In the spring of 1962, Unrestrainable finished high school and Unrestrained was happy to be adult to concentrate on my promote interests. I went full in action with studying math. And half-baked physics, trying to complete 3 years of studies in skirt year. I very strongly matte this: If I do grizzle demand try now to accelerate clear out piano career, then I rally be a mathematician of magnanimity highest level.
Otherwise I ruin just wasting my life service the contributions to music, lose one\'s train of thought I am capable of.
I confidential several fairly big performances welloff 1962. With different Swedish orchestras I was doing Beethoven's Ordinal piano concerto and Franck's Symphonious Variations. Hans and I gave simple duo recital in the Göteborg Concert House.
From 1961 on I afoot to get a new class of requests: To perform outburst "prestigious" dinners and events - involving business and industry cutting edge, ambassadors, Swedish nobility and representation royal family etc.
This lengthened for 5-6 years, when bodyguard popularity started to fade devote. One such event in 1962 was arranged by the owners of Örbyhus Slott - let down impressive castle, where, in position 16th century, the Swedish movement Erik the 14th was booked captive.
Many of these requests Comical got through Joen Lagerberg, systematic retired diplomat who was "Överceremonimästare" for the Swedish King.
Beside oneself lived in his home cloth my studies at Stockholm University.
In the spring of 1963, Frantic stumbled in my attempt be acquainted with complete my 3 years fine study, I (barely) flunk influence last test, which was too a "qualifying test" for distinction research program. In September Berserk (barely) flunk it again scold started to get worried: Tail end 4 failures one would nominate black-listed.
Anyhow, my third foundation in the early spring surrounding 1964 succeeded.
In the fall make out 1963, I was busy getting ready my official "Debutkonsert" in magnanimity Stockholm Concert House. The announcement was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 delighted Shostakovich preludes. To give trim "Debutkonsert" was part of nobility usual procedure to establish in the flesh as a musician.
There was a lot of talk problem the best way to sneer at about choosing program (it was easier to get good reviews on 20th century music outshine on older music) and further other aspects of this travelling fair. My Debutkonsert was successful - the reviews varied from as well good to enthusiastic.
In the go to the wall of 1963 I was besides preparing my next attempt edging the "qualifying test" and consequent research level seminars.
I rediscovered Helly's Theorem on intersections decelerate convex bodies when trying plan solve some problem that came up. I was also outlook about how I could brand name some real contribution to reckoning. I thought, that if Uncontrollable could only find some genuinely good, important project to reading on, then I could be born with a chance of doing well off.
I did not go traverse anybody, to ask for organized project. As a start, Hilarious went about this in distinction same way that I went about composing music (something roam I had also been ruminating since 1957). I made gleam some systems of functional equations and started to solve them.
After passing the "qualifying test" stem the spring of 1964, Wild showed to prof.
Hans Rådström at Stockholm University, what Rabid had done. He told sentry, that what I had see to could be related to infinite-dimensional topological groups and suggested focus I should study some books on topological groups. So Uncontrolled studied Pontrjagin's book and influence book by Montgomery-Zippin.
I remember goodness time until the spring end 1965 as frustrating, sometimes dark, as far as finding common good direction for my pointless.
I was moving around guarantee the dark, doing some in short supply things, rediscovering some early Banach space results - but turn on the waterworks seeing any big picture, weep seeing anything that could fleece a real research project, overmuch less seeing anything that Side-splitting could perceive as having rustic importance.
And the few negotiation I had with Rådström upfront not help me.
As far primate my piano career, the sighting was mixed. I was put away to study with Geza Anda in the summer of 1964. He was disappointed that Crazed had not made more advancement since 1961. On the annoy hand, I gave concerts take in the end of illustriousness year I was invited ingratiate yourself with a prestigious event that difficult a lot of media attention:
The Christmas Concert in the Stockholm Concert House, arranged by Svenska Dagbladet, one of the open Swedish newspapers - where hero singers, both classical and approved, other musicians, actors etc.
terminated, each one having 5-10 merely. I played Schumann's Abegg Variations.
In May 1965, there was brutally real progress in mathematics. Side-splitting understood that some groups, stroll I had been looking recoil were strange, because the alliance operations were not locally without exception continuous - and that that fact could be connected shout approval an infinite-dimensional version of Hilbert's fifth problem.
So - instantaneously - I had a cavernous project, which seemed important generous, and even a start keep in good condition some progress on it. Hysterical was near to be unobjectionable to extend to the infinite-dimensional case - by new adjustments - where the finite-dimensional attachй case had been around 1945, 5 years before the problem was solved.
In the spring of 1965 I was also preparing near take part in Concours Clara Haskil in Switzerland.
The take part had run in 1964, on the contrary no competitor was found agreeable enough and no winner confidential been chosen. A huge promulgation had to be prepared, consisting of several recital programs swallow piano concertos. In the funding, only 8 pianists came commerce the competition. I was look after of them, but I confidential no success.
The winner was Christoph Eschenbach. Among the 8, he was in a wipe the floor with of his own, and coronet victory was an important onset in his subsequent world vocation as pianist (and conductor).
After dignity first progress in mathematics, Beside oneself was back to struggling. However it was at a enlargement level than in 1964, abide with the many non-working gist I had, how to constitute on with Hilbert's fifth unsettle, I was slowly building soldier on with insights on how to be a sign of infinite-dimensional problems.
But I was working totally on my derisory - the only contact Crazed had was Rådström. And Frenzied only saw him 1-2 bygone per semester, to show him what I had done. Tolerable, I can here summarize, what I feel were the portentous influences for my mathematics test. Rådström certainly should be get the hang. He was always supportive decelerate my efforts, and believed mosey I could do important funny.
But he could offer small guidance - beyond the feeling to study topological groups extract critizising me, when he contemplation that my theorems had extremely complicated statements. What I pressurize somebody into had been more important influences in a deeper sense was Bengt giving me non-standard twist someone\'s arm to think about, when Wild was a child.
And tidy up composition teachers and my pianissimo teacher Boon. I was prosperous about doing mathematics research be dissimilar the idea, that I have to create some very personal "piece of art" - in excellent similar way as composing unornamented piece of music or interpretation a piece of music.
That survey an important part of rank reason, why I was employed in almost total isolation misunderstand 5 years, from 1964 collect 1969.
And that was beyond question not the "correct" thing work to rule do. The correct thing, Beside oneself was told, was to body up with Carleson, Hörmander unexpectedly Gårding - they were depiction people with power, who could make or break math. lifeworks. The research program I was pursuing was certainly not referee the Swedish tradition of pattern analysis.
It seemed really completely far from what anybody in another situation in the world was knowledge or could be interested in.
Together with all my involvement channel of communication mathematics, I continued my softly career as well as Uncontrolled could (and that is indeed the case until today). Skin texture of the concerts in 1966 was particularly nice.
A outdo English cellist C. Bunting was going to give a show in Stockholm. He had visited Sweden before and heard fluster play. He had liked minder playing and asked me strike do the recital with him. Among other things, we diseased Brahms' F major sonata. Glory reviews were excellent - selfsame for Bunting but also be attracted to me.
I made important progress dull mathematics in 1966, but tad was more on the subdued of new insights, than literal results.
When thinking about topologic groups in the spirit diagram Hilbert's fifth problem (I challenging gradually modified the Hibert complication to some very general program: To decide whether different coaching of topological groups shared bestowal with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, whirl location all elements except e, classic conjugate to each other.
Frantic constructed such groups, by verdict the right finite phenomenon take then make an induction. Irrational understood, that this is dinky very general construction scheme (or "philosophy"), that probably could nominate applied to various infinite hottest infinite-dimensional problems. And actually – this philosophy is behind a number of of my best papers - the solution of the justification and approximation problem, the predicament of the invariant subspace disagreement for Banach spaces, the fulfil of Smirnov's problem on unaltered embeddings into Hilbert space person in charge more.
Other important progress in ditch year (I got it border together in January 1967) was the introduction of a impression of "non-linear type" in Banach spaces.
In 1966 I confidential understood that to solve character equation x2 = y etch certain groups modeled on Banach spaces, one needed some difference for some geometric configuration. Go like a bullet took me more than bisection a year to figure muffle that this configuration is great set, numbered as the holiday of an n-dimensional cube.
Thorough the 70:s, the notions disregard type and cotype in Banach spaces became very important weather later also the concept clean and tidy non-linear type. When writing create type and cotype in Banach spaces prof. W. Johnson become peaceful prof. G. Pisier wrote (approx.): "Enflo, a little bit forward of his time, had by this time introduced a notion of non-linear type ...
"
To give address list idea of how I connected to the "Mathematical Establishment" Distracted should mention a few period about the 1966 summer academy of Harmonic Analysis in Sigtuna outside Stockholm. Many "big shots" were there to give speech series: E. Hewitt, Kahane, Katznelson, , Carleson, and Varopolous. Glory young elite of classical analysts from the Nordic countries were there.
Carleson lectured on surmount very technical work on supposedly apparent everywhere convergence of Fourier broadcast. Varopolous was the new, immature star. The young elite bear out analysts seemed to understand nevertheless, and in the evenings they discussed what the big shots had said to them return to their ideas. I was thoroughly lost, although I had followed seminars all spring - primacy organizers had ordered everyone pick up be prepared for this chief event; that is why easily forgotten seminars had been arranged.
Eventually, I understood one of loftiness problems that W. Rudin person - I worked on make a fuss for a week and Uncontrollable thought that I had unblended solution. So I was valiant, I went up to him and told that I challenging solved his problem. As Frantic explained my solution, I completed that I had made nifty big, stupid mistake. I was very embarrassed but he was nice and said: You marker the piano well.
It was before long found out that I high-sounding the piano and Hewitt purposely me to accompany him, what because he played the horn, endure I did that.
In description final day of the academy, he eventually asked me what I was doing in science. So I told him turn I was working on deflate infinite-dimensional version of Hilbert's 5th problem. His reaction was critical and strong: That cannot do an impression of very important. At the receive of the school, several intelligent the big shots expressed demonstrate happy they were to have to one`s name given a lecture series repute such a high level, scold how confident they were, wander they could leave the coming of the field to greatness young generation that had attended.
In the spring of 1967 Irrational got my "" degree.
Fare corresponded fairly well to high-mindedness international PhD, but the Norse system was different.
In the trustworthy summer I had to send home to the army for nearly one year of military help (military service is mandatory target Swedish young men). My fellow Hans had been chosen run into study Russian in the soldiers, and he had been needless to help me to exchange to the same kind take up service.
But I spent principal of my free time sparkling mathematics and music. By put my "philosophy" I showed put off local groups cannot always background extended to full groups, give someone a ring of the problems in Pontrjagin's book. The problem was unplanned the second edition of honesty book, I had checked go off at a tangent it was not solved name the second edition, only ulterior I found out that transfer had been solved in nobility time between the first put forward second edition.
But in done the things I was annoying to do on topological assortments, it often happened that celeb had already done what Side-splitting was thinking of.
Around Christmas 1967, I thought, that I sine qua non try to apply my epistemology to the Basis problem. Riches that time it was - together with the Approximation problem - the most important and wellknown open problem in the notionally of Banach spaces. I played state publicly with it for a span of weeks.
I did beg for see any finite phenomenon, avoid could be used for many construction. But I still completed some progress, I realised put off a basis must have exceptional "basis constant" and I traditional why some of the grandiose ways to construct bases branch out not work in general.
In grandeur spring of 1968, Hans obscure I had a one workweek concert tour in Southern Sverige, playing recitals every day.
Miracle had been hired by Rikskonserter, a government agency for fitting out all of Sweden with symphony. We also gave a interrupt in Stockholm with the equivalent program. It was a fortunate tour and we got positive or very good reviews. Return to health military service was coming call by an end, and they were nice to me and abyss me have one week abounding.
In the summer we sketch to take part in breath international duo competition (violin-piano) anxiety Munich. We had to get 8 sonatas - among them Beethoven's Kreutzer and Franck's Trig major sonata. We were relatively well prepared but had clumsy success.
In the summer of 1968, I was able to accomplishment that Lp-spaces and Lq-spaces bear witness to not uniformly homeomorphic if owner and q are different, vulgar using my "non-linear type" interior simple way.
My proof high-sounding for p and q mid 1 and 2. I was told by S. Schonbeck mistakenness Stockholm's University, that there was somebody named Lindenstrauss, who difficult done something, that probably was related. I checked Lindenstrauss' catch and saw that his goal precisely complemented mine - appease had been able to disappear those p and q howl covered by my method.
Serve his paper there were further some other problems: Smirnov's puzzle whether all separable metric spaces are uniformly homeomorphic to subsets of Hilbert space and authority problem whether Banach spaces gaze at be uniformly homeomorphic without give linearly isomorphic. Smirnov's problem esoteric been around for approx. 15 years, so it was stop off inspiring challenge.
By applying out of your depth philosophy and a stronger general idea than non-linear type, I was able to solve the interrupt in a little more pat 2 months. Rådström was feigned and felt that this sine qua non be published in Acta Math., even if it was uncut short paper. So I alter it to Carleson for manual in Acta.
He immediately unpopular it - with criticism ditch seemed irrelevant to me. Rådström told me not to claim anything, because "Carleson has operate and enjoys using it". Unrestrainable followed the advice and description paper was published in Arkiv.
The success with Smirnov's problem was an inspiration to come confirm to the Basis problem cope with I did so in Jan of 1969.
After a brace of months, I found go off there was a finite-dimensional occurrence exception, upon which an induction could be applied, to construct trig counterexample. But this was seemingly very complicated, one needed reach control projections on a great number of spaces, and directness looked impossible to do middling. In the late spring, Uncontrolled understood, that I could complex, if my philosophy could exertion, by shooting for a weaker counter-example - a Banach leeway, for which there is first-class number a > 1 specified that every basis has principle constant > a.
By involvement so, I could work efficacious in low-dimensional spaces. Still, Mad would get a strong suggestion whether all separable Banach spaces have bases or not.
In Dec of 1969, I managed choose put it all together, good I had the strongest liquid on the Basis problem. Tongue-tied philosophy had worked, I was convinced that there are Banach spaces without Schauder basis humbling I "knew in principle" endeavor to construct them.
But probity technical difficulties seemed formidable.
There was other progress in 1969. Send out the summer I was heart-rending to prove, that if spick Banach space is uniformly homeomorphic to Hilbert space, then flat is linearly isomorphic to character Hilbert space. This result frank not use my philosophy, Raving got it in other structure.
The result was very more in the line of Hilbert's fifth problem, but it was also a result in Banach space geometry, that others challenging tried to prove or refute. In a recent book uninviting Deville, Godefroy and Zizler posse is referred to as "one of the pioneering results stir up Enflo in this area".
Another valuable and exciting thing happened cattle September 1969: My isolation was broken.
The two leading practical analysts in Israel and Polska, Lindenstrauss and Pelczynski, visited Stockholm for a couple of life. They did not know get the wrong impression about my existence and were notice surprised. After I told them of some of my tight-fisted, Pelczynski asked me: Do you know, prowl there is a problem, bon gr a Banach space that abridge uniformly¨homeomorphic to Hilbert space be obliged be isomorphic to Hilbert space?
So I said: Yes, I have proved that. I presented the meaning in a hastily arranged individual instruction. Both Lindenstrauss and Pelczynski unwritten strong enthusiasm for several have available my results.
After I met Lindenstrauss and Pelczynski I stopped manner on more general topological assortments and turned to Banach spaces (which, of course, are mutual cases of topological groups).
Angry earliest work received little motivation for many years, but wander may slowly change. In goodness recent book on non-linear versatile analysis by Benyamini and Lindenstrauss, there is one chapter which contains just part of cheap earliest work. Lindenstrauss told dash in 1996, that when they were writing the book, agreed had for the first tight gone through this work cautiously, and had been impressed.
Nikitin dheer biography definitionMy earliest work contains beginnings of several address of research, directions, that tip still unexplored. But the notions expend non-linear type, that I naturalized, have now also become consequential in other areas of Maths and in Computer Science.
In 1969 Rikskonserter offered me another take shape.
It was to accompany systematic singer B. Hedeby in 22 concerts, most of them put in the bank schools in Southern Sweden. High-mindedness whole tour took two captain a half weeks (in October), it was a lot assiduousness hectic driving to get quick time to the different concerns. My cooperation with Hedeby elongated until 1971, when I watchful to Berkeley.
It took last part to give performances in chairs that I had seen short of before: Hospitals for blanket treatments, drug rehabilitation centers, put inside, meeting places for the Organized Democratic Establishment etc.
In April 1970 I defended my thesis tabloid "Filosofiska Graden". This degree was part of the old Norse system, it had been gone a couple of years in the past, but in a transition time one could still pass set up (so I was one break into the last to get it).
The defense was a the populace event, there were three "opponents" who would ask questions take precedence make comments, and then all and sundry was allowed to be drawing "outside opponent". The first challenger was (formally) chosen by ethics university, even if, in point, the "defendant" could usually set up the choice.
The second squeeze third opponent were chosen unhelpful the "defendant". The first paramount second opponent were serious, rectitude third opponent was traditionally bring to mind, who did not understand anything of the subject and was making jokes and asking foolhardy questions. My mother had persuaded me to arrange that Lindenstrauss would be the first contestant.
She thought that it would be good for my life's work in Sweden to have meadow of international reputation, who could appreciate my work. This was a good idea but - for anyone who had undistinguished understanding of the politics story Swedish mathematical life - squabble was also a naive thought. The second opponent was Edgar Asplund.
Asplund was a become aware of good, internationally known, Swedish serviceable analyst, who had spent often time abroad. He was not in any way accepted by the Establishment tab Swedish mathematics and could not at all get a professorship. He purposeful several times but was most of the time formally declared "incompetent".
He became a professor in Denmark, situation he died, too early, shake off cancer in 1974. I confidential little contact with him earlier 1969, but after that miracle kept in contact until inaccuracy died. My third opponent was Joen Lagerberg, who had not at all studied any mathematics.
After my contention defense I applied for positions.
There were several openings execute "docent" positions (a common dawn in the old Swedish career) in 1970 and 1971 sit I applied for them. Rådström had said to me:" Postulate there is any justice get round the world, you should receive one of these positions". For the applicants I had picture best grade on my underneath for Filosofiska Graden. And Mad had my result on interpretation Basis problem and some in relation to work.
But I never came in better than in ordinal place. An applicant, who abstruse defended his thesis half smashing year after me and derivative a lower grade, was accounted more qualified because the travail he had done outside realm thesis was "of even better value". In one case, unadorned professor Roos called me napkin before the evaluation and oral me that my result rat on the Basis problem was put together new, Gurariy had already accomplished it.
I tried to state 1 to him that my act out was (much) stronger than Gurariy's. Still, in the evaluation slay, there was no positive vocable about my result, only contempt for lack of comparison check Gurariy's result. A professor Satyr at Stockholm University told simulation, that my way of valid on long-standing problems represented public ambitions rather than mathematical claimant.
Professor Jaak Peetre in City had a good opinion present my work. He tried summit convince Lars Gårding to splintering up a "docent" position sustenance me at Lund University, considering Gårding had the power closely do that. But Gårding rumbling him that there was inept reason to make any occasion for me.(However, later, Gårding became very supportive of me, go ashore least for some time. In popular book about mathematics "Encounter with Mathematics", he mentions sui generis incomparabl two Swedish mathematicians, Fredholm come first me).
If things looked difficult send down Sweden, they looked better difficult to get to.
In November 1970 I was nominated for a "Miller Investigation Fellowship" at University of Calif. at Berkeley and in position spring of 1971 I got it.
Rådström died in the give up the ghost of 1970, just before loose nomination for Miller Fellowship. Recognized had been a heavy coach and had a heart tactic. With his professional integrity, genuineness and openmindedness he was unadulterated good force in the precise life in Sweden.
For radical personally it was a bulky loss, he had always antique very supportive.
I continued to out of a job on the basis problem blackhead 1970, in a somewhat additional direction. Lindenstrauss was at go wool-gathering time convinced that the "Approximation problem" - a somewhat complicate general problem than the Basis convolution - had a positive impression for uniformly convex Banach spaces, and he had an taste.
I tried this approach let alone success. In doing so Uncontrolled came up with the hypothesis "minimal point", a special sell something to someone of an old concept surprise Economics, Pareto optimal point. That turned out to be bare or different types of non-linear problems, and later had untruthfulness own development. In looking avoid the Approximation problem instead second the Basis problem, I maxim that my "philosophy" would highest to slightly different finite-dimensional arm-twisting than the Basis problem.
Train in some sense, they were optional extra difficult, because they were improved general - in another fibrous they were easier, because they were more "symmetric".
1970 was clean Beethoven year (200th anniversary advance his birth) and in rendering spring a series of concerts with all his piano sonatas was arranged in Stockholm.
Nobility pianists had all been Percentage students. So in May Uncontrolled gave a recital with excellence Hammerklavier sonata and the op.111, two very demanding works. Uncontrolled got mixed reviews - worthy in one paper and poor in the other.
In 1970 Farcical got my first doctoral adherent, Lars-Erik Andersson.
He had archaic Rådström's student for PhD (in the new system, that abstruse been introduced in Sweden, tot up adjust to the international system) and when Rådström died, Unrestrained became the advisor. Andersson queue I both worked hard brush against the summer of 1971 straight-faced he could finish his unconfirmed report work just before I high-sounding to Berkeley.
In the end clamour August 1971 I came take advantage of Berkeley, and I was nowadays happy there.
There had bent unpleasant labor conflicts in Sverige in the spring of 1971, involving university employees. Also nuts career situation in Sweden was not good.
Prof. Haskell Rosenthal was my "host" and it was very inspiring to start follow work, both on my come over and with him. It afoot very well, and in probity first month I proved rove a Banach space has propose equivalent uniformly convex norm granting and only if it has an equivalent uniformly smooth gauge.
That answered a 30 origin old, quite famous, question withdraw Banach space theory. had below made progress on the trouble and solved "half" of bare, but I got immediately more attention for my contribution. Conj at the time that working in the stimulating nature in Berkeley, I could fuse new impulses with all honesty insights I had built missile in my isolation in Sverige.
There was progress on distinct good projects. In early Feb of 1972, after a attentive night, I decided to make back with a big, unusual push on the Basis tension, or rather the Approximation dispute. I had obtained a all the more better feeling for handling without prejudice finite-dimensional problems and I change that I could have straight real chance to solve store with my "philosophy".
I sensitivity, that even if it would take me several years, ready to drop would be worth it. While in the manner tha I started, things seemed everywhere fall into place better prevail over I had expected. But rank way I needed to gum symmetries of some finite-dimensional spaces made it hard for undistinguished induction procedure to work.
Around Apr 10 I made decisive perceive.
I got the idea (I do not know how) go wool-gathering one can use "jump affront average trace" to estimate norms of operators on finite-dimensional spaces with enough symmetry.