Celebratio Mathematica

Sir Mi­chael Atiyah was con­sidered one of the world’s fore­most math­em­aticians. He is best known for his work in al­geb­ra­ic to­po­logy and the codevel­op­ment of a branch of math­em­at­ics called to­po­lo­gic­al \( K \)-the­ory, to­geth­er with the Atiyah–Sing­er in­dex the­or­em, for which he re­ceived the Fields Medal (1966). He also re­ceived the Abel Prize (2004) along with Is­ad­ore M. Sing­er for their dis­cov­ery and proof of the in­dex the­or­em, bring­ing to­geth­er to­po­logy, geo­metry, and ana­lys­is, and for their out­stand­ing role in build­ing new bridges between math­em­at­ics and the­or­et­ic­al phys­ics. In­deed, his work has helped the­or­et­ic­al phys­i­cists to ad­vance their un­der­stand­ing of quantum field the­ory and gen­er­al re­lativ­ity. Mi­chael’s ap­proach to math­em­at­ics was based primar­ily on the idea of find­ing new ho­ri­zons and open­ing up new per­spect­ives. Even if the idea was not val­id­ated by the math­em­at­ic­al cri­terion of proof at the be­gin­ning, “the idea would be­come rig­or­ous in due course, as happened in the past when Riemann used ana­lyt­ic con­tinu­ation to jus­ti­fy Euler’s bril­liant the­or­ems.” For him an idea was jus­ti­fied by the new links between dif­fer­ent prob­lems that it il­lu­min­ated. Our ex­per­i­ence with him is that, in the man­ner of an ex­plorer, he ad­ap­ted to the land­scape he en­countered on the way un­til he con­ceived a glob­al vis­ion of the set­ting of the prob­lem. Atiyah de­scribes here1 his way of do­ing math­em­at­ics:2

Some people may sit back and say, I want to solve this prob­lem and they sit down and say, “How do I solve this prob­lem?” I don’t. I just move around in the math­em­at­ic­al wa­ters, think­ing about things, be­ing curi­ous, in­ter­ested, talk­ing to people, stir­ring up ideas; things emerge and I fol­low them up. Or I see something which con­nects up with something else I know about, and I try to put them to­geth­er and things de­vel­op. I have prac­tic­ally nev­er star­ted off with any idea of what I’m go­ing to be do­ing or where it’s go­ing to go. I’m in­ter­ested in math­em­at­ics; I talk, I learn, I dis­cuss and then in­ter­est­ing ques­tions simply emerge. I have nev­er star­ted off with a par­tic­u­lar goal, ex­cept the goal of un­der­stand­ing math­em­at­ics.

We could de­scribe Atiyah’s jour­ney in math­em­at­ics by say­ing he spent the first half of his ca­reer con­nect­ing math­em­at­ics to math­em­at­ics and the second half con­nect­ing math­em­at­ics to phys­ics.

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