := DX-XD}$ is close to the identity in the sense that

$\displaystyle \| [D,X] - I \|_{op} \leq \varepsilon \ \ \ \ \ (1)$

for some ${\varepsilon > 0}$, then one has the lower bound

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

In the other direction, for any ${0 < \varepsilon < 1}$, there are examples of operators ${D,X \in B(H)}$ obeying (1) such that

$\displaystyle \| X \|_{op} \|D \|_{op} \ll \varepsilon^{-2}. \ \ \ \ \ (3)$

In this paper we improve the upper bound to come closer to the lower bound:

Theorem 1 For any ${0 < \varepsilon < 1/2}$, and any infinite-dimensional ${H}$, there exist operators ${D,X \in B(H)}$ obeying (1) such that

$\displaystyle \| X \|_{op} \|D \|_{op} \ll \log^{16} \frac{1}{\varepsilon}. \ \ \ \ \ (4)$

One can probably improve the exponent ${16}$ somewhat by a modification of the methods, though it does not seem likely that one can lower it all the way to ${1}$ without a substantially new idea. Nevertheless I believe it plausible that the lower bound (2) is close to optimal.

We now sketch the methods of proof. The construction giving (3) proceeded by first identifying ${B(H)}$ with the algebra ${M_2(B(H))}$ of ${2 \times 2}$ matrices that have entries in ${B(H)}$. It is then possible to find two matrices ${D, X \in M_2(B(H))}$ whose commutator takes the form

$\displaystyle [D,X] = \begin{pmatrix} I & u \\ 0 & I \end{pmatrix}$

for some bounded operator ${u \in B(H)}$ (for instance one can take ${u}$ to be an isometry). If one then conjugates ${D, X}$ by the diagonal operator ${\mathrm{diag}(\varepsilon,1)}$, one can eusure that (1) and (3) both hold.

It is natural to adapt this strategy to ${n \times n}$ matrices ${D,X \in M_n(B(H))}$ rather than ${2 \times 2}$ matrices, where ${n}$ is a parameter at one’s disposal. If one can find matrices ${D,X \in M_n(B(H))}$ that are almost upper triangular (in that only the entries on or above the lower diagonal are non-zero), whose commutator ${[D,X]}$ only differs from the identity in the top right corner, thus

$\displaystyle [D, X] = \begin{pmatrix} I & 0 & 0 & \dots & 0 & S \\ 0 & I & 0 & \dots & 0 & 0 \\ 0 & 0 & I & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & I & 0 \\ 0 & 0 & 0 & \dots & 0 & I \end{pmatrix}.$

for some ${S}$, then by conjugating by a diagonal matrix such as ${\mathrm{diag}( \mu^{n-1}, \mu^{n-2}, \dots, 1)}$ for some ${\mu}$ and optimising in ${\mu}$, one can improve the bound ${\varepsilon^{-2}}$ in (3) to ${O_n( \varepsilon^{-\frac{2}{n-1}} )}$; if the bounds in the implied constant in the ${O_n(1)}$ are polynomial in ${n}$, one can then optimise in ${n}$ to obtain a bound of the form (4) (perhaps with the exponent ${16}$ replaced by a different constant).

The task is then to find almost upper triangular matrices ${D, X}$ whose commutator takes the required form. The lower diagonals of ${D,X}$ must then commute; it took me a while to realise then that one could (usually) conjugate one of the matrices, say ${X}$ by a suitable diagonal matrix, so that the lower diagonal consisted entirely of the identity operator, which would make the other lower diagonal consist of a single operator, say ${u}$. After a lot of further lengthy experimentation, I eventually realised that one could conjugate ${X}$ further by unipotent upper triangular matrices so that all remaining entries other than those on the far right column vanished. Thus, without too much loss of generality, one can assume that ${X}$ takes the normal form

$\displaystyle X := \begin{pmatrix} 0 & 0 & 0 & \dots & 0 & b_1 \\ I & 0 & 0 & \dots & 0 & b_2 \\ 0 & I & 0 & \dots & 0 & b_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 0 & b_{n-1} \\ 0 & 0 & 0 & \dots & I & b_n \end{pmatrix}.$

$\displaystyle D := \begin{pmatrix} v & I & 0 & \dots & 0 & b_1 u \\ u & v & 2 I & \dots & 0 & b_2 u \\ 0 & u & v & \dots & 0 & b_3 u \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & v & (n-1) I + b_{n-1} u \\ 0 & 0 & 0 & \dots & u & v + b_n u \end{pmatrix}$

for some ${u,v \in B(H)}$, solving the system of equations

$\displaystyle [v, b_i] + [u, b_{i-1}] + i b_{i+1} + b_i [u, b_n] = 0 \ \ \ \ \ (5)$

for ${i=2,\dots,n-1}$, and also

$\displaystyle [v, b_n] + [u, b_{n-1}] + b_n [u, b_n] = n \cdot 1_{B(H)}. \ \ \ \ \ (6)$

It turns out to be possible to solve this system of equations by a contraction mapping argument if one takes ${u,v}$ to be a “Hilbert’s hotel” pair of isometries as in the previous post, though the contraction is very slight, leading to polynomial losses in ${n}$ in the implied constant.

There is a further question raised in Popa’s paper which I was unable to resolve. As a special case of one of the main theorems (Theorem 2.1) of that paper, the following result was shown: if ${A \in B(H)}$ obeys the bounds

$\displaystyle \|A \| = O(1)$

and

$\displaystyle \| A \| = O( \mathrm{dist}( A, {\bf C} + K(H) )^{2/3} ) \ \ \ \ \ (7)$

(where ${{\bf C} + K(H)}$ denotes the space of all operators of the form ${\lambda I + T}$ with ${\lambda \in {\bf C}}$ and ${T}$ compact), then there exist operators ${D,X \in B(H)}$ with ${\|D\|, \|X\| = O(1)}$ such that ${A = [D,X]}$. (In fact, Popa’s result covers a more general situation in which one is working in a properly infinite ${W^*}$ algebra with non-trivial centre.) We sketch a proof of this result as follows. Suppose that ${\mathrm{dist}(A, {\bf C} + K(H)) = \varepsilon}$ and ${\|A\| = O( \varepsilon^{2/3})}$ for some ${0 < \varepsilon \ll 1}$. A standard greedy algorithm argument (see this paper of Brown and Pearcy) allows one to find orthonormal vectors ${e_n, f_n, g_n}$ for ${n=1,2,\dots}$ such that for each ${n}$, one has ${A e_n = \varepsilon_n f_n + v_n}$ for some ${\varepsilon_n}$ comparable to ${\varepsilon}$, and some ${v_n}$ orthogonal to all of the ${e_n,f_n,g_n}$. After some conjugation (and a suitable identification of ${B(H)}$ with ${M_2(B(H))}$, one can thus place ${A}$ in a normal form

$\displaystyle A = \begin{pmatrix} \varepsilon^{2/3} x & \varepsilon v^* \\ \varepsilon^{2/3} y & \varepsilon^{2/3} z \end{pmatrix}$

where ${v \in B(H)}$ is a isometry with infinite deficiency, and ${x,y,z \in B(H)}$ have norm ${O(1)}$. Setting ${\varepsilon' := \varepsilon^{1/3}}$, it then suffices to solve the commutator equation

$\displaystyle [D,X] = \begin{pmatrix} x & \varepsilon' v^* \\ y & z \end{pmatrix}$

with ${\|D\|_{op} \|X\|_{op} \ll (\varepsilon')^{-2}}$; note the similarity with (3).

By the usual Hilbert’s hotel construction, one can complement ${v}$ with another isometry ${u}$ obeying the “Hilbert’s hotel” identity

$\displaystyle uu^* + vv^* = I$

and also ${u^* u = v^* v = I}$, ${u^* v = v^* u = 0}$. Proceeding as in the previous post, we can try the ansatz

$\displaystyle D = \begin{pmatrix} \frac{1}{2} u^* & 0 \\ a & \frac{1}{2} u^* - v^* \end{pmatrix}, X = \begin{pmatrix} b & \varepsilon' I \\ c & d \end{pmatrix}$

for some operators ${a,b,c,d \in B(H)}$, leading to the system of equations

$\displaystyle [\frac{1}{2} u^*, b] + [\frac{1}{2} u^* - v^*, c] = x+z$

$\displaystyle \varepsilon' a = [\frac{1}{2} u^*, b] - x$

$\displaystyle \frac{1}{2} u^* c + c (\frac{1}{2} u^* - v^*) + ab-da = y.$

Using the first equation to solve for ${b,c}$, the second to then solve for ${a}$, and the third to then solve for ${c}$, one can obtain matrices ${D,X}$ with the required properties.

Thus far, my attempts to extend this construction to larger matrices with good bounds on ${D,X}$ have been unsuccessful. A model problem would be to express

$\displaystyle \begin{pmatrix} I & 0 & \varepsilon v^* \\ 0 & I & 0 \\ 0 & 0 & I \end{pmatrix}$

as a commutator ${[D,X]}$ with ${\|D\| \|X\|}$ significantly smaller than ${O(\varepsilon^{-2})}$. The construction in my paper achieves something like this, but with ${v^*}$ replaced by a more complicated operator. One would also need variants of this result in which one is allowed to perturb the above operator by an arbitrary finite rank operator of bounded operator norm.

Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.

In these notes, random variables will be denoted in boldface.

Definition 1 A real random variable ${\mathbf{X}}$ is said to be normally distributed with mean ${x_0 \in {\bf R}}$ and variance ${\sigma^2 > 0}$ if one has

$\displaystyle \mathop{\bf E} F(\mathbf{X}) = \frac{1}{\sqrt{2\pi} \sigma} \int_{\bf R} e^{-(x-x_0)^2/2\sigma^2} F(x)\ dx$

for all test functions ${F \in C_c({\bf R})}$. Similarly, a complex random variable ${\mathbf{Z}}$ is said to be normally distributed with mean ${z_0 \in {\bf R}}$ and variance ${\sigma^2>0}$ if one has

$\displaystyle \mathop{\bf E} F(\mathbf{Z}) = \frac{1}{\pi \sigma^2} \int_{\bf C} e^{-|z-x_0|^2/\sigma^2} F(z)\ dx dy$

for all test functions ${F \in C_c({\bf C})}$, where ${dx dy}$ is the area element on ${{\bf C}}$.

A real Brownian motion with base point ${x_0 \in {\bf R}}$ is a random, almost surely continuous function ${\mathbf{B}^{x_0}: [0,+\infty) \rightarrow {\bf R}}$ (using the locally uniform topology on continuous functions) with the property that (almost surely) ${\mathbf{B}^{x_0}(0) = x_0}$, and for any sequence of times ${0 \leq t_0 < t_1 < t_2 < \dots < t_n}$, the increments ${\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})}$ for ${i=1,\dots,n}$ are independent real random variables that are normally distributed with mean zero and variance ${t_i - t_{i-1}}$. Similarly, a complex Brownian motion with base point ${z_0 \in {\bf R}}$ is a random, almost surely continuous function ${\mathbf{B}^{z_0}: [0,+\infty) \rightarrow {\bf R}}$ with the property that ${\mathbf{B}^{z_0}(0) = z_0}$ and for any sequence of times ${0 \leq t_0 < t_1 < t_2 < \dots < t_n}$, the increments ${\mathbf{B}^{z_0}(t_i) - \mathbf{B}^{z_0}(t_{i-1})}$ for ${i=1,\dots,n}$ are independent complex random variables that are normally distributed with mean zero and variance ${t_i - t_{i-1}}$.

Remark 2 Thanks to the central limit theorem, the hypothesis that the increments ${\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})}$ be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).

Real and complex Brownian motions exist from any base point ${x_0}$ or ${z_0}$; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

• (i) (Translation invariance) If ${\mathbf{B}^{x_0}}$ is a real Brownian motion with base point ${x_0 \in {\bf R}}$, and ${h \in {\bf R}}$, show that ${\mathbf{B}^{x_0}+h}$ is a real Brownian motion with base point ${x_0+h}$. Similarly, if ${\mathbf{B}^{z_0}}$ is a complex Brownian motion with base point ${z_0 \in {\bf R}}$, and ${h \in {\bf C}}$, show that ${\mathbf{B}^{z_0}+c}$ is a complex Brownian motion with base point ${z_0+h}$.
• (ii) (Dilation invariance) If ${\mathbf{B}^{0}}$ is a real Brownian motion with base point ${0}$, and ${\lambda \in {\bf R}}$ is non-zero, show that ${t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})}$ is also a real Brownian motion with base point ${0}$. Similarly, if ${\mathbf{B}^0}$ is a complex Brownian motion with base point ${0}$, and ${\lambda \in {\bf C}}$ is non-zero, show that ${t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})}$ is also a complex Brownian motion with base point ${0}$.
• (iii) (Real and imaginary parts) If ${\mathbf{B}^0}$ is a complex Brownian motion with base point ${0}$, show that ${\sqrt{2} \mathrm{Re} \mathbf{B}^0}$ and ${\sqrt{2} \mathrm{Im} \mathbf{B}^0}$ are independent real Brownian motions with base point ${0}$. Conversely, if ${\mathbf{B}^0_1, \mathbf{B}^0_2}$ are independent real Brownian motions of base point ${0}$, show that ${\frac{1}{\sqrt{2}} (\mathbf{B}^0_1 + i \mathbf{B}^0_2)}$ is a complex Brownian motion with base point ${0}$.

The next lemma is a special case of the optional stopping theorem.

Lemma 4 (Optional stopping identities)

• (i) (Real case) Let ${\mathbf{B}^{x_0}}$ be a real Brownian motion with base point ${x_0 \in {\bf R}}$. Let ${\mathbf{t}}$ be a bounded stopping time – a bounded random variable with the property that for any time ${t \geq 0}$, the event that ${\mathbf{t} \leq t}$ is determined by the values of the trajectory ${\mathbf{B}^{x_0}}$ for times up to ${t}$ (or more precisely, this event is measurable with respect to the ${\sigma}$ algebra generated by this proprtion of the trajectory). Then

$\displaystyle \mathop{\bf E} \mathbf{B}^{x_0}(\mathbf{t}) = x_0$

and

$\displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^2 - \mathbf{t} = 0$

and

$\displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^4 = O( \mathop{\bf E} \mathbf{t}^2 ).$

• (ii) (Complex case) Let ${\mathbf{B}^{z_0}}$ be a real Brownian motion with base point ${z_0 \in {\bf R}}$. Let ${\mathbf{t}}$ be a bounded stopping time – a bounded random variable with the property that for any time ${t \geq 0}$, the event that ${\mathbf{t} \leq t}$ is determined by the values of the trajectory ${\mathbf{B}^{x_0}}$ for times up to ${t}$. Then

$\displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t}) = z_0$

$\displaystyle \mathop{\bf E} (\mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0$

$\displaystyle \mathop{\bf E} (\mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0$

$\displaystyle \mathop{\bf E} \mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) \mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) = 0$

$\displaystyle \mathop{\bf E} |\mathbf{B}^{x_0}(\mathbf{t})-z_0|^4 = O( \mathop{\bf E} \mathbf{t}^2 ).$

Proof: (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take ${x_0=0}$. Let ${T}$ be an upper bound for ${\mathbf{t}}$. Since ${\mathbf{B}^0(T)}$ is a real normally distributed variable with mean zero and variance ${T}$, we have

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T ) = 0$

and

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^2 = T$

and

$\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^4 = 3T^2.$

By the law of total expectation, we thus have

$\displaystyle \mathop{\bf E} \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 0$

and

$\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = T$

and

$\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 3T^2$

where the inner conditional expectations are with respect to the event that ${\mathbf{t}, \mathbf{B}^{0}(\mathbf{t})}$ attains a particular point in ${S}$. However, from the independent increment nature of Brownian motion, once one conditions ${(\mathbf{t}, \mathbf{B}^{0}(\mathbf{t}))}$ to a fixed point ${(t, x)}$, the random variable ${\mathbf{B}^0(T)}$ becomes a real normally distributed variable with mean ${x}$ and variance ${T-t}$. Thus we have

$\displaystyle \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})$

and

$\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^2 + T - \mathbf{t}$

and

$\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^4 + 6(T - \mathbf{t}) \mathbf{B}^{z_0}(\mathbf{t})^2 + 3(T - \mathbf{t})^2$

which give the first two claims, and (after some algebra) the identity

$\displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t})^4 - 6 \mathbf{t} \mathbf{B}^{z_0}(\mathbf{t})^2 + 3 \mathbf{t}^2 = 0$

which then also gives the third claim. $\Box$

Exercise 5 Prove the second part of Lemma 4.

This is the ninth “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 now tentatively improved the upper bound of the de Bruijn-Newman constant to ${\Lambda \leq 0.22}$. Among the technical improvements in our approach, we now are able to use Taylor expansions to efficiently compute the approximation ${A+B}$ to ${H_t(x+iy)}$ for many values of ${x,y}$ in a given region, thus speeding up the computations in the barrier considerably. Also, by using the heuristic that ${H_t(x+iy)}$ behaves somewhat like the partial Euler product ${\prod_p (1 - \frac{1}{p^{\frac{1+y-ix}{2}}})^{-1}}$, we were able to find a good location to place the barrier in which ${H_t(x+iy)}$ is larger than average, hence easier to keep away from zero.

The main remaining bottleneck is that of computing the Euler mollifier bounds that keep ${A+B}$ bounded away from zero for larger values of ${x}$ beyond the barrier. In going below ${0.22}$ we are beginning to need quite complicated mollifiers with somewhat poor tail behavior; we may be reaching the point where none of our bounds will succeed in keeping ${A+B}$ bounded away from zero, so we may be close to the natural limits of our methods.

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

We now approach conformal maps from yet another perspective. Given an open subset ${U}$ of the complex numbers ${{\bf C}}$, define a univalent function on ${U}$ to be a holomorphic function ${f: U \rightarrow {\bf C}}$ that is also injective. We will primarily be studying this concept in the case when ${U}$ is the unit disk ${D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}$.

Clearly, a univalent function ${f: D(0,1) \rightarrow {\bf C}}$ on the unit disk is a conformal map from ${D(0,1)}$ to the image ${f(D(0,1))}$; in particular, ${f(D(0,1))}$ is simply connected, and not all of ${{\bf C}}$ (since otherwise the inverse map ${f^{-1}: {\bf C} \rightarrow D(0,1)}$ would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset ${V \subsetneq {\bf C}}$ of the complex numbers is the image of a univalent function on ${D(0,1)}$. Furthermore, if ${V}$ contains the origin, then the univalent function ${f: D(0,1) \rightarrow {\bf C}}$ with this image becomes unique once we normalise ${f(0) = 0}$ and ${f'(0) > 0}$. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions ${f: D(0,1) \rightarrow {\bf C}}$ with ${f(0)=0}$ and ${f'(0)>0}$. We will focus particular attention on the univalent functions ${f: D(0,1) \rightarrow {\bf C}}$ with the normalisation ${f(0)=0}$ and ${f'(0)=1}$; such functions will be called schlicht functions.

One basic example of a univalent function on ${D(0,1)}$ is the Cayley transform ${z \mapsto \frac{1+z}{1-z}}$, which is a Möbius transformation from ${D(0,1)}$ to the right half-plane ${\{ \mathrm{Re}(z) > 0 \}}$. (The slight variant ${z \mapsto \frac{1-z}{1+z}}$ is also referred to as the Cayley transform, as is the closely related map ${z \mapsto \frac{z-i}{z+i}}$, which maps ${D(0,1)}$ to the upper half-plane.) One can square this map to obtain a further univalent function ${z \mapsto \left( \frac{1+z}{1-z} \right)^2}$, which now maps ${D(0,1)}$ to the complex numbers with the negative real axis ${(-\infty,0]}$ removed. One can normalise this function to be schlicht to obtain the Koebe function

$\displaystyle f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)$

which now maps ${D(0,1)}$ to the complex numbers with the half-line ${(-\infty,-1/4]}$ removed. A little more generally, for any ${\theta \in {\bf R}}$ we have the rotated Koebe function

$\displaystyle f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)$

that is a schlicht function that maps ${D(0,1)}$ to the complex numbers with the half-line ${\{ -re^{-i\theta}: r \geq 1/4\}}$ removed.

Every schlicht function ${f: D(0,1) \rightarrow {\bf C}}$ has a convergent Taylor expansion

$\displaystyle f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots$

for some complex coefficients ${a_1,a_2,\dots}$ with ${a_1=1}$. For instance, the Koebe function has the expansion

$\displaystyle f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n$

and similarly the rotated Koebe function has the expansion

$\displaystyle f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.$

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients ${a_n}$ should obey the bound ${|a_n| \leq n}$ for all ${n}$. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions ${g: D(0,1) \rightarrow {\bf C}}$ that are odd, thus ${g(-z)=-g(z)}$ for all ${z}$, and the Taylor expansion now reads

$\displaystyle g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots$

for some complex coefficients ${b_1,b_3,\dots}$ with ${b_1=1}$. One can transform a general schlicht function ${f: D(0,1) \rightarrow {\bf C}}$ to an odd schlicht function ${g: D(0,1) \rightarrow {\bf C}}$ by observing that the function ${f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}$, after removing the singularity at zero, is a non-zero function that equals ${1}$ at the origin, and thus (as ${D(0,1)}$ is simply connected) has a unique holomorphic square root ${(f(z^2)/z^2)^{1/2}}$ that also equals ${1}$ at the origin. If one then sets

$\displaystyle g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)$

it is not difficult to verify that ${g}$ is an odd schlicht function which additionally obeys the equation

$\displaystyle f(z^2) = g(z)^2. \ \ \ \ \ (4)$

Conversely, given an odd schlicht function ${g}$, the formula (4) uniquely determines a schlicht function ${f}$.

For instance, if ${f}$ is the Koebe function (1), ${g}$ becomes

$\displaystyle g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)$

which maps ${D(0,1)}$ to the complex numbers with two slits ${\{ \pm iy: y > 1/2 \}}$ removed, and if ${f}$ is the rotated Koebe function (2), ${g}$ becomes

$\displaystyle g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)$

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let ${n \geq 1}$ be a natural number.

• (i) (Robertson conjecture) If ${g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}$ is an odd schlicht function, then

$\displaystyle \sum_{k=1}^n |b_{2k-1}|^2 \leq n.$

• (ii) (Bieberbach conjecture) If ${f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}$ is a schlicht function, then

$\displaystyle |a_n| \leq n.$

It is easy to see that the Robertson conjecture for a given value of ${n}$ implies the Bieberbach conjecture for the same value of ${n}$. Indeed, if ${f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}$ is schlicht, and ${g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}$ is the odd schlicht function given by (3), then from extracting the ${z^{2n}}$ coefficient of (4) we obtain a formula

$\displaystyle a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}$

for the coefficients of ${f}$ in terms of the coefficients of ${g}$. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of ${n}$ from the Robertson conjecture for the same value of ${n}$. We remark that Littlewood and Paley had conjectured a stronger form ${|b_{2k-1}| \leq 1}$ of Robertson’s conjecture, but this was disproved for ${k=3}$ by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of ${\log \frac{f(z)}{z}}$, known as the Milin conjecture. Next, one continuously enlarges the image ${f(D(0,1))}$ of the schlicht function to cover all of ${{\bf C}}$; done properly, this places the schlicht function ${f}$ as the initial function ${f = f_0}$ in a sequence ${(f_t)_{t \geq 0}}$ of univalent maps ${f_t: D(0,1) \rightarrow {\bf C}}$ known as a Loewner chain. The functions ${f_t}$ obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term ${\mu_t}$ (or ${\theta(t)}$, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of ${f(z)}$, ${g(z)}$, or ${\log \frac{f(z)}{z}}$. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of ${n}$) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

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 ${\{ w: |w - f(z_0)| = |f'(z_0)| \varepsilon \}}$.

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).

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?

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).