# Ahlfors and Beurling

This is an exposition on Ahlfors and Beurling, two mathematicians I admire. There is nothing original at all. Everything below is collected from Beckman’s excellent book [B], Wikipedia, and papers which I found on the internet using google searches.

The universe consists of Nine Worlds which are resting on an immense tree called Yggdrasil. Asgard is one of the Nine Worlds and is the country of the Norse Gods. Asgard is a land more fertile than any other, blessed also with a great abundance and its people excelled beyond all others in strength, beauty and talent. Odin is a major god and the ruler of Asgard. Odin has many great sons, among them, the brothers Baldur and Tyr.  – Nortic legend

The tree Yggdrasil

Lars Ahlfors and Arne Beurling are among the greatest mathematicians in history. Eventually, they became became collaborators in research, and the best of friends.

Ahlfors

Baldur (or Baldr, Balder) is the god of innocence and piety. So bright and fair he is that light shines from his features. He is wise, eloquent, gentle, and righteous to such a degree that his judgments stand always unshaken. – Norse Legend

Finland was a part of Sweden from the 12th to 19th century and from 1809 to 1917 was part of the Russian Empire, although given some autonomy. The Parliament of Finland passed a Declaration of Independence from Russia in 1917, approved a month or so later by the Russian government. Finland fought World War II as essentially three separate conflicts:

• the Winter War (1939-40), against the Soviet Union which forced Finland to cede 11% of its pre-war territory and 30% of its economic assets to the Soviet Union,
• the Continuation War (1941-44), against the Soviet Union which forced Finland to pay $226,500,000 in reparations to the Soviet Union, while ultimately retaining its independence, and • the Lapland War (1944-45), against Nazi Germany. Lars Ahlfors was born in Helsinki, Finland in 1907 during a time when Finland was under Russian rule. Sadly, his mother died in childbirth. His father was a Professor of Engineering at the Helsinki University of Technology. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna. (Nevanlinna replaced Weyl in Zurich for the session 1928/29 while Weyl was on leave, and Ahlfors went to Zurich with him.) He completed his doctorate from the University of Helsinki in 1930. After he presented his doctoral thesis in 1930, he made a number of visits to Paris and other European centers from 1930-1932. He also taught (in Swedish) at Aaboe Academy in Turku from 1929 to 1932. Ahlfors worked as an Associate Professor at the University of Helsinki from 1933 to 1936. In 1935-1938 Ahlfors visited Harvard University. From 1938-1943 he was a professor at the University of Helsinki. As a Finn, he was required to serve in the military, for which he worked in a bomb shelter. World War II presented many problems for Ahlfors and his wife and children. Due to travel restrictions and threats to his family, his wife and children left for Sweden early in the war, leaving Ahlfors behind. He left when he could and was professor at the Swiss Federal Institute of Technology at Zurich from 1945-1946 (he was only there for 1 year, though he was offered the post earlier). From 1946-1977, Ahlfors was professor at Harvard. He was the William Caspar Graustein Professor of Mathematics of Harvard from 1964-1977. In 1936, the year the first Fields medals were awarded, he was one of the first two people to be given the Medal. He was an invited speaker at the ICM in 1962 and in 1978. He was selected for the Wihuri Sibelius Prize [W] in 1968, the Wolf Prize in Mathematics in 1981, and the AMS Steele Prize in 1982. For a student of mathematics, a common introduction to Ahlfors was by reading his classic textbook on complex analysis which is still very highly regarded. It was Complex Analysis: an Introduction to the Theory of Analytic Functions of One Complex Variable. Ahlfors’ first famous result was his proof of Denjoy’s conjecture, discussed next. If $f:{\mathbb{C}} \to {\mathbb{C}}$ is a function, the function $M(r, f) = \max_{0\leq \theta\leq 2\pi} |f(re^{i\theta} )|$ is called the maximum modulus of $f$. The order of $f$ is $\rho= \limsup_{r\to\infty} \frac{\log\log M(r,f)}{\log r}.$ For example, if $f(z) = e^{z^k}$ then we have $\rho=k$. We say $f$ has an asymptotic value $\alpha$ if there is a curve $\Gamma$ connecting $0$ to $\infty$ such that $f (z) \to \alpha$ as $z \to \infty$ on $\Gamma$. Denjoy (1907) asked the question “if $f$ has $k$ distinct finite asymptotic values, is the order of $f$ greater than or equal to $k/2$?” In his PhD thesis, Ahlfors proved the answer is yes. Ahlfors also worked with Beurling on several papers, which shall be discussed below. Later in his career, Ahlfors worked on Kleinian groups, sometimes in collaboration with Lipman Bers. A Kleinian group $G$ is a discrete subgroup of $PSL(2, {\mathbb{C}}).$ The group $PSL(2, {\mathbb{C}})$ of $2 \times 2$ complex matrices of determinant $1$ modulo its center may be represented as a group of conformal transformations of the Riemann sphere. The boundary of the closed ball is called the sphere at infinity, and is denoted $S_\infty$. Consider $G$ as orientation preserving conformal maps of the open unit ball $S_\infty$ in ${\mathbb{R}}^3$ to itself. The orbit $Gp$ of a point $p$ will typically accumulate on the boundary of the closed ball. The set of accumulation points of $Gp$ in $S_\infty$ is called the limit set of $G$. A Kleinian group is said to be of type $1$ if the limit set is the whole Riemann sphere, and of type $2$ otherwise. One of Ahlfors’ most well-known results is his “finiteness theorem,” which says that the ordinary set $\Omega$ of a (finitely generated) Kleinian group $G$ factored by the action of the group is an orbifold $\Omega/G$ of finite type. In other words, $\Omega/G$ has finitely many “marked” points and can be compactified by adding a finite number of points. Ahlfors has remarked that perhaps of greater interest than the theorems that he has been able to prove were the ones he was not able to prove. For example, there is the assertion that the limit set of a finitely generated Kleinian group has two-dimensional Lebesgue measure zero. This has become known as the Ahlfors measure zero conjecture. It is still unsolved. Beurling Tyr is the god of war and he does nothing whatever for the promotion of concord. Tyr is bold and courageous – men call upon him in battle, and he gives them courage and heroism. The eastern half of Sweden, present-day Finland, was lost to Russia in 1809. The last war in which Sweden was directly involved was in 1814, when Sweden by military means forced Norway into a personal union. Since then, Sweden has been at peace, practicing “non-participation in military alliances during peacetime and neutrality during wartime.” During the German invasion of the Soviet Union, Sweden allowed the Wehrmacht to use Swedish railways to transport (June-July 1941) the German 163rd Infantry Division along with howitzers, tanks and anti-aircraft weapons and associated ammunition, from Norway to Finland. German soldiers traveling on leave between Norway and Germany were allowed passage through Sweden. Iron ore was sold to Germany throughout the war. And for the Allies, Sweden shared military intelligence and helped to train soldiers made up of refugees from Denmark and Norway. It also allowed the Allies to use Swedish airbases between 1944 and 1945. Arne Carl-August Beurling (February 3, 1905 – November 20, 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937-1954) and later at the Institute for Advanced Study in Princeton, New Jersey. He was awarded the Swedish Academy of Sciences Prize 1937 and 1946, the Celsius Gold Medal 1961, and the University of Yeshiva Science Award 1963. In his honor a “Beurling Year” was held at the Mittag-Leffler Institute (Beurling was offered the directorship of the MLI but turned it down…) in Stockholm 1976/77. Family: Father – Konrad Beurling; Mother – Elsa Raab Beurling (who divorced Konrad in 1908 and went back to using Baroness Elsa Raab); Great grandfather (on father’s side) – Pehl Beurling, whose father was a famous clock-maker. According to Beckman [B], Konrad had 14 children in wedlock, and “an unknown number” out of wedlock. (I don’t know if “unknown” means Konrad didn’t know or Beckman didn’t know:-) Arne graduated high school in 1924 and went to the university in Uppsala. He got his undergaduate degree in 1926, masters in 1928. In 1925 the Swedish General Staff contacted a commercial company to design a machine that would be superior to the German Enigma. Hagelin developed a prototype for evaluation called the B21. The B21 was approved for the Swedish General Staff and Hagelin also sold the machine to several other countries. While working on his PhD, Beurling did his military service, excelling in a course in military cryptanalysis. In fact, this is a considerable understatement. Towards the end of the course, the instructor, a Naval officer named Anderberg, brought in a commercial crypto-machine, the Hagelin B21. The graduate student Beurling modestly asked if he could examine the machine over the weekend. The instructor okayed this simple request. The following Monday, Beurling told Anderberg that the machine had a weakness. Anderberg denied this claim and challenged him with a long crib. Beurling promply decrypted it, stunning Anderberg. Some of Beurling’s other cryptographic feats: • Beurling broke the Geheimschreiber without having a copy of the machine (see pp. 43-86 of Beckman [B]). (As the Geheimschreiber encrypted very high-level Nazi traffic, the military significance of this cannot be underestimated.) • Beurling broke several encrypted telegrams in Czech, a language he did not know (see pp. 109-113 in Beckman [B]). In mid-June 1941 Sweden told Great Britian of Germany’s plan to attack the Soviet Union, the attack to commence in late June 1941 (“Operation Barbarossa” – see pp. 120-121 in Beckman [B]). A spy then communicated this information to the Soviet Union (see Ulfving and Weierud [UF], page 22). At the time, the Soviets did not believe the information, thinking the British were only trying to trick them into attacking the Germans. Beurling started teaching in the mathematics department of Uppsala University in 1932. His PhD was defended in 1933, though most of it was written much earlier. (He went on a long hunting trip with his father to Panama. Also, he did his military service at this time.) Beurling’s thesis contained a proof of the Denjoy conjecture, whose proof was found independently and published in his PhD by Ahlfors a few years earlier. Except for visiting positions, he stayed at Uppsala until he left to go to the Institute for Advanced Study in the 1950s. A photo of Beurling taken by Annette-Marie Yxkull. Consider a plane region $D$ and let $E$ be a subset of the boundary of $D$. Consider a harmonic function in $D$, denoted by $\omega (z,E,D)$ and known as the harmonic measure at $z$ of $E$ with respect to $D$. It is defined by having boundary values $1$ on $E$ and $0$ on the rest of the boundary. If it is known that $|\log f(z)|\leq \omega$ on the whole boundary, then, by the maximum principle, $|\log f(z_0)|\leq \omega(z_0,E, D)$, for every interior point $z_0$. A difficult and important result is Beurling’s estimate for the harmonic measure expressed through the inequality $\omega \leq \exp(-\lambda^2+1),$ where $\lambda =\lambda (z_0, E, D)$ is the extremal distance between $z_0$ and the boundary set $E$. If $E$ is conformally equivalent to an arc on the unit circle there is also an opposite inequality $\omega \geq \exp(-\lambda^2).$ These inequalities were strong enough for a proof of the Denjoy conjecture (independent of that of Ahlfors) concerning the number of asymptotic values of an entire function of finite order. This may be Beurling’s most famous theorem: Let $H^2$ be the Hilbert space of holomorphic functions in the unit disk which belong to $L^2$ on $|z|= 1$. Let $E$ be a closed subspace, invariant under multiplication by $z$. Then there exists an inner function $\phi(z)$, i.e. $|\phi(z)|<1$, $|\phi(e^{i\theta})|=1$ a.e., such that $f\in E$ if and only if $f=\phi\cdot f_0$, with$f_0\in H^2\$.

Beurling’s first paper in harmonic analysis is an extension of the prime number theorem to generalized primes. Let $P: 1 be the given sequence of "primes" and let

${\mathbb{Z}}_P = \{ p_1^{a_1}\dots p_k^{a_k}\ |\ a_i\in {\mathbb{Z}}\}$
be the "integers". Let $\pi_P (x)$ and $N_P(x)$ be the corresponding counting functions. Then $|N_P(x)-ax| 0$, implies the prime-number theorem $\pi_P(x)\sim x/\log x$ if $\gamma >3/2$ but in a sense not if $\gamma <3/2$.

If $P$ satisfies $\pi_P(x)\sim x/\log x$ then we call $P$ a Beurling prime number system. A question raised by Beurling remains open: what functions $E(x)$ are such that $|N_P(x)-x|0$, then Olofsson [O] conjectures that

$\limsup_{x\to\infty} \frac{N_P(x)-x}{\ln x} >0,$
provided $P$ is different from the rational primes. Olofsson conjectures that Beurling’s question is addressed by any function $E(x)$ satisfying $E(x)=o(\ln x)$.

This is just a small sampling of Beurling’s brilliant work.

Ahlfors and Beurling

Ahlfors once wrote “Arne Beurling was the best friend I ever had.” Ahlfors enjoyed the quiet life of a writer and mathematician, whereas Beurling loved the outdoors, sailing, hunting. They first met in 1934 at the Scandanavian Congress of Mathematicians held in Stockholm. They were both still single, although Ahlfors would marry the following year. Beurling had just gotten his PhD the previous year. After the 1934 Congress, while Ahlfors was still at Mittag-Leffler, he would often take the train up to Uppsala to visit Beurling. Much later, when Ahlfors was at Harvard, Beurling’s girlfriend Annette-Marie Yxhull wrote Ahlfors a latter, explaining how unhappy Beurling was at Uppsala (see Beckman [B]). Ahlfors acted quickly, obtaining a visiting position for his friend at Harvard. Later, Beurling was given a permanent position at the Institute for Advanced Study, no doubt with Ahlfors’ strong recommendation.

A photo of Ahlfors taken by A. Beurling.

Beurlings work with Ahlfors was in the field of quasiconformal mappings. Let $f:D \to D'$ be an orientation-preserving homeomorphism between open sets in the plane. If $f$ is continuously differentiable, then it is $K$-quasiconformal if the derivative of at every point maps circles to ellipses with eccentricity bounded by $K$. We say $f$ is quasiconformal if $f$ is $K$-quasiconformal for some finite $K$.

A quasiconformal map.

According to Ahlfors, in their joint paper [AB], the decisive idea was entirely due to Beurling. It deals with the boundary values of quasiconformal mappings. If $h(x)$ increases for $-\infty ,

$1/\rho \leq \frac{h(x+t)-h(x)}{h(x)-h(x-t)} \leq \rho,$
for all $x$ and $t$.

Bibliography

[AB] L. V. Ahlfors and A. Beurling, The boundary correspondence under
quasi-conformal mappings,
Acta Math., 96 (1956), 125-142.

[B] B. Beckman, Codebreakers, AMS, 2002.

[O] R. Olofsson, Properties of the Beurling generalized primes, preprint, 2008.

[UF] The Geheimschreiber Secret, by Lars Ulfving, Frode Weierud, in Coding Theory and Cryptography: From Enigma and Geheimschreiber to Quantum Theory, Springer-Verlag, 2000.
paper in pdf or link at Frode’s cryptocellar