Celebratio Mathematica

I had heard of Al­bert Schwarz and his fam­ous, re­volu­tion­ary ideas cre­at­ing to­po­lo­gic­al field the­ory be­fore I had the chance to meet him. My en­coun­ters with him re­main as shin­ing memor­ies and were truly up to my ex­pect­a­tions! In fact two of them had a re­mark­ably pro­duct­ive ef­fect on me.

I first met Al­bert in Les Houches in the middle of the 90s. He gave a lec­ture in which he presen­ted “an­oth­er kind of non­com­mut­at­ive geo­metry” (in his own words) as re­pla­cing the usu­al poly­no­mi­al equa­tions of al­geb­ra­ic geo­metry by equa­tions in­volving non­com­mut­ing vari­ables and in par­tic­u­lar matrices. In his talk, which was in the late af­ter­noon, Al­bert for­mu­lated the prob­lem of find­ing a good proof that the for­mula for the sum of roots of a poly­no­mi­al equa­tion con­tin­ues to hold in the non­com­mut­at­ive case. After listen­ing to his talk I went to din­ner with my wife, Danye, and she no­ticed that I had the usu­al ab­sent-minded look char­ac­ter­ist­ic of think­ing hard about a prob­lem; and in­deed at 10pm I was back in the con­fer­ence build­ing and I wrote the proof I had found dur­ing the din­ner on a small piece of pa­per which I pinned in the en­trance. The idea is the same as in the proof of the Bott peri­od­icity and uses “matrices of matrices” as a key device. That was our first en­counter and we wrote a joint pa­per called “Mat­rix Vi­eta The­or­em re­vis­ited” which ap­peared in Let­ters in Math­em­at­ic­al Phys­ics [1].

My second in­ter­ac­tion with Al­bert pro­duced the pa­per cited more of­ten by far than any of my oth­er pa­pers.

It was an amaz­ing cir­cum­stance and happened as fol­lows: Al­bert and I were walk­ing to­geth­er from the main build­ing to the tea room at the IHES where Al­bert was vis­it­ing. On the (very short) way to the tea room Al­bert for­mu­lated a ques­tion which he had ex­trac­ted from the con­struc­tion of IKKT mod­els in phys­ics. It turned out that the work I had done in 1980 on the non­com­mut­at­ive tor­us and con­nec­tions on the vec­tor bundles I had con­struc­ted there was giv­ing the full an­swer to his ques­tion! The in­ter­ac­tion las­ted a bit more than on the short path to tea, and Mi­chael Douglas and Al­bert Schwarz kindly asked me to co­sign the pa­per they de­vised us­ing the above con­struc­tion. This pa­per, “Non­com­mut­at­ive geo­metry and mat­rix the­ory,” which ap­peared in the Journ­al of High En­ergy Phys­ics in 1998 [2], has been cited so many times that when I turned 60, Al­bert offered me as a present a thick book­let (which I still have) in which he col­lec­ted and prin­ted the thou­sand cita­tions of that time!

In my mind he, Al­bert, is like a pure dia­mond, with such a power­ful and un­com­prom­ising mind whose cre­ativ­ity sheds light around him in a unique man­ner.