#### by Terry McConnell

In 1983 Don Burkholder was visiting Cornell where I had my post-doc. Knowing how much he loved walks, I had driven him out to nearby Taughannock Falls to pass some time before his colloquium talk. Among my cohort of Illinois graduate students Don had a reputation for being friendly and approachable, but he was not the easiest person to talk to one-on-one for extended periods. He gave the impression of being very well read on a wide range of topics both inside and outside of mathematics, and it was an intimidating problem for an intellectually inexperienced young person to introduce a topic that might interest him.

Signs of his lack of interest were subtle, but unmistakable, and his
students quickly learned to recognize them. If his eyes started to
glaze or to seek the
non-existent view from the window of his spartan
office, it probably meant that you were off on the wrong track. Quiet
attentiveness was a very good omen, and on rare occasions you might
aspire to the ultimate sign that you had piqued his interest — an
entry on one of his __\( 3\times5 \)__ cards. He always kept a stack of them
on him and would sometimes withdraw one during a memorable seminar and
write down a word or two before returning it to his hip pocket.

Unfailingly polite, Don always feigned interest in my conversational gambits, but it seldom lasted very long, and I found myself on the walk back from the falls casting about in some desperation for a suitable topic. “Much of your most recent work”, I said, “seems to involve finding the smallest possible constants in known inequalities.” I had the analysis student’s casual contempt for constants in inequalities. It was a badge of honor among us to prove results with enormous constants in them, as if deep and powerful processes are needed to produce such large numbers. His answer, quoted above, is probably my second favorite from him.

It was my privilege and extreme good luck to become Don’s student at a time when his work was taking off in a bold new direction. Over the decade prior to my enrollment in 1979 as a University of Illinois graduate student, Don’s name had become associated with martingale inequalities, particularly ones that explain how the size of a martingale transform is controlled by the original martingale. He also explored analogous results in other fields, such as singular integrals (harmonic analysis), and the relationship between a harmonic function and its conjugate (complex analysis). In all this work, certain quadratic expressions (“square functions”) seemed to play an essential role.

One day, having come to his office to report on my latest lack of
progress, I found him quite excited over a new idea he had gotten from
a letter, received from
Béla Bollabás,
that sketched a new
approach to some known
inequalities for the martingale square function.
The method involved inventing a function of two variables having
certain convexity properties, and Don noticed that if a similar
function could be constructed on __\( B\times B \)__, where __\( B \)__ is a Banach
space, then it could be used to extend some of the known inequalities
for real-valued martingale transforms to
__\( B \)__-valued martingale
transforms. General
Banach spaces have no multiplication (and thus no square functions),
so the new method provided an end-run around the use of these
quadratic expressions.

Over the following decade Don refined and extended his methods in a
series a brilliant papers, including a monumental work for the *Annals
of Probability* that returned to real-valued inequalities and showed
that the new methods often yielded the best constants in that context.
(This was the work I had in mind on our walk back from the falls.) It
is striking that a method capable of handling the greatest level of
generality, when specialized to a classical setting, can produce the
best possible results there as well. This, and the limpid clarity of
the arguments, makes the conclusion inevitable: he had indeed found
the best proofs.

It is curious that no clear consensus has emerged on a name for either
the method or the functions it uses. Some probabilists have called
them *special functions*, or *Burkholder’s functions*; harmonic
analysts use the term *Bellman functions* in honor of
Richard Bellman, who introduced functions with similar properties in control
theory. Burkholder himself never really had a name for them. In the
first paper he called the crucial function __\( \zeta \)__, and spaces having
such functions __\( \zeta \)__-*convex*, but the name never
caught on, and it was quickly replaced by UMD
(for *unconditional martingale differences*).

Whatever you call them, Burkholder’s special functions are devilishly difficult to find. Once found, it is simple enough to verify that they have the required properties, and the resulting proofs based on them unwind as effortlessly as a spring-coiled toy snake released from its can. One day, while walking together across the quad and down the tree-lined street to his house, I complained about my own complete failure to apply his method to any new problem of my own. “How on Earth do you come up with the formula for these functions?”, I asked in frustration.

“Lots and lots of scrap paper.”

*Terry R. McConnell received his PhD in Mathematics from the University of Illinois in 1981. He was an H.C. Wang Research Instructor at Cornell University from 1981 to 1984 and has been a member of the Syracuse University Department of Mathematics since 1984. He was an Alfred P. Sloan Foundation fellow from 1985–1987 and served as an Associate Editor for The Annals of Probability from 1991–1996.*