Celebratio Mathematica

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.