I feel though that there could be more that could be done with this sort of framework (e.g., improved GUI, modification to other logics, developing the ability to write one’s own texts and libraries, exploring mathematical theories such as Peano arithmetic, etc.). But writing this text (particularly the first-order logic sections) has brought me close to the limit of my programming ability, as the number of bugs introduced with each new feature implemented has begun to grow at an alarming rate. I would like to repackage the code so that it can be re-used by more adept programmers for further possible applications, though I have never done something like this before and would appreciate advice on how to do so. The code is already available under a Creative Commons licence, but I am not sure how readable and modifiable it will be to others currently.

[One thing I noticed is that I would probably have to make more of a decoupling between the GUI elements, the underlying logical elements, and the interactive text. For instance, at some point I made the decision (convenient at the time) to use some GUI elements to store some of the state variables of the text, e.g. the exercise buttons are currently storing the status of what exercises are unlocked or not. This is presumably not an example of good programming practice, though it would be relatively easy to fix. More seriously, due to my inability to come up with a good general-purpose matching algorithm (or even specification of such an algorithm) for the the laws of first-order logic, many of the laws have to be hard-coded into the matching routine, so one cannot currently remove them from the text. It may well be that the best thing to do in fact is to rework the entire codebase from scratch using more professional software design methods.]

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After the two previous congresses in 2010 and 2014, I wrote blog posts describing some of the work of each of the winners. This time, though, I happened to be a member of the Fields Medal selection committee, and as such had access to a large number of confidential letters and discussions about the candidates with the other committee members; in order to have the opinions and discussion as candid as possible, it was explicitly understood that these communications would not be publicly disclosed. Because of this, I will unfortunately not be able to express much of a comment or opinion on the candidates or the process as an individual (as opposed to a joint statement of the committee). I can refer you instead to the formal citations of the laureates (which, as a committee member, I was involved in crafting, and then signing off on), or the profiles of the laureates by Quanta magazine; see also the short biographical videos of the laureates by the Simons Foundation that accompanied the formal announcements of the winners. I am sure, though, that there will be plenty of other mathematicians who will be able to present the work of each of the medalists (for instance, there was a *laudatio* given at the ICM for each of the winners, which should eventually be made available at this link).

I know that there is a substantial amount of interest in finding out more about the inner workings of the Fields Medal selection process. For the reasons stated above, I as an individual will unfortunately be unable to answer any questions about this process (e.g., I cannot reveal any information about other nominees, or of any comparisons between any two candidates or nominees). I think I can safely express the following two personal opinions though. Firstly, while I have served on many prize committees in the past, the process for the Fields Medal committee was by far the most thorough and deliberate of any I have been part of, and I for one learned an astonishing amount about the mathematical work of all of the shortlisted nominees, which was an absolutely essential component of the deliberations, in particular giving the discussions a context which would have been very difficult to obtain for an individual mathematician not in possession of all the confidential letters, presentations, and other information available to the committee (in particular, some of my preconceived impressions about the nominees going into the process had to be corrected in light of this more complete information). Secondly, I believe the four medalists are all extremely deserving recipients of the prize, and I fully stand by the decision of the committee to award the Fields medals this year to these four.

I’ll leave the comments to this post open for anyone who wishes to discuss the work of the medalists. But, for the reasons above, I will not participate in the discussion myself.

*[Edit, Aug 1: looks like the ICM site is (barely) up and running now, so links have been added. At this time of writing, there does not seem to be an online announcement of the composition of the committee, but this should appear in due course. -T.]*

*[Edit, Aug 9: the composition of the Fields Medal Committee for 2018 (which included myself) can be found here. -T.]*

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Since then, I have thought a couple times about whether there were other parts of mathematics which could be gamified in a similar fashion. Shortly after my first blog posts on this topic, I experimented with a similar gamification of Lewis Carroll’s classic list of logic puzzles, but the results were quite clunky, and I was never satisfied with the results.

Over the last few weeks I returned to this topic though, thinking in particular about how to gamify the rules of inference of propositional logic, in a manner that at least vaguely resembles how mathematicians actually go about making logical arguments (e.g., splitting into cases, arguing by contradiction, using previous result as lemmas to help with subsequent ones, and so forth). The rules of inference are a list of a dozen or so deductive rules concerning propositional sentences (things like “( AND ) OR (NOT )”, where are some formulas). A typical such rule is Modus Ponens: if the sentence is known to be true, and the implication “ IMPLIES ” is also known to be true, then one can deduce that is also true. Furthermore, in this deductive calculus it is possible to temporarily introduce some unproven statements as an assumption, only to discharge them later. In particular, we have the deduction theorem: if, after making an assumption , one is able to derive the statement , then one can conclude that the implication “ IMPLIES ” is true without any further assumption.

It took a while for me to come up with a workable game-like graphical interface for all of this, but I finally managed to set one up, now using Javascript instead of Scratch (which would be hopelessly inadequate for this task); indeed, part of the motivation of this project was to finally learn how to program in Javascript, which turned out to be not as formidable as I had feared (certainly having experience with other C-like languages like C++, Java, or lua, as well as some prior knowledge of HTML, was very helpful). The main code for this project is available here. Using this code, I have created an interactive textbook in the style of a computer game, which I have titled “QED”. This text contains thirty-odd exercises arranged in twelve sections that function as game “levels”, in which one has to use a given set of rules of inference, together with a given set of hypotheses, to reach a desired conclusion. The set of available rules increases as one advances through the text; in particular, each new section gives one or more rules, and additionally each exercise one solves automatically becomes a new deduction rule one can exploit in later levels, much as lemmas and propositions are used in actual mathematics to prove more difficult theorems. The text automatically tries to match available deduction rules to the sentences one clicks on or drags, to try to minimise the amount of manual input one needs to actually make a deduction.

Most of one’s proof activity takes place in a “root environment” of statements that are known to be true (under the given hypothesis), but for more advanced exercises one has to also work in sub-environments in which additional assumptions are made. I found the graphical metaphor of nested boxes to be useful to depict this tree of sub-environments, and it seems to combine well with the drag-and-drop interface.

The text also logs one’s moves in a more traditional proof format, which shows how the mechanics of the game correspond to a traditional mathematical argument. My hope is that this will give students a way to understand the underlying concept of forming a proof in a manner that is more difficult to achieve using traditional, non-interactive textbooks.

I have tried to organise the exercises in a game-like progression in which one first works with easy levels that train the player on a small number of moves, and then introduce more advanced moves one at a time. As such, the order in which the rules of inference are introduced is a little idiosyncratic. The most powerful rule (the law of the excluded middle, which is what separates classical logic from intuitionistic logic) is saved for the final section of the text.

Anyway, I am now satisfied enough with the state of the code and the interactive text that I am willing to make both available (and open source; I selected a CC-BY licence for both), and would be happy to receive feedback on any aspect of the either. In principle one could extend the game mechanics to other mathematical topics than the propositional calculus – the rules of inference for first-order logic being an obvious next candidate – but it seems to make sense to focus just on propositional logic for now.

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A measure-preserving system is said to be ergodic if all the invariant sets are either zero measure or full measure. An equivalent form of this statement is that any measurable function which is *locally essentially constant* in the sense that for -almost every , is necessarily *globally essentially constant* in the sense that there is a constant such that for -almost every . A basic consequence of ergodicity is the mean ergodic theorem: if , then the averages converge in norm to the mean . (The mean ergodic theorem also applies to other spaces with , though it is usually proven first in the Hilbert space .) Informally: in ergodic systems, time averages are asymptotically equal to space averages. Specialising to the case of indicator functions, this implies in particular that converges to for any measurable set .

In this short note I would like to use the mean ergodic theorem to show that ergodic systems also have the property that “somewhat locally constant” functions are necessarily “somewhat globally constant”; this is not a deep observation, and probably already in the literature, but I found it a cute statement that I had not previously seen. More precisely:

Corollary 1Let be an ergodic measure-preserving system, and let be measurable. Suppose thatfor some . Then there exists a constant such that for in a set of measure at least .

Informally: if is locally constant on pairs at least of the time, then is globally constant at least of the time. Of course the claim fails if the ergodicity hypothesis is dropped, as one can simply take to be an invariant function that is not essentially constant, such as the indicator function of an invariant set of intermediate measure. This corollary can be viewed as a manifestation of the general principle that ergodic systems have the same “global” (or “space-averaged”) behaviour as “local” (or “time-averaged”) behaviour, in contrast to non-ergodic systems in which local properties do not automatically transfer over to their global counterparts.

*Proof:* By composing with (say) the tangent function, we may assume without loss of generality that is bounded. Let , and partition as , where is the level set

For each , only finitely many of the are non-empty. By (1), one has

Using the ergodic theorem, we conclude that

On the other hand, . Thus there exists such that , thus

By the Bolzano-Weierstrass theorem, we may pass to a subsequence where converges to a limit , then we have

for infinitely many , and hence

The claim follows.

]]> for all . A map is an *affine* homomorphism if one has

for all *additive quadruples* in , by which we mean that and . The two notions are closely related; it is easy to verify that is an affine homomorphism if and only if is the sum of a homomorphism and a constant.

Now suppose that also has a translation-invariant metric . A map is said to be a quasimorphism if one has

for all , where denotes a quantity at a bounded distance from the origin. Similarly, is an *affine quasimorphism* if

for all additive quadruples in . Again, one can check that is an affine quasimorphism if and only if it is the sum of a quasimorphism and a constant (with the implied constant of the quasimorphism controlled by the implied constant of the affine quasimorphism). (Since every constant is itself a quasimorphism, it is in fact the case that affine quasimorphisms are quasimorphisms, but now the implied constant in the latter is not controlled by the implied constant of the former.)

“Trivial” examples of quasimorphisms include the sum of a homomorphism and a bounded function. Are there others? In some cases, the answer is no. For instance, suppose we have a quasimorphism . Iterating (2), we see that for any integer and natural number , which we can rewrite as for non-zero . Also, is Lipschitz. Sending , we can verify that is a Cauchy sequence as and thus tends to some limit ; we have for , hence for positive , and then one can use (2) one last time to obtain for all . Thus is the sum of the homomorphism and a bounded sequence.

In general, one can phrase this problem in the language of group cohomology (discussed in this previous post). Call a map a *-cocycle*. A *-cocycle* is a map obeying the identity

for all . Given a -cocycle , one can form its *derivative* by the formula

Such functions are called *-coboundaries*. It is easy to see that the abelian group of -coboundaries is a subgroup of the abelian group of -cocycles. The quotient of these two groups is the first group cohomology of with coefficients in , and is denoted .

If a -cocycle is bounded then its derivative is a bounded -coboundary. The quotient of the group of bounded -cocycles by the derivatives of bounded -cocycles is called the *bounded first group cohomology* of with coefficients in , and is denoted . There is an obvious homomorphism from to , formed by taking a coset of the space of derivatives of bounded -cocycles, and enlarging it to a coset of the space of -coboundaries. By chasing all the definitions, we see that all quasimorphism from to are the sum of a homomorphism and a bounded function if and only if this homomorphism is injective; in fact the quotient of the space of quasimorphisms by the sum of homomorphisms and bounded functions is isomorphic to the kernel of .

In additive combinatorics, one is often working with functions which only have additive structure a fraction of the time, thus for instance (1) or (3) might only hold “ of the time”. This makes it somewhat difficult to directly interpret the situation in terms of group cohomology. However, thanks to tools such as the Balog-Szemerédi-Gowers lemma, one can upgrade this sort of -structure to -structure – at the cost of restricting the domain to a smaller set. Here I record one such instance of this phenomenon, thus giving a tentative link between additive combinatorics and group cohomology. (I thank Yuval Wigderson for suggesting the problem of locating such a link.)

Theorem 1Let , be additive groups with , let be a subset of , let , and let be a function such thatfor additive quadruples in . Then there exists a subset of containing with , a subset of with , and a function such that

for all (thus, the derivative takes values in on ), and such that for each , one has

Presumably the constants and can be improved further, but we have not attempted to optimise these constants. We chose as the domain on which one has a bounded derivative, as one can use the Bogulybov lemma (see e.g, Proposition 4.39 of my book with Van Vu) to find a large Bohr set inside . In applications, the set need not have bounded size, or even bounded doubling; for instance, in the inverse theory over a small finite fields , one would be interested in the situation where is the group of matrices with coefficients in (for some large , and being the subset consisting of those matrices of rank bounded by some bound .

*Proof:* By hypothesis, there are triples such that and

Thus, there is a set with such that for all , one has (6) for pairs with ; in particular, there exists such that (6) holds for values of . Setting , we conclude that for each , one has

Consider the bipartite graph whose vertex sets are two copies of , and and connected by a (directed) edge if and (7) holds. Then this graph has edges. Applying (a slight modification of) the Balog-Szemerédi-Gowers theorem (for instance by modifying the proof of Corollary 5.19 of my book with Van Vu), we can then find a subset of with with the property that for any , there exist triples such that the edges all lie in this bipartite graph. This implies that, for all , there exist septuples obeying the constraints

and for . These constraints imply in particular that

Also observe that

Thus, if and are such that , we see that

for octuples in the hyperplane

By the pigeonhole principle, this implies that for any fixed , there can be at most sets of the form with , that are pairwise disjoint. Using a greedy algorithm, we conclude that there is a set of cardinality , such that each set with , intersects for some , or in other words that

This implies that there exists a subset of with , and an element for each , such that

for all . Note we may assume without loss of generality that and .

By construction of , and permuting labels, we can find 16-tuples such that

and

for . We sum this to obtain

and hence by (8)

where . Since

we see that there are only possible values of . By the pigeonhole principle, we conclude that at most of the sets can be disjoint. Arguing as before, we conclude that there exists a set of cardinality such that

whenever (10) holds.

For any , write arbitrarily as for some (with if , and if ) and then set

Then from (11) we have (4). For we have , and (5) then follows from (9).

]]>on the zeroes of a time-dependent family of polynomials , with a particular focus on the case when the polynomials 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 , with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when is the numerator of the zeta function of a curve.

More precisely, let be a natural number. We will say that a polynomial

of degree (so that ) obeys the *functional equation* if the are all real and

for all , thus

and

for all non-zero . This means that the zeroes of (counting multiplicity) lie in and are symmetric with respect to complex conjugation and inversion across the circle . We say that this polynomial *obeys the Riemann hypothesis* if all of its zeroes actually lie on the circle . For instance, in the case, the polynomial obeys the Riemann hypothesis if and only if .

Such polynomials arise in number theory as follows: if is a projective curve of genus over a finite field , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the form

where is a degree polynomial obeying both the functional equation and the Riemann hypothesis. In the case that is an elliptic curve, then and takes the form , where is the number of -points of minus . The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

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

Given a polynomial of degree with coefficients

we can evolve it in time by the formula

thus for . Informally, as one increases , this evolution accentuates the effect of the extreme monomials, particularly, and at the expense of the intermediate monomials such as , and conversely as one decreases . This family of polynomials obeys the heat-type equation

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 , and should also be tied to some sort of “” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if obeys the functional equation, then so does for any other time . Now we investigate the evolution of the zeroes. Suppose at some time that the zeroes of are distinct, then

From the inverse function theorem we see that for times sufficiently close to , the zeroes of continue to be distinct (and vary smoothly in ), with

Differentiating this at any not equal to any of the , we obtain

and

and

Inserting these formulae into (2) (expanding as ) and canceling some terms, we conclude that

for sufficiently close to , and not equal to . Extracting the residue at , we conclude that

which we can rearrange as

If we make the change of variables (noting that one can make depend smoothly on for sufficiently close to ), this becomes

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

then the above equation becomes a gradient flow

which implies in particular that is non-increasing in time. This shows that as one evolves time forward from , there is a uniform lower bound on the separation between the phases , and hence the equation can be solved indefinitely; in particular, obeys the Riemann hypothesis for all if it does so at time . Our argument here assumed that the zeroes of were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial obeying the functional equation, the rescaled polynomials converge locally uniformly to as . By Rouche’s theorem, we conclude that the zeroes of converge to the equally spaced points on the circle . Together with the symmetry properties of the zeroes, this implies in particular that obeys the Riemann hypothesis for all sufficiently large positive . In the opposite direction, when , the polynomials converge locally uniformly to , so if , of the zeroes converge to the origin and the other converge to infinity. In particular, fails the Riemann hypothesis for sufficiently large negative . Thus (if ), there must exist a real number , which we call the *de Bruijn-Newman constant* of the original polynomial , such that obeys the Riemann hypothesis for and fails the Riemann hypothesis for . The situation is a bit more complicated if vanishes; if is the first natural number such that (or equivalently, ) does not vanish, then by the above arguments one finds in the limit that of the zeroes go to the origin, go to infinity, and the remaining zeroes converge to the equally spaced points . In this case the de Bruijn-Newman constant remains finite except in the degenerate case , in which case .

For instance, consider the case when and for some real with . Then the quadratic polynomial

has zeroes

and one easily checks that these zeroes lie on the circle when , and are on the real axis otherwise. Thus in this case we have (with if ). Note how as increases to , the zeroes repel each other and eventually converge to , while as decreases to , 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 of degree that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to , basically because the average spacing is and hence by (3) the typical velocity of the zeroes should be comparable to , and the diameter of the unit circle is comparable to , thus requiring time comparable to to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant should typically take on values comparable to (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 given previously) to explore this further.

]]>for some , then one has the lower bound

In the other direction, for any , there are examples of operators obeying (1) such that

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

Theorem 1For any , and any infinite-dimensional , there exist operators obeying (1) such that

One can probably improve the exponent somewhat by a modification of the methods, though it does not seem likely that one can lower it all the way to 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 with the algebra of matrices that have entries in . It is then possible to find two matrices whose commutator takes the form

for some bounded operator (for instance one can take to be an isometry). If one then conjugates by the diagonal operator , one can eusure that (1) and (3) both hold.

It is natural to adapt this strategy to matrices rather than matrices, where is a parameter at one’s disposal. If one can find matrices that are almost upper triangular (in that only the entries on or above the lower diagonal are non-zero), whose commutator only differs from the identity in the top right corner, thus

for some , then by conjugating by a diagonal matrix such as for some and optimising in , one can improve the bound in (3) to ; if the bounds in the implied constant in the are polynomial in , one can then optimise in to obtain a bound of the form (4) (perhaps with the exponent replaced by a different constant).

The task is then to find almost upper triangular matrices whose commutator takes the required form. The lower diagonals of must then commute; it took me a while to realise then that one could (usually) conjugate one of the matrices, say 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 . After a lot of further lengthy experimentation, I eventually realised that one could conjugate 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 takes the normal form

for some , solving the system of equations

It turns out to be possible to solve this system of equations by a contraction mapping argument if one takes 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 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 obeys the bounds

(where denotes the space of all operators of the form with and compact), then there exist operators with such that . (In fact, Popa’s result covers a more general situation in which one is working in a properly infinite algebra with non-trivial centre.) We sketch a proof of this result as follows. Suppose that and for some . A standard greedy algorithm argument (see this paper of Brown and Pearcy) allows one to find orthonormal vectors for such that for each , one has for some comparable to , and some orthogonal to all of the . After some conjugation (and a suitable identification of with , one can thus place in a normal form

where is a isometry with infinite deficiency, and have norm . Setting , it then suffices to solve the commutator equation

with ; note the similarity with (3).

By the usual Hilbert’s hotel construction, one can complement with another isometry obeying the “Hilbert’s hotel” identity

and also , . Proceeding as in the previous post, we can try the ansatz

for some operators , leading to the system of equations

Using the first equation to solve for , the second to then solve for , and the third to then solve for , one can obtain matrices with the required properties.

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

as a commutator with significantly smaller than . The construction in my paper achieves something like this, but with 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.

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

Definition 1A real random variable is said to be normally distributed with mean and variance if one hasfor all test functions . Similarly, a complex random variable is said to be normally distributed with mean and variance if one has

for all test functions , where is the area element on .

A

real Brownian motionwith base point is a random, almost surely continuous function (using the locally uniform topology on continuous functions) with the property that (almost surely) , and for any sequence of times , the increments for are independent real random variables that are normally distributed with mean zero and variance . Similarly, acomplex Brownian motionwith base point is a random, almost surely continuous function with the property that and for any sequence of times , the increments for are independent complex random variables that are normally distributed with mean zero and variance .

Remark 2Thanks to the central limit theorem, the hypothesis that the increments 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 or ; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

- (i) (Translation invariance) If is a real Brownian motion with base point , and , show that is a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and , show that is a complex Brownian motion with base point .
- (ii) (Dilation invariance) If is a real Brownian motion with base point , and is non-zero, show that is also a real Brownian motion with base point . Similarly, if is a complex Brownian motion with base point , and is non-zero, show that is also a complex Brownian motion with base point .
- (iii) (Real and imaginary parts) If is a complex Brownian motion with base point , show that and are independent real Brownian motions with base point . Conversely, if are independent real Brownian motions of base point , show that is a complex Brownian motion with base point .

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

Lemma 4 (Optional stopping identities)

- (i) (Real case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to (or more precisely, this event is measurable with respect to the algebra generated by this proprtion of the trajectory). Then
and

and

- (ii) (Complex case) Let be a real Brownian motion with base point . Let be a bounded stopping time – a bounded random variable with the property that for any time , the event that is determined by the values of the trajectory for times up to . Then

*Proof:* (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take . Let be an upper bound for . Since is a real normally distributed variable with mean zero and variance , we have

and

and

By the law of total expectation, we thus have

and

and

where the inner conditional expectations are with respect to the event that attains a particular point in . However, from the independent increment nature of Brownian motion, once one conditions to a fixed point , the random variable becomes a real normally distributed variable with mean and variance . Thus we have

and

and

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

which then also gives the third claim.

Exercise 5Prove the second part of Lemma 4.

** — 1. Conformal invariance of Brownian motion — **

Let be an open subset of , and a point in . We can define the *complex Brownian motion with base point restricted to * to be the restriction of a complex Brownian motion with base point to the first time in which the Brownian motion exits (or if no such time exists). We have a fundamental conformal invariance theorem of Lévy:

Theorem 6 (Lévy’s theorem on conformal invariance of Brownian motion)Let be a conformal map between two open subsets of , and let be a complex Brownian motion with base point restricted to . Define a rescaling byNote that this is almost surely a continuous strictly monotone increasing function. Set (so that is a homeomorphism from to ), and let be the function defined by the formula

Then is a complex Brownian motion with base point restricted to .

Note that this significantly generalises the translation and dilation invariance of complex Brownian motion.

*Proof:* (Somewhat informal – to do things properly one should first set up Ito calculus) To avoid technicalities we will assume that is bounded above and below on , so that the map is uniformly bilipschitz; the general case can be obtained from this case by a limiting argument that is not detailed here. With this assumption, we see that almost surely extends continuously to the endpoint time if this time is finite. Once one conditions on the value of and up to this time , we then extend this motion further (if ) by declaring for to be a complex Brownian motion with base point , translated in time by . Now is defined on all of , and it will suffice to show that this is a complex Brownian motion based at . The basing is clear, so it suffices to show for all times , the random variable is normally distributed with mean and variance .

Let be a test function. It will suffice to show that

If we define the field

for and , with , then it will suffice to prove the more general claim

for all and (with the convention that is just Brownian motion based at if lies outside of ), where

As is well known, is smooth on and solves the backwards heat equation

on this domain. The strategy will be to show that also solves this equation.

Let and . If then clearly . If instead and , then is a Brownian motion and then we have . Now suppose that be small enough that , where is an upper bound for on . Let be the first time such that either or

Then if we let be the quantity

then and . Let us now condition on a specific value of , and on the trajectory up to time . Then the (conditional) distribution of is that of , and hence the conditional expectation is . By the law of total expectation, we conclude the identity

Next, we obtain the analogous estimate

From Taylor expansion we have

Taking expectations and applying Lemma 4, (2) and Hölder’s inequality (which can interpolate between the bounds and to conclude ), we obtain the desired claim (3). Subtracting, we now have

The expression in the expectation vanishes unless , hence by the triangle inequality

Iterating this using the fact that vanishes at , and sending to zero (noting that the cumulative error term will go to zero since ), we conclude that for all , giving the claim.

One can use Lévy’s theorem (or variants of this theorem) to prove various results in complex analysis rather efficiently. As a quick example, we sketch a Brownian motion-based proof of Liouville’s theorem (omitting some technical steps). Suppose for contradiction that we have a nonconstant bounded entire function . If is a complex Brownian motion based at , then a variant of Levy’s theorem can be used to show that the image is a time parameterisation of Brownian motion. But it is easy to show that Brownian motion is almost surely unbounded, so the image cannot be bounded.

If is an open subset of whose complement contains an arc, then one can show that for any , the complex Brownian motion based at will hit the boundary of in a finite time . The location where this motion first hits the boundary is then a random variable in ; the law of this variable is called the *harmonic measure* of with base point , and we will denote it by ; it is a probability measure on . The reason for the terminology “harmonic measure” comes from the following:

Theorem 7Let be a bounded open subset of , and let be a harmonic (or holomorphic) function that extends continuously to . Then for any , one has the representation formula

*Proof:* (Informal) For simplicity let us assume that extends smoothly to some open neighbourhood of . Let be the motion that is equal to up to time , and then is constant at for all later times. A variant of the Taylor expansion argument used to prove Lévy’s theorem shows that

for any , which on iterating and sending to zero implies that is independent of time. Since this quantity converges to as and to as , the claim follows.

This theorem can also extend to unbounded domains provided that does not grow too fast at infinity (for instance if is bounded, basically thanks to the neighbourhood recurrent properties of complex Brownian motion); we do not give a precise statement here. Among other things, this theorem gives an immediate proof of the maximum principle for harmonic functions, since if on the boundary then from the triangle inequality one has for all . It also gives an alternate route to Liouville’s theorem: if is entire and bounded, then applying the maximum principle to the complement of a small disk we see that for all distinct .

When the boundary is sufficiently nice (e.g. analytic), the harmonic measure becomes absolutely continuous with respect to one-dimensional Lebesgue measure; however, we will not pay too much attention to these sorts of regularity issues in this set of notes.

From Levy’s theorem on the conformal invariance of Brownian motion we deduce the conformal invariance of harmonic measure, thus for any conformal map that extends continuously to the boundaries and any , the harmonic measure of with base point is the pushforward of the harmonic measure of with base point , thus

for any continuous compactly supported test function , and also

for any (Borel) measurable .

- (i) If and , show that the measure on the unit circle is given by
where is arclength measure. In particular, when , then is the uniform measure on the unit circle.

- (ii) If and , show that the measure on the real line is given by
(For this exercise one can assume that harmonic measure is well defined for unbounded domains, and that the representation formula (4) continues to hold for bounded harmonic or holomorphic functions.)

Exercise 9 (Brownian motion description of conformal mapping)Let be the region enclosed by a Jordan curve , and let be three distinct points on in anticlockwise order. Let be three distinct points on the boundary of the unit disk , again traversed in anticlockwise order. Let be the conformal map that takes to for (the existence and uniqueness of this map follows from the Riemann mapping theorem). Let , and for , let be the probability that the terminal point of Brownian motion at with base point lies in the arc between and (here we use the fact that the endpoints are hit with probability zero, or in other words that the harmonic measure is continuous; see Exercise 15 below). Thus are non-negative and sum to . Let be the complex numbers , , . Show the crossratio identityIn principle, this allows one to describe conformal maps purely in terms of Brownian motion.

We remark that the link between Brownian motion and conformal mapping can help gain an intuitive understanding of the Carathéodory kernel theorem (Theorem 12 from Notes 3). Consider for instance the example in Exercise 13 from those notes. It is intuitively clear that a Brownian motion based at the origin will very rarely pass through the slit beween and , instead hitting the right side of the boundary of first. As such, the harmonic measure of the left side of the bounadry should be very small, and in fact one can use this to show that the preimage under of the region to the left of the boundary goes to zero in diameter as , which helps explain why the limiting function does not map to this region at all.

Exercise 10 (Brownian motion description of conformal radius)

- (i) Let and with . Show that the probability that the Brownian motion hits the circle before it hits is equal to . (
Hint:is harmonic away from the origin.)- (ii) Let be a simply connected proper subset of , let be a point in , and let be the conformal radius of around . Show that for small , the probability that a Brownian motion based at a point with will hit the circle before it hits the boundary is equal to , where denotes a quantity that goes to zero as .

Exercise 11Let be a connected subset of , let be a Brownian motion based at the origin, and let be the first time this motion exits . Show that the probability that hits is at least for some absolute constant . (Hint:one can control the event that makes a “loop” around a point in at radius less than , which is enough to force intersection with , at least if one works some distance away from the boundary of the disk.)

We now sketch the proof of a basic Brownian motion estimate that is useful in applications. We begin with a lemma that says, roughly speaking, that “folding” a set reduces the probability of it being hit by Brownian motion.

Lemma 12Let , and let be a closed subset of the unit disk . Write and , and write (i.e. reflected onto the upper half-plane). Let be a complex Brownian motion based at , and let be the first time this motion hits the boundary of . Then

*Proof:* (Informal) To illustrate the argument at a heuristic level, let us make the (almost surely false) assumption that the Brownian motion only crosses the real axis at a finite set of times before hitting the disk. Then the Brownian motion would split into subcurves for , with the convention that . Each subcurve would lie in either the upper half-plane or the lower half-plane, with equal probability of each; furthermore, one could arbitrarily apply complex conjugation to one or more of these subcurves and still obtain a motion with the same law. Observe that if one conditions on the Brownian motion up to time , and the subcurve has a probability of hitting when it lies in the upper half-plane, and a probability of hitting when it lies in the lower half-plane, then it will have a probability of at most of hitting when it lies in the upper half-plane, and probability of hitting when it lies in the lower half-plane; thus the probability of this subcurve hitting is less than or equal to that of it hitting . In principle, the lemma now follows from repeatedly applying the law of total expectation.

This naive argument does not quite work because a Brownian motion starting at a real number will in fact almost surely cross the real axis an infinite number of times. However it is possible to adapt this argument by redefining the so that after each time , the Brownian motion is forced to move some small distance before one starts looking for the next time it hits the real axis. See the proof of Lemma 6.1 of these notes of Lawler for a complete proof along these lines.

This gives an inequality similar in spirit to the Grötzsch modulus estimate from Notes 2:

Corollary 13 (Beurling projection theorem)Let , and let be a compact connected subset the annulus that intersects both boundary circles of the annulus. Let be a complex Brownian motion based at , and let be the first time this motion hits the outer boundary of the annulus. Then the probability that intersects is greater than or equal to the probability that intersects the interval .

*Proof:* (Sketch) One can use the above lemma to fold around the real axis without increasing the probability of being hit by Brownian motion. By rotation, one can similarly fold around any other line through the origin. By repeatedly folding in this fashion to reduce its angular variation, one can eventually replace with a set that lies inside the sector for any . However, by the monotone convergence theorem, the probability that intersects this sector converges to the probability that it intersects in the limit , and the claim follows.

Exercise 14With the notation as the above corollary, show that the probability that intersects the interval is . (Hint:apply a square root conformal map to the disk with removed, and then compare with the half-plane harmonic measure from Exercise 8(ii).)

The following consequence of the above estimate, giving a sort of Hölder regularity of Brownian measure, is particularly useful in applications.

Exercise 15 (Beurling estimate)Let be an open set not containing , with the property that the connected component of containing intersects the unit circle . Let be such that . Then for any , one has ; that is to say, the probability that a Brownian motion based at exits at a point within from the origin is . (Hint:one can use conformal mapping to show that the probability appearing at the end of Corollary 13 is .) Conclude in particular that harmonic measures are always continuous (they assign zero to any point).

Exercise 16Let be a region bounded by a Jordan curve, let , let be the Brownian motion based at , and let be the first time this motion exits . Then for any , show that the probability that the curve has diameter at least is at most .

Exercise 17Let be a conformal map with , and let be a curve with and for . Show that(Hint: use Exercise 11.)

** — 2. Half-plane capacity — **

One can use Brownian motion to construct other close relatives of harmonic measure, such Green’s functions, excursion measures. See for instance these lecture notes of Lawler for more details. We will focus on one such use of Brownian motion, to interpret the concept of *half-plane capacity*; this is a notion that is particularly well adapted to the study of chordal Loewner equations (it plays a role analogous to that of conformal radius for the radial Loewner equation).

Let be the upper half-plane. A subset of the upper half-plane is said to be a *compact hull* if it is bounded, closed in , and the complement is simply connected. By the Riemann mapping theorem, for any compact hull , there is a unique conformal map which is normalised at infinity in the sense that

for some complex numbers . The quantity is particularly important and will be called the *half-plane capacity* of and denoted .

In general, we have the following Brownian motion characterisation of half-plane capacity:

Proposition 19Let be a compact hull, with conformal map and half-plane capacity .

- (i) If is complex Brownian motion based at some point , and is the first time this motion exits , then
- (ii) We have

*Proof:* (Sketch) Part (i) follows from applying Theorem 7 to the bounded harmonic function . Part (ii) follows from part (i) by setting for a large , rearranging, and sending using (5).

Among other things, this proposition demonstrates that for all , and that the half-plane capacity is always non-negative (in fact it is not hard to show from the above proposition that it is strictly positive as long as is non-empty).

If are two compact hulls with , then will map conformally to the complement of in . Thus is also a convex hull, and by the uniqueness of Riemann maps we have the identity

which on comparing Laurent expansions leads to the further identity

In particular we have the monotonicity , with equality if and only if . One may verify that these claims are consistent with Exercise 18.

Exercise 20 (Submodularity of half-plane capacity)Let be two compact hulls.

- (i) If , show that
(

Hint:use Proposition 19, and consider how the times in which a Brownian motion exits , , , and are related.)- (ii) Show that

Exercise 21Let be a compact hull bounded in a disk . For any , show thatas , where is complex Brownian motion based at and is the first time it exits . Similarly, for any , show that

This formula gives a Brownian motion interpretation for on the portion of the boundary of . It can be used to give useful quantitative estimates for in this region; see Section 3.4 of Lawler’s book.

** — 3. The chordal Loewner equation — **

We now develop (in a rather informal fashion) the theory of the chordal Loewner equation, which roughly speaking is to conformal maps from the upper half-plane to the complement of complex hulls as the radial Loewner equation is to conformal maps from the unit disk to subsets of the complex plane. A more rigorous treatment can be found in Lawler’s book.

Suppose one has a simple curve such that and . There are important and delicate issues regarding the regularity hypotheses on this curve (which become particularly important in SLE, when the regularity is quite limited), but for this informal discussion we will ignore all of these issues.

For each time , the set forms a compact hull, and so has some half-plane capacity . From the monotonicity of capacity, this half-plane capacity is increasing in . It is traditional to normalise the curve so that

this is analogous to normalising the Loewner chains from Notes 3 to have conformal radius at time . A basic example of such normalised curves would be the curves for some fixed , since the normalisation follows from (6).

Let be the conformal maps associated to these compact hulls. From (8) we will have

for any and , where is the conformal map associated to the compact hull . From (9) this hull has half-plane capacity , thus we have the Laurent expansion

It can be shown (using the Beurling estimate) that extends continuously to the tip of the curve , and attains a real value at that point; furthermore, depends continuously on . See Lemma 4.2 of Lawler’s book. As such, should be a short arc (of length ) starting at . If , it is possible to use a quantitative version of Exercise 21 (again using the Beurling estimate) to obtain an estimate basically of the form

for any fixed . If is non-zero, we instead have

For instance, if , then for all , and from Exercise 18 we have the exact formula

Inserting (12) into (11) and using the chain rule, we obtain

and we then arrive at the *(chordal) Loewner equation*

for all and . This equation can be justified rigorously for any simple curve : see Proposition 4.4 of Lawler’s book. Note that the imaginary part of is negative, which is consistent with the observation made previously that the imaginary part of is decreasing in .

We have started with a chain of compact hulls associated to a simple curve, and shown that the resulting conformal maps obey the Loewner equation for some continuous driving term . Conversely, suppose one is given a continuous driving term . It follows from Picard existence and uniqueness theorem that for each there is a unique maximal time of existence such that the ODE (13) with initial data can be solved for time , one can show that for each time , is a conformal map from to with the Laurent expansion

hence the complement are an increasing sequence of compact hulls with half-plane capacity . Proving complex differentiability of can be done from first principles, and the Laurent expansion near infinity is also not hard; the main difficulty is to show that the map is surjective, which requires solving (13) backwards in time (and here one can do this indefinitely as now one is moving away from the real axis instead of towards it). See Theorem 4.6 of Lawler’s book for details (in fact a more general theorem is proven, in which the single point is replaced by a probability measure, analogously to how the radial Loewner equation uses Herglotz functions instead of a single driving function when not restricted to slit domains). However, there is a subtlety, in that the hulls are not necessarily the image of simple curves . This is often the case for short times if the driving function does not oscillate too wildly, but it can happen that the curve that one would expect to trace out eventually intersects itself, in which case the region it then encloses must be absorbed into the hull (cf. the “pinching off” phenomenon in the Carathéodory kernel theorem). Nevertheless, it is still possible to have Loewner chains that are “generated” by non-simple paths , in the sense that consists of the unbounded connected component of the complement .

There are some symmetries of the transform from the to the . If one translates by a constant, , then the resulting domains are also translated, , and . Slightly less trivially, for any , if one performs a rescaled dilation , then one can check using (13) that , and the corresponding conformal maps are given by . On the other hand, just performing a scalar multiple on the driving force can transform the behavior of dramatically; the transform from to is very definitely not linear!

** — 4. Schramm-Loewner evolution — **

In the previous section, we have indicated that every continuous driving function gives rise to a family of conformal maps obeying the Loewner equation (13). The (chordal) Schramm-Loewner evolution () with parameter is the special case in which the driving function takes the form for some real Brownian motion based at the origin. Thus is now a random conformal map from a random domain , defined by solving the Schramm-Loewner equation

with initial condition for , and with defined as the set of all for which the above ODE can be solved up to time taking values in . The parameter cannot be scaled away by simple renormalisations such as scaling, and in fact the behaviour of is rather sensitive to the value of , with special behaviour or significance at various values such as playing particularly special roles; there is also a duality relationship between and which we will not discuss here.

The case is rather boring, in which is deterministic, and is just with the line segment between and removed. The cases are substantially more interesting. It is a non-trivial theorem (particularly at the special value ) that is almost surely generated by some random path ; see Theorem 6.3 of Lawler’s book. The nature of this path is sensitive to the choice of parameter :

- For , the path is almost surely simple and goes to infinity as ; it also avoids the real line (except at time ).
- For ; it also has non-trivial intersection with the real line.
- For , the path is almost surely space-filling (which of course also implies that ), and also hits every point on .

See Section 6.2 of Lawler’s book. The path becomes increasingly fractal as increases: it is a result of Rohde and Schramm and Beffara that the image almost surely has Hausdorff dimension .

We have asserted that defines a random path in that starts at the origin and generally “wanders off” to infinity (though for it keeps recurring back to bounded sets infinitely often). By the Riemann mapping theorem, we can now extend this to other domains. Let be a simply connected open proper subset of whose boundary we will assume for simplicity to be a Jordan curve (this hypothesis can be relaxed). Let be two distinct points on the boundary . By the Riemann mapping theorem and Carathéodory’s theorem (Theorem 20 from Notes 2), there is a conformal map whose continuous extension maps and to and respectively; this map is unique up to rescalings for . One can then define the Schramm-Loewner evolution on from to to be the family of conformal maps for , where is the usual Schramm-Loewner evolution with parameter . The Schramm-Loewner evolution on is well defined up to a time reparameterisation . The Markovian and stationary nature of Brownian motion translates to an analogous Markovian and conformally invariant property of . Roughly speaking, it is the following: if is any reasonable domain with two boundary points , is on this domain from to with associated path , and is any time, then after conditioning on the path up to time , the remainder of the path has the same image as the path on the domain from to . Conversely, under suitable regularity hypotheses, the processes are the *only* random path processes on domains with this property (much as Brownian motion is the only Markovian stationary process, once one normalises the mean and variance). As a consequence, whenever one now a random path process that is known or suspected to enjoy some conformal invariance properties, it has become natural to conjecture that it obeys the law of (though in some cases it is more natural to work with other flavours of SLE than the chordal SLE discussed here, such as radial SLE or whole-plane SLE). For instance, in the pioneering work of Schramm, this line of reasoning was used to conjecture that the loop-erased random walk in a domain has the law of (radial) ; this conjecture was then established by Lawler, Schramm, and Werner. Many further processes have since been either proven or conjectured to be linked to one of the SLE processes, such as the limiting law of a uniform spanning tree (proven to be ), interfaces of the Ising model (proven to be ), or the scaling limit of self-avoiding random walks (conjectured to be ). Further discussion of these topics is beyond the scope of this course, and we refer the interested reader to Lawler’s book for more details.

We have now tentatively improved the upper bound of the de Bruijn-Newman constant to . Among the technical improvements in our approach, we now are able to use Taylor expansions to efficiently compute the approximation to for many values of in a given region, thus speeding up the computations in the barrier considerably. Also, by using the heuristic that behaves somewhat like the partial Euler product , we were able to find a good location to place the barrier in which 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 bounded away from zero for larger values of beyond the barrier. In going below 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 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.

]]>Clearly, a univalent function on the unit disk is a conformal map from to the image ; in particular, is simply connected, and not all of (since otherwise the inverse map would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset of the complex numbers is the image of a univalent function on . Furthermore, if contains the origin, then the univalent function with this image becomes unique once we normalise and . 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 with and . We will focus particular attention on the univalent functions with the normalisation and ; such functions will be called schlicht functions.

One basic example of a univalent function on is the Cayley transform , which is a Möbius transformation from to the right half-plane . (The slight variant is also referred to as the Cayley transform, as is the closely related map , which maps to the upper half-plane.) One can square this map to obtain a further univalent function , which now maps to the complex numbers with the negative real axis removed. One can normalise this function to be schlicht to obtain the Koebe function

which now maps to the complex numbers with the half-line removed. A little more generally, for any we have the *rotated Koebe function*

that is a schlicht function that maps to the complex numbers with the half-line removed.

Every schlicht function has a convergent Taylor expansion

for some complex coefficients with . For instance, the Koebe function has the expansion

and similarly the rotated Koebe function has the expansion

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 should obey the bound for all . 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 that are *odd*, thus for all , and the Taylor expansion now reads

for some complex coefficients with . One can transform a general schlicht function to an odd schlicht function by observing that the function , after removing the singularity at zero, is a non-zero function that equals at the origin, and thus (as is simply connected) has a unique holomorphic square root that also equals at the origin. If one then sets

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

Conversely, given an odd schlicht function , the formula (4) uniquely determines a schlicht function .

For instance, if is the Koebe function (1), becomes

which maps to the complex numbers with two slits removed, and if is the rotated Koebe function (2), becomes

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 be a natural number.

- (i) (Robertson conjecture) If is an odd schlicht function, then
- (ii) (Bieberbach conjecture) If is a schlicht function, then

It is easy to see that the Robertson conjecture for a given value of implies the Bieberbach conjecture for the same value of . Indeed, if is schlicht, and is the odd schlicht function given by (3), then from extracting the coefficient of (4) we obtain a formula

for the coefficients of in terms of the coefficients of . Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of from the Robertson conjecture for the same value of . We remark that Littlewood and Paley had conjectured a stronger form of Robertson’s conjecture, but this was disproved for 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 , known as the *Milin conjecture*. Next, one continuously enlarges the image of the schlicht function to cover all of ; done properly, this places the schlicht function as the initial function in a sequence of univalent maps known as a Loewner chain. The functions obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term (or , 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 , , or . 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 ) 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.

** — 1. The area theorem and its consequences — **

We begin with the area theorem of Grönwall.

Theorem 2 (Grönwall area theorem)Let be a univalent function with a convergent Laurent expansionThen

*Proof:* By shifting we may normalise . By hypothesis we have for any ; by replacing with and using a limiting argument, we may assume without loss of generality that the have some exponential decay as (in order to justify some of the manipulations below).

Let be a large parameter. If , then and . The area enclosed by the simple curve is equal to

crucially, the error term here goes to zero as . Meanwhile, by the change of variables formula (using monotone convergence if desired to work in compact subsets of the annulus initially) and Plancherel's theorem, the area of the region is

Comparing these bounds we conclude that

sending to infinity, we obtain the claim.

Exercise 3Let be a univalent function with Taylor expansionShow that the area of is equal to . (In particular, has finite area if and only if .)

Corollary 4 (Bieberbach inequality)

- (i) If is an odd schlicht function, then .
- (ii) If is a schlicht function, then .

*Proof:* For (i), we apply Theorem 2 to the univalent function defined by , which has a Laurent expansion , to give the claim. For (ii), apply part (i) to the square root of with first term .

Exercise 5Show that equality occurs in Corollary 4(i) if and only if takes the form for some , and in Corollary 4(ii) if and only if takes the form of a rotated Koebe function for some .

The Bieberbach inequality can be rescaled to bound the second coefficient of univalent functions:

Exercise 6 (Rescaled Bieberbach inequality)If is a univalent function, show thatWhen does equality hold?

The Bieberbach inequality gives a useful lower bound for the image of a univalent function, known as the Koebe quarter theorem:

Corollary 7 (Koebe quarter theorem)Let be a univalent function. Then contains the disk .

*Proof:* By applying a translation and rescaling, we may assume without loss of generality that is a schlicht function, with Taylor expansion

Our task is now to show that for every , the equation has a solution in . If this were not the case, then the function is invertible on , with inverse being univalent and having the Taylor expansion

Applying Exercise 6 we then have

while from the Bieberbach inequality one also has . Hence by the triangle inequality , which is incompatible with the hypothesis .

Exercise 8Show that the radius is best possible in Corollary 7 (thus, does not contain any disk with ) if and only if takes the form for some complex numbers and real .

Remark 9The univalence hypothesis is crucial in the Koebe quarter theorem. Consider for instance the functions defined by . These are locally univalent functions (since is holomorphic with non-zero derivative) and , , but avoids the point .

Exercise 10 (Koebe distortion theorem)Let be a schlicht function, and let have magnitude .

- (i) Show that
(

Hint:compose on the right with a Möbius automorphism of that sends to and then apply the rescaled Bieberbach inequality.)- (ii) Show that
(

Hint:use (i) to control the radial derivative of .)- (iii) Show that
- (iv) Show that
(This cannot be directly derived from (ii) and (iii). Instead, compose on the right with a Mobius automorphism that sends to and to , rescale it to be schlicht, and apply (iii) to this function at .)

- (v) Show that the space of schlicht functions is a normal family. In other words, if is any sequence of schlicht functions, then there is a subsequence that converges locally uniformly on compact sets.
- (vi) (Qualitative Bieberbach conjecture) Show that for each natural number there is a constant such that whenever is a schlicht function with Taylor expansion

Exercise 11 (Conformal radius)If is a non-empty simply connected open subset of that is not all of , and is a point in , define theconformal radiusof at to be the quantity , where is any conformal map from to that maps to (the existence and uniqueness of this radius follows from the Riemann mapping theorem). Thus for instance the disk has conformal radius around .

- (i) Show that the conformal radius is strictly monotone in : if are non-empty simply connected open subsets of , and , then the conformal radius of around is strictly greater than that of .
- (ii) Show that the conformal radius of a disk around an element of the disk is given by the formula .
- (iii) Show that the conformal radius of around lies between and , where is the radius of the maximal disk that is contained in .
- (iv) If the conformal radius of around is equal to , show that for all sufficiently small , the ring domain has modulus , where denotes a quantity that goes to zero as , and the modulus of a ring domain was defined in Notes 2.

We can use the distortion theorem to obtain a nice criterion for when univalent maps converge to a given limit, known as the Carathéodory kernel theorem.

Theorem 12 (Carathéodory kernel theorem)Let be a sequence of simply connected open proper subsets of containing the origin, and let be a further simply connected open proper subset of containing . Let and be the conformal maps with and (the existence and uniqueness of these maps are given by the Riemann mapping theorem). Then the following are equivalent:

- (i) converges locally uniformly on compact sets to .
- (ii) For every subsequence of the , is the set of all such that there is an open connected set containing and that is contained in for all sufficiently large .

If conclusion (ii) holds, is known as the *kernel* of the domains .

*Proof:* Suppose first that converges locally uniformly on compact sets to . If , then for some . If , then the holomorphic functions converge uniformly on to the function , which is not identically zero but has a zero in . By Hurwitz’s theorem we conclude that also has a zero in for all sufficiently large ; indeed the same argument shows that one can replace by any element of a small neighbourhood of to obtain the same conclusion, uniformly in . From compactness we conclude that for sufficiently large , has a zero in for all , thus for sufficiently large . Since is open connected and contains and , we see that is contained in the set described in (ii).

Conversely, suppose that is a subsequence of the and is such that there is an open connected set containing and that is contained in for sufficiently large . The inverse maps are holomorphic and bounded, hence form a normal family by Montel’s theorem. By refining the subsequence we may thus assume that the converge locally uniformly to a holomorphic limit . The function takes values in , but by the open mapping theorem it must in fact map to . In particular, . Since converges to , and converges locally uniformly to , we conclude that converges to , thus and hence . This establishes the derivation of (ii) from (i).

Now suppose that (ii) holds. It suffices to show that every subsequence of has a further subsequence that converges locally uniformly on compact sets to (this is an instance of the *Urysohn subsequence principle*). Then (as contains ) in particular there is a disk that is contained in the for all sufficiently large ; on the other hand, as is not all of , there is also a disk which is *not* contained in the for all sufficiently large . By Exercise 11, this implies that the conformal radii of the around zero is bounded above and below, thus is bounded above and below.

By Exercise 10(v), and rescaling, the functions then form a normal family, thus there is a subsequence of the that converges locally uniformly on compact sets to some limit . Since is positive and bounded away from zero, is also positive, so is non-constant. By Hurwitz’s theorem, is therefore also univalent, and thus maps to some region . By the implication of (ii) from (i) (with replaced by ) we conclude that is the set of all such that there is an open connected set containing and that is contained in for all sufficiently large ; but by hypothesis, this set is also . Thus , and then by the uniqueness part of the Riemann mapping theorem, as desired.

The condition in Theorem 12(ii) indicates that “converges” to in a rather complicated sense, in which large parts of are allowed to be “pinched off” from and disappear in the limit. This is illustrated in the following explicit example:

Exercise 13 (Explicit example of kernel convergence)Let be the function from (5), thus is a univalent function from to with the two vertical rays from to , and from to , removed. For any natural number , let and let , and define the transformed functions .

- (i) Show that is a univalent function from to with the two vertical rays from to , and from to , removed, and that and .
- (ii) Show that converges locally uniformly to the function , and that this latter map is a univalent map from to the half-plane . (
Hint:one does not need to compute everything exactly; for instance, any terms of the form can be written using the notation instead of expanded explicitly.)- (iii) Explain why these facts are consistent with the Carathéodory kernel theorem.

As another illustration of the theorem, let be two distinct convex open proper subsets of containing the origin, and let be the associated conformal maps from to respectively with and . Then the alternating sequence does not converge locally uniformly to any limit. The set is the set of all points that lie in a connected open set containing the origin that eventually is contained in the sequence ; but if one passes to the subsequence , this set of points enlarges to , and so the sequence does not in fact have a kernel.

However, the kernel theorem simplifies significantly when the are monotone increasing, which is already an important special case:

Corollary 14 (Monotone increasing case of kernel theorem)Let the notation and assumptions be as in Theorem 12. Assume furthermore thatand that . Then converges locally uniformly on compact sets to .

Loewner observed that the kernel theorem can be used to approximate univalent functions by functions mapping into slit domains. More precisely, define a *slit domain* to be an open simply connected subset of formed by deleting a half-infinite Jordan curve connecting some finite point to infinity; for instance, the image of the Koebe function is a slit domain.

Theorem 15 (Loewner approximation theorem)Let be a univalent function. Then there exists a sequence of univalent functions whose images are slit domains, and which converge locally uniformly on compact subsets to .

*Proof:* First suppose that extends to a univalent function on a slightly larger disk for some . Then the image of the unit circle is a Jordan curve enclosing the region in the interior. Applying the Jordan curve theorem (and the Möbius inversion ), one can find a half-infinite Jordan curve from to infinity that stays outside of . For any , one can concatenate this curve with the arc to obtain another half-infinite Jordan curve , whose complement is a slit domain which has as kernel (why?). If we let be the conformal maps from to with and , we conclude from the Carathéodory kernel theorem that converges locally uniformly on compact sets to .

If is just univalent on , then it is the locally uniform limit of the dilations , which are univalent on the slightly larger disks . By the previous arguments, each is in turn the locally uniform limit of univalent functions whose images are slit domains, and the claim now follows from a diagonalisation argument.

** — 2. Loewner chains — **

The material in this section is based on these lecture notes of Contreras.

An important tool in analysing univalent functions is to study one-parameter families of univalent functions, parameterised by a time parameter , in which the images are increasing in ; roughly speaking, these families allow one to study an arbitrary univalent function by “integrating” along such a family from back to . Traditionally, we normalise these families into (radial) Loewner chains, which we now define:

Definition 16 (Loewner chain)A (radial) Loewner chain is a family of univalent maps with and (so in particular is schlicht), such that for all . (In these notes we use the prime notation exclusively for differentiation in the variable; we will use later for differentiation in the variable.)

A key example of a Loewner chain is the family

of dilated Koebe functions; note that the image of each is the slit domain , which is clearly monotone increasing in . More generally, we have the rotated Koebe chains

Whenever one has a family of simply connected proper open subsets of containing with for , and . By definition, is then the conformal radius of around , which is a strictly increasing function of by Exercise 11. If this conformal radius is equal to at and increases continuously to infinity as , then one can reparameterise the variable so that , at which point one obtains a Loewner chain.

From the Koebe quarter theorem we see that each image in a Loewner chain contains the disk . In particular the increase to fill out all of : .

Let be a Loewner chain, Let . The relation is sometimes expressed as the assertion that is *subordinate* to . It has the consequence that one has a composition law of the form

for a univalent function , uniquely defined as , noting taht is well-defined on . By construction, we have and

as well as the composition laws

for . We will refer to the as *transition functions*.

From the Schwarz lemma, we have

for , with strict inequality when . In particular, if we introduce the function

for and , then (after removing the singularity at infinity and using (10)) we see that is a holomorphic map to the right half-plane , normalised so that

Define a Herglotz function to be a holomorphic function , thus is a Herglotz function for all . A key family of examples of a Herglotz function are the Möbius transforms for . In fact, all other Herglotz functions are basically just averages of this one:

Exercise 17 (Herglotz representation theorem)Let be a Herglotz function, normalised so that .

- (i) For any , show that
for . (

Hint:The real part of is harmonic, and so has a Poisson kernel representation. Alternatively, one can use a Taylor expansion of .)- (ii) Show that there exists a (Radon) probability measure on such that
for all . (One will need a measure-theoretic tool such as Prokhorov’s theorem, the Riesz representation theorem, or the Helly selection principle.) Conversely, show that every probability measure on generates a Herglotz function with by the above formula.

- (iii) Show that the measure constructed on (ii) is unique.

This has a useful corollary, namely a version of the Harnack inequality:

Exercise 18 (Harnack inequality)Let be a Herglotz function, normalised so that . Show thatfor all .

This gives some useful Lipschitz regularity properties of the transition functions and univalent functions in the variable:

Lemma 19 (Lipschitz regularity)Let be a compact subset of , and let . Use to denote a quantity bounded in magnitude by , where depends only on .

- (i) For any and , one has
- (ii) For any and , one has

One can make the bounds much more explicit if desired (see e.g. Lemma 2.3 of these notes of Contreras), but for our purposes any Lipschitz bound will suffice.

*Proof:* To prove (i), it suffices from (11) and the Schwarz-Pick lemma (Exercise 13 from Notes 2) to establish this claim when . We can also assume that since the claim is trivial when . From the Harnack inequality one has

for , which by (12) and some computation gives

Now we prove (ii). We may assume without loss of generality that is convex. From Exercise 10 (normalising to be schlicht) we see that for , and hence has a Lipschitz constant of on . Since , the claim now follows from (13).

As a first application of this we show that every schlicht function starts a Loewner chain.

Lemma 20Let be schlicht. Then there exists a Loewner chain with .

*Proof:* This will be similar to the proof of Theorem 15. First suppose that extends to be univalent on for some , then is a Jordan curve. Then by Carathéodory’s theorem (Theorem 20 of Notes 2) (and the Möbius inversion ) one can find a conformal map from the exterior of to the exterior of that sends infinity to infinity. If we define for to be the region enclosed by the Jordan curve , then the are increasing in with conformal radius going to infinity as . If one sets to be the conformal maps with and , then (by the uniqueness of Riemann mapping) and by the Carathéodory kernel theorem, converges locally uniformly to as . In particular, the conformal radii are continuous in . Reparameterising in one can then obtain the required Loewner chain.

Now suppose is only univalent of . As in the proof of Theorem 15, one can express as the locally uniform limit of schlicht functions , each of which extends univalently to some larger disk . By the preceding discussion, each of the extends to a Loewner chain . From the Lipschitz bounds (and the Koebe distortion theorem) one sees that these chains are locally uniformly equicontinuous in and , uniformly in , and hence by Arzela-Ascoli we can pass to a subsequence that converges locally uniformly in to a limit ; one can also assume that the transition functions converge locally uniformly to limits . It is then not difficult by Hurwitz theorem to verify the limiting relations (9), (11), and that is a Loewner chain with as desired.

Suppose that are close to each other: . Then one heuristically has the approximations

and hence by (12) and some rearranging

and hence on applying , (9), and the Newton approximation

This suggests that the should obey the *Loewner equation*

for some Herglotz function . This is essentially the case:

Theorem 21 (Loewner equation)Let be a Loewner chain. Then, for outside of an exceptional set of Lebesgue measure zero, the functions are differentiable in time for each , and obey the equation (14) for all and , and some Herglotz function for each with . Furthermore, the maps are measurable for every .

*Proof:* Let be a countable dense subset of . From Lemma 19, the function is Lipschitz continuous, and thus differentiable almost everywhere, for each . Thus there exists a Lebesgue measure zero set such that is differentiable in outside of for each . From the Koebe distortion theorem is also locally Lipschitz (hence locally uniformly equicontinuous) in the variable, so in fact is differentiable in outside of for all . Without loss of generality we may assume contains zero.

Let , and let . Then as approaches from below, we have

uniformly; from (9) and Newton approximation we thus have

which implies that

Also we have

and hence by (12)

Taking limits, we see that the function is Herglotz with , giving the claim. It is also easy to verify the measurability (because derivatives of Lipschitz functions are measurable)

Example 22The Loewner chain (7) solves the Loewner equation with the Herglotz function . With the rotated Koebe chains (8), we instead have .

Although we will not need it in this set of notes, there is also a converse implication that for every family of Herglotz functions depending measurably on , one can associate a Loewner chain.

Let us now Taylor expand a Loewner chain at each time as

as , we have . As is differentiable in almost every for each , and is locally uniformly continuous in , we see from the Cauchy integral formulae that the are also differentiable almost everywhere in . If we similarly write

for all outside of , then , and we obtain the equations

and so forth. For instance, for the Loewner chain (7) one can verify that and for solve these equations. For (8) one instead has and .

We have the following bounds on the first few coefficients of :

Exercise 23Let be a Herglotz function with . Let be the measure coming from the Herglotz representation theorem.

- (i) Show that for all . In particular, for all . Use this to give an alternate proof of the upper bound in the Harnack inequality.
- (ii) Show that .

We can use this to establish the first two cases of the Bieberbach conjecture:

Theorem 24 ( cases of Bieberbach)If is schlicht, then and .

The bound is not new, and indeed was implicitly used many times in the above arguments, but we include it to illustrate the use of the equations (15), (16).

*Proof:* By Lemma 20, we can write (and ) for some Loewner chain .

We can write (15) as . On the other hand, from the Koebe distortion theorem applied to the schlicht functions , we have , so in particular goes to zero at infinity. We can integrate from to infinity to obtain

From Harnack’s inequality we have , giving the required bound .

In a similar vein, writing (16) as

we obtain

As , we may integrate from to infinity to obtain the identity

Taking real parts using Exercise 23(ii) and (17), we have

Since , we thus have

where . By Cauchy-Schwarz, we have , and from the bound , we thus have

Replacing by the schlicht function (which rotates by ) and optimising in , we obtain the claim .

Exercise 25Show that equality in the above bound is only attained when is a rotated Koebe function.

The Loewner equation (14) takes a special form in the case of slit domains. Indeed, let be a slit domain not containing the origin, with conformal radius around , and let be the Loewner chain with . We can parameterise so that the sets have conformal radius around for every , in which case we see that must be the unique conformal map from to with and . For instance, for the chain (7) we would have .

Theorem 26 (Loewner equation for slit domains)In the above situation, we have the Loewner equation holding with

*Proof:* Let be a time where the Loewner equation holds. For , the function extends continuously to the boundary, and is two-to-one on the split , except at the tip where there is a single preimage on the unit circle; this can be seen by taking a holomorphic square root of , using a Möbius transformation to map the resulting image to a set bounded by a Jordan curve, and applying Carathéodory's theorem (Theorem 20 from Notes 2) to the resulting conformal map. The image is then with a Jordan arc removed, where is a point on the boundary of the sphere. Applying Carathéodory’s theorem to a holomorphic square root of , we see that extends continuously to be a map from to , with an arc on the boundary mapping (in two-to-one fashion) to the arc , and the endpoints of this arc mapping to . From this and (12), we see that converges to zero outside of the arc , which by the Herglotz representation theorem implies that the measure associated to is supported on the arc . An inspection of the proof of Carathéodory’s theorem also reveals that the are equicontinuous on as , and thus converge uniformly to (which is the identity function) as . This implies that must converge to the point as approaches , and so converges vaguely to the Dirac mass at . Since converges locally uniformly to , we conclude the formula (18). As depends measurably in , we conclude that does also.

In fact one can show that extends to a continuous function , and that the Loewner equation holds for all , but this is a bit trickier to show (it requires some further distortion estimates on conformal maps, related to the arguments used to prove Carathéodory’s theorem in the previous notes) and will not be done here. One can think of the function as “driving force” that incrementally enlarges the slit via the Loewner equation; this perspective is often used when studying the Schramm-Loewner evolution, which is the topic of the next (and final) set of notes.

** — 3. The Bieberbach conjecture — **

We now turn to the resolution of the Bieberbach (and Robertson) conjectures. We follow the simplified treatment of de Branges’ original proof, due to FitzGerald and Pommerenke, though we omit the proof of one key ingredient, namely the non-negativity of a certain hypergeometric function.

The first step is to work not with the Taylor coefficients of a schlicht function or with an odd schlicht function , but rather with the (normalised) logarithm of a schlicht function , as the coefficients end up obeying more tractable equations. To transfer to this setting we need the following elementary inequalities relating the coefficients of a power series with the coefficients of its exponential.

Lemma 27 (Second Lebedev-Milin inequality)Let be a formal power series with complex coefficients and no constant term, and let be its formal exponential, thus

*Proof:* If we formally differentiate (19) in , we obtain the identity

extracting the coefficient for any , we obtain the formula

By Cauchy-Schwarz, we thus have

Using and telescoping series, it thus suffices to prove the identity

But this follows from observing that

and that

for all .

Exercise 28Show that equality holds in (20) for a given if and only if there is such that for all .

Exercise 29 (First Lebedev-Milin inequality)With the notation as in the above lemma, and under the additional assumption , prove that(

Hint:using the Cauchy-Schwarz inequality as above, first show that the power series is bounded term-by-term by the power series of .) When does equality occur?

Exercise 30 (Third Lebedev-Milin inequality)With the notation as in the above lemma, show that(

Hint:use the second Lebedev-Milin inequality and (21), together with the calculus inequality for all .) When does equality occur?

Using these inequalities, one can reduce the Robertson and Bieberbach conjectures to the following conjecture of Milin, also proven by de Branges:

Theorem 31 (Milin conjecture)Let be a schlicht function. Let be the branch of the logarithm of that equals at the origin, thus one hasfor some complex coefficients . Then one has

for all .

Indeed, if

is an odd schlicht function, let be the schlicht function given by (4), then

Applying Lemma 27 with , we obtain the Robertson conjecture, and the Bieberbach conjecture follows.

Example 32If is the Koebe function (1), thenso in this case and . Similarly, for the rotated Koebe function (2) one has and again . If one works instead with the dilated Koebe function , we have , thus the time parameter only affects the constant term in . This is already a hint that the coefficients of could be worth studying further in this problem.

To prove the Milin conjecture, we use the Loewner chain method. It suffices by Theorem 15 and a limiting argument to do so in the case that is a slit domain. Then, by Theorem 26, is the initial function of a Loewner chain that solves the Loewner equation

for all and almost every , and some function .

We can transform this into an equation for . Indeed, for non-zero we may divide by to obtain

(for any local branch of the logarithm) and hence

Since , is equal to at the origin (for an appropriate branch of the logarithm). Thus we can write

The are locally Lipschitz in (basically thanks to Lemma 19) and for almost every we have the Taylor expansions

and

Comparing coefficients, we arrive at the system of ordinary differential equations

Fix (we will not need to use any induction on here). We would like to use the system (22) to show that

The most naive attempt to do this would be to show that one has a monotonicity formula

for all , and that the expression goes to zero as , as the claim would then follow from the fundamental theorem of calculus. This turns out to not quite work; however it turns out that a slight modification of this idea does work. Namely, we introduce the quantities

where for each , is a continuously differentiable function to be chosen later. If we have the initial condition

for all , then the Milin conjecture is equivalent to asking that . On the other hand, if we impose a boundary condition

for , then we also have as , since is schlicht and hence is a normal family, implying that the are bounded in for each . Thus, to solve the Milin, Robertson, and Bieberbach conjectures, it suffices to find a choice of weights obeying the initial and boundary conditions (23), (24), and such that

for almost every (note that will be Lipschitz, so the fundamental theorem of calculus applies).

Let us now try to establish (25) using (22). We first write , and drop the explicit dependence on , thus

for . To simplify this equation, we make a further transformation, introducing the functions

(with the convention ); then we can write the above equation as

We can recover the from the by the formula

It may be worth recalling at this point that in the example of the rotated Koebe Loewner chain (2) one has , , and , for some real constant . Observe that has a simpler form than in this example, suggesting again that the decision to transform the problem to one about the rather than the is on the right track.

We now calculate

Conveniently, the unknown function no longer appears explicitly! Some simple algebra shows that

and hence by summation by parts

with the convention .

In the example of the rotated Koebe function, with , the factors and both vanish, which is consistent with the fact that vanishes in this case regardless of the choice of weights . So these two factors look to be related to each other. On the other hand, for more general choices of , these two expressions do not have any definite sign. For comparison, the quantity also vanishes when , and has a definite sign. So it is natural to see of these three factors are related to each other. After a little bit of experimentation, one eventually discovers the following elementary identity giving such a connection:

Inserting this identity into the above equation, we obtain

which can be rearranged as

We can kill the first summation by fiat, by imposing the requirement that the obey the system of differential equations

Hence if we also have the non-negativity condition

for all and , we will have obtained the desired monotonicity (25).

To summarise, in order to prove the Milin conjecture for a fixed value of , we need to find functions obeying the initial condition (23), the boundary condition (24), the differential equation (26), and the nonnegativity condition (27), with the convention . This is a significant reduction to the problem, as one just has to write down an explicit formula for such functions and verify all the properties.

Let us work out some simple cases. First consider the case . Now our task is to solve the system

for all . This is easy: we just take (indeed this is the unique choice). This gives the case of the Milin conjecture (which corresponds to the case of Bieberbach).

Next consider the case . The system is now

Again, a routine computation shows that there is a unique solution here, namely and . This gives the case of the Milin conjecture (which corresponds to the case of Bieberbach). One should compare this argument to that in Theorem 24, in particular one should see very similar weight functions emerging.

Let us now move on to . The system is now

A slightly lengthier calculation gives the unique explicit solution

to the above conditions.

These simple cases already indicate that there is basically only one candidate for the weights that will work. A calculation can give the explicit formula:

Exercise 33Let .

- (i) Show there is a unique choice of continuously differentiable functions that solve the differential equations (26) with initial condition (23), with the convention . (Use the Picard existence theorem.)
- (ii) For any , show that the expression
is equal to when is even and when is odd.

- (iii) Show that the functions
for obey the properties (23), (26), (24). (

Hint:for (23), first use (ii) to show that is equal to when is even and when is odd, then use (26).)

The Bieberbach conjecture is then reduced to the claim that

for any and . This inequality can be directly verified for any fixed ; for general it follows from general inequalities on Jacobi polynomials by Askey and Gasper, with an alternate proof given subsequently by Gasper. A further proof of (28), based on a variant of the above argument due to Weinstein that avoids explicit use of (28), appears in this article of Koepf. We will not detail these arguments here.

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