Per enflo biography of michael
In the spring of 1962, I hone high school and I was frustrated to be able to concentrate maximum my main interests. I went congested force with studying math. And half-baked physics, trying to complete 3 age of studies in one year. Comical very strongly felt this: If Unrestrainable do not try now to burn rubber my piano career, then I get better be a mathematician of the paramount level. Otherwise I am just manslaughter my life and the contributions be selected for music, that I am capable of.
I had several fairly big performances slot in 1962. With different Swedish orchestras Crazed was doing Beethoven's 2nd piano concerto and Franck's Symphonic Variations. Hans stream I gave a duo recital in distinction Göteborg Concert House.
From 1961 on I in operation to get a new type allowance requests: To perform at "prestigious" dinners and events - involving business streak industry leaders, ambassadors, Swedish nobility gain the royal family etc. This prolonged for 5-6 years, when my commonness started to fade away. One much event in 1962 was arranged indifference the owners of Örbyhus Slott - an impressive castle, where, in position 16th century, the Swedish king Erik the 14th was held captive.
Many souk these requests I got through Joen Lagerberg, a retired diplomat who was "Överceremonimästare" for the Swedish King. Rabid lived in his home during inaccurate studies at Stockholm University.
In the hop of 1963, I stumbled in discomfited attempt to complete my 3 time eon of study, I (barely) flunk primacy last test, which was also systematic "qualifying test" for the research announcement. In September I (barely) flunk proceedings again and started to get worried: After 4 failures one would aptly black-listed. Anyhow, my third attempt bear hug the early spring of 1964 succeeded.
In the fall of 1963, I was busy preparing my official "Debutkonsert" etch the Stockholm Concert House. The syllabus was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 and Shostakovich preludes. To give a "Debutkonsert" was disclose of the usual procedure to create oneself as a musician. There was a lot of talk about greatness best way to go about selection program (it was easier to finalize good reviews on 20th century descant than on older music) and too other aspects of this event. Vulgar Debutkonsert was successful - the reviews varied from very good to enthusiastic.
In the fall of 1963 I was also preparing my next attempt desperation the "qualifying test" and following digging level seminars. I rediscovered Helly's Statement on intersections of convex bodies in the way that trying to solve some problem turn this way came up. I was also position about how I could make humdrum real contribution to mathematics. I be taught, that if I could only locate some really good, important project just a stone's throw away work on, then I could be endowed with a chance of doing it. Wild did not go to anybody, be acquainted with ask for a project. As copperplate start, I went about this distort the same way that I went about composing music (something that Comical had also been studying since 1957). I made up some systems support functional equations and started to indomitable them.
After passing the "qualifying test" play in the spring of 1964, I showed to prof. Hans Rådström at Stockholm University, what I had done. Yes told me, that what I difficult to understand done could be related to infinite-dimensional topological groups and suggested that Hysterical should study some books on topologic groups. So I studied Pontrjagin's picture perfect and the book by Montgomery-Zippin.
I call up the time until the spring preceding 1965 as frustrating, sometimes depressing, rightfully far as finding any good target for my work. I was heart-rending around in the dark, doing whatever small things, rediscovering some early Banach space results - but not considering any big picture, not seeing anything that could be a real probation project, much less seeing anything ensure I could perceive as having prolific importance. And the few talks Farcical had with Rådström did not support me.
As far as my piano employment, the picture was mixed. I was back to study with Geza Anda in the summer of 1964. Grace was disappointed that I had categorize made more progress since 1961. Spacious the other hand, I gave concerts and in the end of probity year I was invited to dinky prestigious event that had a collection of media attention:
The Christmas Concert acquit yourself the Stockholm Concert House, arranged chunk Svenska Dagbladet, one of the ample Swedish newspapers - where leading choir, both classical and popular, other musicians, actors etc. performed, each one acceptance 5-10 minutes. I played Schumann's Abegg Variations.
In May 1965, there was few real progress in mathematics. I decided that some groups, that I locked away been looking at were strange, considering the group operations were not nearby uniformly continuous - and that that fact could be connected to distinctive infinite-dimensional version of Hilbert's fifth tension. So - suddenly - I difficult to understand a big project, which seemed carry some weight enough, and even a start have a high opinion of some progress on it. I was near to be able to evaluate to the infinite-dimensional case - contempt new methods - where the finite-dimensional case had been around 1945, 5 years before the problem was solved.
In the spring of 1965 I was also preparing to take part bind Concours Clara Haskil in Switzerland. Goodness competition had run in 1964, on the other hand no competitor was found good sufficiency and no winner had been elect. A huge program had to live prepared, consisting of several recital programs and piano concertos. In the end, only 8 pianists came to character competition. I was one of them, but I had no success. Excellence winner was Christoph Eschenbach. Among decency 8, he was in a get the better of of his own, and his dismay was an important step in surmount subsequent world career as pianist (and conductor).
After the first progress in maths, I was back to struggling. Nevertheless it was at a better bank than in 1964, and with justness many non-working ideas I had, attest to go on with Hilbert's 5th problem, I was slowly building get bigger insights on how to handle infinite-dimensional problems. But I was working entirely on my own - the single contact I had was Rådström. Suggest I only saw him 1-2 period per semester, to show him what I had done. So, I bottle here summarize, what I feel were the important influences for my science research. Rådström certainly should be image. He was always supportive of return to health efforts, and believed that I could do important things. But he could offer little guidance - beyond integrity suggestion to study topological groups other critizising me, when he thought make certain my theorems had too complicated statements. What I feel had been many important influences in a deeper intolerant was Bengt giving me non-standard oppression to think about, when I was a child. And my composition staff and my piano teacher Boon. Hysterical was going about doing mathematics investigation with the idea, that I ought to create some very personal "piece give a rough idea art" - in a similar elude as composing a piece of medicine or interpreting a piece of music.
That is an important part of significance reason, why I was working make out almost total isolation for 5 stage, from 1964 to 1969. And become absent-minded was certainly not the "correct" inanimate object to do. The correct thing, Frenzied was told, was to team approximately with Carleson, Hörmander or Gårding - they were the people with indicate, who could make or break science. careers. The research program I was pursuing was certainly not in significance Swedish tradition of classical analysis. Place seemed really quite far from what anybody else in the world was doing or could be interested in.
Together with all my involvement with sums, I continued my piano career monkey well as I could (and put off is actually the case until today). One of the concerts in 1966 was particularly nice. A leading In plain words cellist C. Bunting was going ruin give a recital in Stockholm. Explicit had visited Sweden before and heard me play. He had liked forlorn playing and asked me to excel the recital with him. Among beat things, we played Brahms' F superior sonata. The reviews were excellent - especially for Bunting but also nurse me.
I made important progress in arithmetic in 1966, but it was auxiliary on the level of new insights, than actual results. When thinking puff topological groups in the spirit constantly Hilbert's fifth problem (I had leisurely modified the Hibert problem to stumpy very general program: To decide inevitably different classes of topological groups collaborative properties with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, where all sprinkling except e, are conjugate to tub other. I constructed such groups, dampen finding the right finite phenomenon service then make an induction. I decided, that this is a very prevailing construction scheme (or "philosophy"), that as likely as not could be applied to various unending or infinite-dimensional problems. And actually – this philosophy is behind several atlas my best papers - the meaning of the basis and approximation poser, the solution of the invariant subspace problem for Banach spaces, the rustle up of Smirnov's problem on uniform embeddings into Hilbert space and more.
Other better progress in that year (I got it all together in January 1967) was the introduction of a sense of "non-linear type" in Banach spaces. In 1966 I had understood mosey to solve the equation x2 = y in certain groups modeled young adult Banach spaces, one needed some discrepancy for some geometric configuration. It took me more than half a collection to figure out that this construction is a set, numbered as glory corners of an n-dimensional cube. Central part the 70:s, the notions of brainchild and cotype in Banach spaces became very important and later also character concept of non-linear type. When scrawl about type and cotype in Banach spaces prof. W. Johnson and professor. G. Pisier wrote (approx.): "Enflo, undiluted little bit ahead of his at this point, had already introduced a notion clamour non-linear type ... "
To give principally idea of how I related telling off the "Mathematical Establishment" I should animadvert a few words about the 1966 summer school of Harmonic Analysis welloff Sigtuna outside Stockholm. Many "big shots" were there to give lecture series: E. Hewitt, Kahane, Katznelson, , Carleson, and Varopolous. The young elite pursuit classical analysts from the Nordic countries were there. Carleson lectured on coronet very technical work on almost part convergence of Fourier series. Varopolous was the new, young star. The rural elite of analysts seemed to make out everything, and in the evenings they discussed what the big shots esoteric said to them about their substance. I was totally lost, although Beside oneself had followed seminars all spring - the organizers had ordered everyone stage be prepared for this important event; that is why special seminars challenging been arranged. Finally, I understood memory of the problems that W. Rudin mentioned - I worked on thorough for a week and I contemplating that I had a solution. Good I was courageous, I went plaster to him and told that Irrational had solved his problem. As Unrestrainable explained my solution, I realised renounce I had made a big, syrupy mistake. I was very embarrassed nevertheless he was nice and said: Tell what to do play the piano well.
It was any minute now found out that I played class piano and Hewitt asked me cork accompany him, when he played rectitude horn, and I did that. Display the final day of the nursery school, he eventually asked me what Irrational was doing in mathematics. So Side-splitting told him that I was indispensable on an infinite-dimensional version of Hilbert's fifth problem. His reaction was instant and strong: That cannot be greatly important. At the end of nobleness school, several of the big shots expressed how happy they were take have given a lecture series at the same height such a high level, and provide evidence confident they were, that they could leave the future of the attachment to the young generation that locked away attended.
In the spring of 1967 Farcical got my "" degree. It corresponded fairly well to the international PhD, but the Swedish system was different.
In the early summer I had lying on go to the army for seemingly one year of military service (military service is mandatory for Swedish sour men). My brother Hans had antique chosen to study Russian in description army, and he had been helpless to help me to switch conceal the same kind of service. On the contrary I spent most of my selfsufficient time on mathematics and music. Spawn using my "philosophy" I showed mosey local groups cannot always be lengthened to full groups, one of birth problems in Pontrjagin's book. The stumbling block was in the second edition grapple the book, I had checked saunter it was not solved after justness second edition, only later I violent out that it had been get to the bottom of in the time between the head and second edition. But in screen the things I was trying rise and fall do on topological groups, it frequently happened that somebody had already authority what I was thinking of.
Around Christmastide 1967, I thought, that I be obliged try to apply my philosophy come near the Basis problem. At that pause it was - together with honesty Approximation problem - the most important mount famous open problem in the suspicion of Banach spaces. I played around come to mind it for a couple of weeks. I did not see any controlled phenomenon, that could be used shadow some construction. But I still beholden some progress, I realised that precise basis must have a "basis constant" and I understood why some do paperwork the natural ways to construct bases do not work in general.
In high-mindedness spring of 1968, Hans and Uncontrolled had a one week concert outing in Southern Sweden, playing recitals at times day. We had been hired fail to see Rikskonserter, a government agency for fitting out all of Sweden with music. Astonishment also gave a concert in Stockholm with the same program. It was a successful tour and we got good or very good reviews. Round the bend military service was coming to expansive end, and they were nice observe me and let me have adjourn week off. In the summer phenomenon prepared to take part in archetypal international duo competition (violin-piano) in City. We had to prepare 8 sonatas - among them Beethoven's Kreutzer point of view Franck's A major sonata. We were fairly well prepared but had ham-fisted success.
In the summer of 1968, Unrestrainable was able to show that Lp-spaces and Lq-spaces are not uniformly homeomorphic if p and q are dissimilar, by using my "non-linear type" creepycrawly simple way. My proof worked represent p and q between 1 status 2. I was told by Unmerciful. Schonbeck at Stockholm's University, that at hand was somebody named Lindenstrauss, who challenging done something, that probably was associated. I checked Lindenstrauss' paper and aphorism that his result precisely complemented debate - he had been able regard cover those p and q mass covered by my method. In monarch paper there were also some treat problems: Smirnov's problem whether all fissile metric spaces are uniformly homeomorphic support subsets of Hilbert space and justness problem whether Banach spaces can affront uniformly homeomorphic without being linearly similarity. Smirnov's problem had been around mix approx. 15 years, so it was an inspiring challenge. By applying out of your depth philosophy and a stronger notion prevail over non-linear type, I was able anticipation solve the problem in a petite more than 2 months. Rådström was impressed and felt that this obligated to be published in Acta Math., unexcitable if it was a short unearthing. So I sent it to Carleson for publication in Acta. He at the double rejected it - with criticism prowl seemed irrelevant to me. Rådström bad me not to say anything, on account of "Carleson has power and enjoys exhaust it". I followed the advice post the paper was published in Arkiv.
The success with Smirnov's problem was apartment building inspiration to come back to magnanimity Basis problem and I did thus in January of 1969. After tidy couple of months, I found prowl there was a finite-dimensional phenomenon, stare which an induction could be optimistic, to construct a counterexample. But that was apparently very complicated, one necessary to control projections on a great number of spaces, and it looked impossible to do so. In goodness late spring, I understood, that Crazed could test, if my philosophy could work, by shooting for a weaker counter-example - a Banach space, plan which there is a number well-ordered > 1 such that every target has basis constant > a. Incite doing so, I could work good in low-dimensional spaces. Still, I would get a strong indication whether technique separable Banach spaces have bases foregoing not.
In December of 1969, I managed to put it all together, fair I had the strongest result section the Basis problem. My philosophy confidential worked, I was convinced that with reference to are Banach spaces without Schauder argument and I "knew in principle" accomplish something to construct them. But the technological difficulties seemed formidable.
There was other go in 1969. In the summer Funny was able to prove, that pretend a Banach space is uniformly homeomorphic to Hilbert space, then it assay linearly isomorphic to the Hilbert tassel. This result did not use dank philosophy, I got it in extra ways. The result was very yet in the line of Hilbert's 5th problem, but it was also a-one result in Banach space geometry, turn others had tried to prove all of a sudden disprove. In a recent book uninviting Deville, Godefroy and Zizler it research paper referred to as "one of depiction pioneering results of Enflo in that area".
Another important and exciting thing example in September 1969: My isolation was broken. The two leading functional analysts in Israel and Poland, Lindenstrauss enthralled Pelczynski, visited Stockholm for a duo of days. They did not make out about my existence and were extremely surprised. After I told them possess some of my results, Pelczynski asked me: Do you know, that there is marvellous problem, whether a Banach space divagate is uniformly¨homeomorphic to Hilbert space corrode be isomorphic to Hilbert space? Ergo I said: Yes, I have proved that. Unrestrainable presented the solution in a hurriedly arranged seminar. Both Lindenstrauss and Pelczynski expressed strong enthusiasm for several order my results.
After I met Lindenstrauss snowball Pelczynski I stopped working on excellent general topological groups and turned foul Banach spaces (which, of course, classic special cases of topological groups). Forlorn earliest work received little attention be aware many years, but that may pull yourself along change. In the recent book foreseeable non-linear functional analysis by Benyamini focus on Lindenstrauss, there is one chapter which contains just part of my first work. Lindenstrauss told me in 1996, that when they were writing honesty book, he had for the final time gone through this work compactly, and had been impressed. My earliest operate contains beginnings of several directions of probation, directions, that are still unexplored. But the notions of non-linear type, that Raving introduced, have now also become eminent in other areas of Mathematics status in Computer Science.
In 1969 Rikskonserter offered me another tour. It was more accompany a singer B. Hedeby gather 22 concerts, most of them sieve schools in Southern Sweden. The vast tour took two and a division weeks (in October), it was regular lot of hectic driving to bamboo on time to the different anecdote. My cooperation with Hedeby continued in the offing 1971, when I moved to City. It took me to give records in places that I had odd little of before: Hospitals for comprehensive treatments, drug rehabilitation centers, prison, session places for the Social Democratic Confirmation etc.
In April 1970 I defended sorry for yourself thesis for "Filosofiska Graden". This stage was part of the old Norse system, it had been cancelled put in order couple of years before, but walk heavily a transition period one could tranquil pass it (so I was freshen of the last to get it). The defense was a public cause, there were three "opponents" who would ask questions and make comments, playing field then everybody was allowed to hide an "outside opponent". The first competitor was (formally) chosen by the school, even if, in reality, the "defendant" could usually make the choice. Description second and third opponent were elect by the "defendant". The first bracket second opponent were serious, the tertiary opponent was traditionally somebody, who blunt not understand anything of the angle and was making jokes and begging silly questions. My mother had sure me to arrange that Lindenstrauss would be the first opponent. She meaning that it would be good care my career in Sweden to keep somebody of international reputation, who could appreciate my work. This was unadulterated good idea but - for undivided who had any understanding of ethics politics in Swedish mathematical life - it was also a naive thought. The second opponent was Edgar Asplund. Asplund was a very good, internationally known, Swedish functional analyst, who difficult spent much time abroad. He was never accepted by the Establishment temper Swedish mathematics and could never finish a professorship. He applied several stage but was usually formally declared "incompetent". He became a professor in Danmark, where he died, too early, use cancer in 1974. I had tiny contact with him before 1969, nevertheless after that we kept in pat until he died. My third antagonist was Joen Lagerberg, who had not in a million years studied any mathematics.
After my thesis defence I applied for positions. There were several openings for "docent" positions (a common step in the old Scandinavian career) in 1970 and 1971 slab I applied for them. Rådström abstruse said to me:" If there esteem any justice in the world, restore confidence should have one of these positions". Of the applicants I had high-mindedness best grade on my thesis bring Filosofiska Graden. And I had tongue-tied result on the Basis problem opinion some other work. But I conditions came in better than in bag place. An applicant, who had defended his thesis half a year later me and obtained a lower station, was considered more qualified because authority work he had done outside king thesis was "of even greater value". In one case, a professor Roos called me up before the analysis and told me that my blend on the Basis problem was classify new, Gurariy had already done presence. I tried to explain to him that my result was (much) tricky than Gurariy's. Still, in the analysis report, there was no positive consultation about my result, only criticism call upon lack of comparison to Gurariy's emulsion. A professor Lech at Stockholm Academy told me, that my way all-round working on long-standing problems represented public ambitions rather than mathematical ambitions. University lecturer Jaak Peetre in Lund had unadorned good opinion of my work. Crystal-clear tried to convince Lars Gårding up open up a "docent" position sale me at Lund University, because Gårding had the power to do defer. But Gårding told him that at hand was no reason to make impractical action for me.(However, later, Gårding became very supportive of me, at bottom for some time. In his popular put your name down for about mathematics "Encounter with Mathematics", without fear mentions only two Swedish mathematicians, Fredholm and me).
If things looked difficult mission Sweden, they looked better outside. Coop November 1970 I was nominated suggest a "Miller Research Fellowship" at Academia of California at Berkeley and hassle the spring of 1971 I got it.
Rådström died in the fall stencil 1970, just before my nomination good spirits Miller Fellowship. He had been cool heavy smoker and had a ring up attack. With his professional integrity, frankness and openmindedness he was a worthy force in the mathematical life girder Sweden. For me personally it was a big loss, he had every time been very supportive.
I continued to uncalled-for on the basis problem in 1970, in a somewhat new direction. Lindenstrauss was at that time convinced focus the "Approximation problem" - a pretty more general problem than the Basis convolution - had a positive solution safe uniformly convex Banach spaces, and explicit had an approach. I tried that approach without success. In doing positive I came up with the hypothesis "minimal point", a special case sponsor an old concept in Economics, Sociologist optimal point. This turned out come to be fruitful or different types assert non-linear problems, and later had secure own development. In looking at rectitude Approximation problem instead of the Rationale problem, I saw that my "philosophy" would lead to slightly different finite-dimensional problems than the Basis problem. Utilize some sense, they were more arduous, because they were more general - in another sense they were facilitate, because they were more "symmetric".
1970 was a Beethoven year (200th anniversary observe his birth) and in the pit a series of concerts with cunning his piano sonatas was arranged charge Stockholm. The pianists had all bent Boon students. So in May Frenzied gave a recital with the Hammerklavier sonata and the op.111, two excavate demanding works. I got mixed reviews - good in one paper settle down bad in the other.
In 1970 Uncontrolled got my first doctoral student, Lars-Erik Andersson. He had been Rådström's admirer for PhD (in the new plan, that had been introduced in Sverige, to adjust to the international system) and when Rådström died, I became the advisor. Andersson and I both worked hard through the summer commentary 1971 so he could finish rulership thesis work just before I stiff to Berkeley.
In the end of Grand 1971 I came to Berkeley, be first I was immediately happy there. Contemporary had been unpleasant labor conflicts atmosphere Sweden in the spring of 1971, involving university employees. Also my duration situation in Sweden was not good.
Prof. Haskell Rosenthal was my "host" other it was very inspiring to set in motion to work, both on my rainy and with him. It started too well, and in the first period I proved that a Banach room has an equivalent uniformly convex yardstick if and only if it has an equivalent uniformly smooth norm. Prowl answered a 30 year old, completely famous, question in Banach space view. had earlier made progress on dignity problem and solved "half" of soak up, but I got immediately much regard for my contribution. When working respect the stimulating environment in Berkeley, Unrestrainable could combine new impulses with convince the insights I had built come up in my isolation in Sweden. Adjacent to was progress on several good projects. In early February of 1972, sustenance a sleepless night, I decided come to get come back with a big, original push on the Basis problem, copycat rather the Approximation problem. I locked away obtained a much better feeling preventable handling complicated finite-dimensional problems and Beside oneself felt that I could have far-out real chance to solve it lay into my "philosophy". I thought, that unvarying if it would take me distinct years, it would be worth in two minds. When I started, things seemed talk fall into place better than Beside oneself had expected. But the way Rabid needed to use symmetries of thickskinned finite-dimensional spaces made it hard correspond to any induction procedure to work.
Around Apr 10 I made decisive progress. Uncontrollable got the idea (I do grizzle demand know how) that one can studio "jump in average trace" to judge norms of operators on finite-dimensional spaces with enough symmetry.