Celebratio Mathematica

Al­bert was my ad­visor at UC Dav­is where I stud­ied for my PhD in the late nineties. I really en­joyed my stud­ies there and learned a lot from the courses and sem­inars but the main thing was work­ing with Al­bert. I would say that the best word that char­ac­ter­ises what that time meant for me is ap­pren­tice­ship. Through Al­bert’s ini­ti­at­ive we em­barked on a num­ber of pro­jects on di­verse top­ics in math­em­at­ic­al phys­ics re­lated to string the­ory, con­form­al field the­ory, su­per­geo­metry and non­com­mut­at­ive geo­metry. To me it was quite a roller coast­er ride through so many things I had not known (and of­ten had no idea what they were) but had to learn quickly. It was an ex­cit­ing time in math­em­at­ic­al phys­ics with many break­throughs in string the­ory re­lated to du­al­it­ies, D-branes and M-the­ory. Al­bert was on top of all of these de­vel­op­ments, con­stantly try­ing new ideas, re­think­ing them in his own way. The work we did to­geth­er res­ul­ted in sev­er­al pub­lic­a­tions, but also there were a few pro­jects which were after some time aban­doned (or per­haps put on a back burn­er) but from which I learned a great deal non­ethe­less. I was learn­ing how to work by ob­serving a great mas­ter and by help­ing him pur­sue spe­cif­ic trains of thought.

I think Al­bert has a very pe­cu­li­ar style of think­ing and work­ing and I was priv­ileged to have a great deal of ex­pos­ure to it. I spent a lot of time in his house, of­ten on week­ends, dis­cuss­ing the pro­jects. He liked to think out loud and write things while hav­ing me by his side. Of­ten I played the role of dev­il’s ad­voc­ate ask­ing lots of ques­tions, of­ten na­ive ones, but through an­swer­ing the ques­tions we made pro­gress. I al­ways found great hos­pit­al­ity at Al­bert’s house with his wife, Ly­ubov Matveevna, mak­ing sure we had re­fresh­ments by our side when work­ing and of­ten treat­ing me to din­ner.

Be­sides play­ing dev­il’s ad­voc­ate dur­ing our dis­cus­sions, I had to fig­ure out many things on my own which in­cluded do­ing cal­cu­la­tions. In the­or­et­ic­al phys­ics a lot of work re­volves around par­tic­u­lar cal­cu­la­tions. I ob­served many times how vari­ous phys­i­cists tend to first do a cal­cu­la­tion and then try to fig­ure out what the res­ult means con­cep­tu­ally. That is def­in­itely not Al­bert’s style. The centre of his pro­jects were al­ways about con­cep­tu­al un­der­stand­ing. Cal­cu­la­tions were sec­ond­ary, serving al­most a dec­or­at­ive pur­pose. Be­fore sug­gest­ing a cal­cu­la­tion he thought a lot about it un­til he would come up with what he per­ceived was the most eco­nom­ic­al way to do it. Some­how his in­tu­ition would al­ways be strong enough to cap­ture many fea­tures of the an­swer be­fore com­put­ing it. I have also learned one oth­er im­port­ant tip from Al­bert re­gard­ing cal­cu­la­tions: when search­ing for er­rors one should nev­er try to re­peat the cal­cu­la­tion fol­low­ing the same steps but design a dif­fer­ent route or prefer­ably sev­er­al dif­fer­ent ap­proaches. In­deed an er­ror of­ten springs from some false as­sump­tion or step that we ha­bitu­ally re­peat when re­do­ing the cal­cu­la­tion while a dif­fer­ent ap­proach gets us off the beaten track.

Of­ten, in dif­fer­ent pro­jects, our re­search would ar­rive at an im­passe of a kind. Al­bert used some rather pe­cu­li­ar phrases to punc­tu­ate these im­passes. For ex­ample, when the over­all har­mony of our un­der­stand­ing of a con­struc­ted math­em­at­ic­al world seemed to be sud­denly threatened by some ugly counter­example or some mis­match, Al­bert would say a phrase in Rus­si­an that can be trans­lated more or less as “sharks do not ex­ist”. The phrase it­self, as I have learned years later, is bor­rowed from a book which is a col­lec­tion of funny stor­ies about small chil­dren. In one of the stor­ies a child claims (us­ing a slightly in­cor­rect lan­guage) that sharks do not ex­ist in the world. Be­sides mark­ing the im­passe, the phrase con­veyed some re­as­sur­ance that the obstacle would dis­solve and har­mony be re­stored.

An­oth­er phrase Al­bert of­ten used is more wide­spread in ver­nacu­lar Rus­si­an and can be trans­lated as: “Was there a boy?” Wiki­pe­dia de­scribes the ori­gin of this ex­pres­sion to be a phrase spoken by a char­ac­ter in one of Gorky’s nov­els, al­though an­oth­er ver­sion I heard claims it to be re­lated to the Beil­is af­fair (an in­fam­ous anti-semit­ic tri­al that took place in Rus­sia).1 Al­bert would use the phrase when we would be forced to doubt the ex­ist­ence of some math­em­at­ic­al ob­ject or con­struc­tion.

It is not the phrases them­selves that I am im­pressed with, al­though they sound amus­ing, but the work of Al­bert’s in­tu­ition: he would sense a par­tic­u­lar ten­sion on the bound­ary of known and un­known, and would then take the de­cision about which way to change the course of our re­search. Sim­il­ar mo­ments con­stantly arise in my own work and I do say to my­self the same phrases, re­call­ing how Al­bert used to do it.

Anato­ly Konechny was born and grew up in Rus­sia. He re­ceived his BSc in Math­em­at­ic­al Phys­ics from the In­de­pend­ent Uni­versity of Mo­scow in 1995. He got his PhD in Math­em­at­ics from UC Dav­is in 1999. After postdoc­tor­al po­s­i­tions at UC Berke­ley, The Hebrew Uni­versity of Jer­u­s­alem and Rut­gers, he moved to Ed­in­burgh in 2006 where he is cur­rently an as­so­ci­ate pro­fess­or at Heri­ot-Watt Uni­versity.