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While talking mathematics with a postdoc here at UCLA (March Boedihardjo) we came across the following matrix problem which we managed to solve, but the proof was cute and the process of discovering it was fun, so I thought I would present the problem here as a puzzle without revealing the solution for now.

The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group $SO(3)$ of real $3 \times 3$ matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the Banach-Tarski paradox).  Given any abstract word $w$ of two generators $x,y$ and their inverses (i.e., an element of the free group ${\bf F}_2$), one can define the word map $w: SO(3) \times SO(3) \to SO(3)$ simply by substituting a pair of matrices in $SO(3)$ into these generators.  For instance, if one has the word $w = x y x^{-2} y^2 x$, then the corresponding word map $w: SO(3) \times SO(3) \to SO(3)$ is given by

$\displaystyle w(A,B) := ABA^{-2} B^2 A$

for $A,B \in SO(3)$.  Because $SO(3)$ contains a copy of the free group, we see the word map is non-trivial (not equal to the identity) if and only if the word itself is nontrivial.

Anyway, here is the problem:

Problem. Does there exist a sequence $w_1, w_2, \dots$ of non-trivial word maps $w_n: SO(3) \times SO(3) \to SO(3)$ that converge uniformly to the identity map?

To put it another way, given any $\varepsilon > 0$, does there exist a non-trivial word $w$ such that $\|w(A,B) - 1 \| \leq \varepsilon$ for all $A,B \in SO(3)$, where $\| \|$ denotes (say) the operator norm, and $1$ denotes the identity matrix in $SO(3)$?

As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.

Note: this post is not required reading for this course, or for the sequel course in the winter quarter.

In a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes equations. We did not quite follow Leray’s original proof, in that the notes relied more heavily on the machinery of Littlewood-Paley projections, which have become increasingly common tools in modern PDE. On the other hand, we did use the same “exploiting compactness to pass to weakly convergent subsequence” strategy that is the standard one in the PDE literature used to construct weak solutions.

As I discussed in a previous post, the manipulation of sequences and their limits is analogous to a “cheap” version of nonstandard analysis in which one uses the Fréchet filter rather than an ultrafilter to construct the nonstandard universe. (The manipulation of generalised functions of Columbeau-type can also be comfortably interpreted within this sort of cheap nonstandard analysis.) Augmenting the manipulation of sequences with the right to pass to subsequences whenever convenient is then analogous to a sort of “lazy” nonstandard analysis, in which the implied ultrafilter is never actually constructed as a “completed object“, but is instead lazily evaluated, in the sense that whenever membership of a given subsequence of the natural numbers in the ultrafilter needs to be determined, one either passes to that subsequence (thus placing it in the ultrafilter) or the complement of the sequence (placing it out of the ultrafilter). This process can be viewed as the initial portion of the transfinite induction that one usually uses to construct ultrafilters (as discussed using a voting metaphor in this post), except that there is generally no need in any given application to perform the induction for any uncountable ordinal (or indeed for most of the countable ordinals also).

On the other hand, it is also possible to work directly in the orthodox framework of nonstandard analysis when constructing weak solutions. This leads to an approach to the subject which is largely equivalent to the usual subsequence-based approach, though there are some minor technical differences (for instance, the subsequence approach occasionally requires one to work with separable function spaces, whereas in the ultrafilter approach the reliance on separability is largely eliminated, particularly if one imposes a strong notion of saturation on the nonstandard universe). The subject acquires a more “algebraic” flavour, as the quintessential analysis operation of taking a limit is replaced with the “standard part” operation, which is an algebra homomorphism. The notion of a sequence is replaced by the distinction between standard and nonstandard objects, and the need to pass to subsequences disappears entirely. Also, the distinction between “bounded sequences” and “convergent sequences” is largely eradicated, particularly when the space that the sequences ranged in enjoys some compactness properties on bounded sets. Also, in this framework, the notorious non-uniqueness features of weak solutions can be “blamed” on the non-uniqueness of the nonstandard extension of the standard universe (as well as on the multiple possible ways to construct nonstandard mollifications of the original standard PDE). However, many of these changes are largely cosmetic; switching from a subsequence-based theory to a nonstandard analysis-based theory does not seem to bring one significantly closer for instance to the global regularity problem for Navier-Stokes, but it could have been an alternate path for the historical development and presentation of the subject.

In any case, I would like to present below the fold this nonstandard analysis perspective, quickly translating the relevant components of real analysis, functional analysis, and distributional theory that we need to this perspective, and then use it to re-prove Leray’s theorem on existence of global weak solutions to Navier-Stokes.

The celebrated decomposition theorem of Fefferman and Stein shows that every function ${f \in \mathrm{BMO}({\bf R}^n)}$ of bounded mean oscillation can be decomposed in the form

$\displaystyle f = f_0 + \sum_{i=1}^n R_i f_i \ \ \ \ \ (1)$

modulo constants, for some ${f_0,f_1,\dots,f_n \in L^\infty({\bf R}^n)}$, where ${R_i := |\nabla|^{-1} \partial_i}$ are the Riesz transforms. A technical note here a function in BMO is defined only up to constants (as well as up to the usual almost everywhere equivalence); related to this, if ${f_i}$ is an ${L^\infty({\bf R}^n)}$ function, then the Riesz transform ${R_i f_i}$ is well defined as an element of ${\mathrm{BMO}({\bf R}^n)}$, but is also only defined up to constants and almost everywhere equivalence.

The original proof of Fefferman and Stein was indirect (relying for instance on the Hahn-Banach theorem). A constructive proof was later given by Uchiyama, and was in fact the topic of the second post on this blog. A notable feature of Uchiyama’s argument is that the construction is quite nonlinear; the vector-valued function ${(f_0,f_1,\dots,f_n)}$ is defined to take values on a sphere, and the iterative construction to build these functions from ${f}$ involves repeatedly projecting a potential approximant to this function to the sphere (also, the high-frequency components of this approximant are constructed in a manner that depends nonlinearly on the low-frequency components, which is a type of technique that has become increasingly common in analysis and PDE in recent years).

It is natural to ask whether the Fefferman-Stein decomposition (1) can be made linear in ${f}$, in the sense that each of the ${f_i, i=0,\dots,n}$ depend linearly on ${f}$. Strictly speaking this is easily accomplished using the axiom of choice: take a Hamel basis of ${\mathrm{BMO}({\bf R}^n)}$, choose a decomposition (1) for each element of this basis, and then extend linearly to all finite linear combinations of these basis functions, which then cover ${\mathrm{BMO}({\bf R}^n)}$ by definition of Hamel basis. But these linear operations have no reason to be continuous as a map from ${\mathrm{BMO}({\bf R}^n)}$ to ${L^\infty({\bf R}^n)}$. So the correct question is whether the decomposition can be made continuously linear (or equivalently, boundedly linear) in ${f}$, that is to say whether there exist continuous linear transformations ${T_i: \mathrm{BMO}({\bf R}^n) \rightarrow L^\infty({\bf R}^n)}$ such that

$\displaystyle f = T_0 f + \sum_{i=1}^n R_i T_i f \ \ \ \ \ (2)$

modulo constants for all ${f \in \mathrm{BMO}({\bf R}^n)}$. Note from the open mapping theorem that one can choose the functions ${f_0,\dots,f_n}$ to depend in a bounded fashion on ${f}$ (thus ${\|f_i\|_{L^\infty} \leq C \|f\|_{BMO}}$ for some constant ${C}$, however the open mapping theorem does not guarantee linearity. Using a result of Bartle and Graves one can also make the ${f_i}$ depend continuously on ${f}$, but again the dependence is not guaranteed to be linear.

It is generally accepted folklore that continuous linear dependence is known to be impossible, but I had difficulty recently tracking down an explicit proof of this assertion in the literature (if anyone knows of a reference, I would be glad to know of it). The closest I found was a proof of a similar statement in this paper of Bourgain and Brezis, which I was able to adapt to establish the current claim. The basic idea is to average over the symmetries of the decomposition, which in the case of (1) are translation invariance, rotation invariance, and dilation invariance. This effectively makes the operators ${T_0,T_1,\dots,T_n}$ invariant under all these symmetries, which forces them to themselves be linear combinations of the identity and Riesz transform operators; however, no such non-trivial linear combination maps ${\mathrm{BMO}}$ to ${L^\infty}$, and the claim follows. Formal details of this argument (which we phrase in a dual form in order to avoid some technicalities) appear below the fold.

In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time do not have high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit ${\nu \rightarrow 0}$.

However, it is possible to construct “weak” solutions which lack many of the desirable features of strong solutions (notably, uniqueness, propagation of regularity, and conservation laws) but can often be constructed globally in time even when one us unable to do so for strong solutions. Broadly speaking, one usually constructs weak solutions by some sort of “compactness method”, which can generally be described as follows.

1. Construct a sequence of “approximate solutions” to the desired equation, for instance by developing a well-posedness theory for some “regularised” approximation to the original equation. (This theory often follows similar lines to those in the previous set of notes, for instance using such tools as the contraction mapping theorem to construct the approximate solutions.)
2. Establish some uniform bounds (over appropriate time intervals) on these approximate solutions, even in the limit as an approximation parameter is sent to zero. (Uniformity is key; non-uniform bounds are often easy to obtain if one puts enough “mollification”, “hyper-dissipation”, or “discretisation” in the approximating equation.)
3. Use some sort of “weak compactness” (e.g., the Banach-Alaoglu theorem, the Arzela-Ascoli theorem, or the Rellich compactness theorem) to extract a subsequence of approximate solutions that converge (in a topology weaker than that associated to the available uniform bounds) to a limit. (Note that there is no reason a priori to expect such limit points to be unique, or to have any regularity properties beyond that implied by the available uniform bounds..)
4. Show that this limit solves the original equation in a suitable weak sense.

The quality of these weak solutions is very much determined by the type of uniform bounds one can obtain on the approximate solution; the stronger these bounds are, the more properties one can obtain on these weak solutions. For instance, if the approximate solutions enjoy an energy identity leading to uniform energy bounds, then (by using tools such as Fatou’s lemma) one tends to obtain energy inequalities for the resulting weak solution; but if one somehow is able to obtain uniform bounds in a higher regularity norm than the energy then one can often recover the full energy identity. If the uniform bounds are at the regularity level needed to obtain well-posedness, then one generally expects to upgrade the weak solution to a strong solution. (This phenomenon is often formalised through weak-strong uniqueness theorems, which we will discuss later in these notes.) Thus we see that as far as attacking global regularity is concerned, both the theory of strong solutions and the theory of weak solutions encounter essentially the same obstacle, namely the inability to obtain uniform bounds on (exact or approximate) solutions at high regularities (and at arbitrary times).

For simplicity, we will focus our discussion in this notes on finite energy weak solutions on ${{\bf R}^d}$. There is a completely analogous theory for periodic weak solutions on ${{\bf R}^d}$ (or equivalently, weak solutions on the torus ${({\bf R}^d/{\bf Z}^d)}$ which we will leave to the interested reader.

In recent years, a completely different way to construct weak solutions to the Navier-Stokes or Euler equations has been developed that are not based on the above compactness methods, but instead based on techniques of convex integration. These will be discussed in a later set of notes.

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.

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 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.

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 https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/. 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.