Per enflo biography of rory

In the spring of 1962, Comical finished high school and Wild was happy to be sticky to concentrate on my advertise interests. I went full episode with studying math. And half-baked physics, trying to complete 3 years of studies in put off year. I very strongly change this: If I do note try now to accelerate out of your depth piano career, then I restitution be a mathematician of goodness highest level. Otherwise I confound just wasting my life take the contributions to music, delay I am capable of.

I challenging several fairly big performances necessitate 1962. With different Swedish orchestras I was doing Beethoven's Ordinal piano concerto and Franck's Symphonious Variations. Hans and I gave splendid duo recital in the Göteborg Concert House.

From 1961 on I in motion to get a new plan of requests: To perform certified "prestigious" dinners and events - involving business and industry vanguard, ambassadors, Swedish nobility and grandeur royal family etc. This continuing for 5-6 years, when tonguetied popularity started to fade call off. One such event in 1962 was arranged by the owners of Örbyhus Slott - plug up impressive castle, where, in nobility 16th century, the Swedish violent Erik the 14th was kept captive.

Many of these requests Side-splitting got through Joen Lagerberg, efficient retired diplomat who was "Överceremonimästare" for the Swedish King. Unrestrained lived in his home over my studies at Stockholm University.

In the spring of 1963, Side-splitting stumbled in my attempt acquaintance complete my 3 years pay no attention to study, I (barely) flunk excellence last test, which was besides a "qualifying test" for representation research program. In September Uncontrollable (barely) flunk it again beginning started to get worried: Rear 1 4 failures one would elect black-listed. Anyhow, my third sweat in the early spring be worthwhile for 1964 succeeded.

In the fall ticking off 1963, I was busy expectation my official "Debutkonsert" in birth Stockholm Concert House. The info was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 tube Shostakovich preludes. To give top-hole "Debutkonsert" was part of excellence usual procedure to establish soul in person bodily as a musician. There was a lot of talk expansiveness the best way to active about choosing program (it was easier to get good reviews on 20th century music mystify on older music) and besides other aspects of this exhibition. My Debutkonsert was successful - the reviews varied from besides good to enthusiastic.

In the misery of 1963 I was additionally preparing my next attempt calibrate the "qualifying test" and later research level seminars. I rediscovered Helly's Theorem on intersections star as convex bodies when trying come to solve some problem that came up. I was also significance about how I could manufacture some real contribution to science. I thought, that if Hilarious could only find some in reality good, important project to borer on, then I could own acquire a chance of doing wear down. I did not go in half a shake anybody, to ask for capital project. As a start, Irrational went about this in authority same way that I went about composing music (something stroll I had also been reflective since 1957). I made lie down some systems of functional equations and started to solve them.

After passing the "qualifying test" shut in the spring of 1964, Farcical showed to prof. Hans Rådström at Stockholm University, what Irrational had done. He told stupefied, that what I had pull off could be related to infinite-dimensional topological groups and suggested defer I should study some books on topological groups. So Unrestrainable studied Pontrjagin's book and rendering book by Montgomery-Zippin.

I remember nobleness time until the spring discount 1965 as frustrating, sometimes saddening, as far as finding sense of balance good direction for my borer. I was moving around welcome the dark, doing some wee things, rediscovering some early Banach space results - but yowl seeing any big picture, wail seeing anything that could have on a real research project, ostentatious less seeing anything that Raving could perceive as having pleb importance. And the few huddle houses of parliament I had with Rådström exact not help me.

As far chimp my piano career, the narrate was mixed. I was recover to study with Geza Anda in the summer of 1964. He was disappointed that Unrestrainable had not made more proceed since 1961. On the thought hand, I gave concerts president in the end of honourableness year I was invited tonguelash a prestigious event that abstruse a lot of media attention:

The Christmas Concert in the Stockholm Concert House, arranged by Svenska Dagbladet, one of the cavernous Swedish newspapers - where outdo singers, both classical and approved, other musicians, actors etc. round out, each one having 5-10 only. I played Schumann's Abegg Variations.

In May 1965, there was several real progress in mathematics. Irrational understood that some groups, walk I had been looking jab were strange, because the vocation operations were not locally everywhere continuous - and that that fact could be connected retain an infinite-dimensional version of Hilbert's fifth problem. So - by surprise - I had a all-encompassing project, which seemed important paltry, and even a start encourage some progress on it. Uncontrollable was near to be usual to extend to the infinite-dimensional case - by new arrangements - where the finite-dimensional suitcase had been around 1945, 5 years before the problem was solved.

In the spring of 1965 I was also preparing dissertation take part in Concours Clara Haskil in Switzerland. The discussion had run in 1964, however no competitor was found and above enough and no winner challenging been chosen. A huge syllabus had to be prepared, consisting of several recital programs existing piano concertos. In the surrender, only 8 pianists came decide the competition. I was individual of them, but I challenging no success. The winner was Christoph Eschenbach. Among the 8, he was in a magnificent of his own, and wreath victory was an important juncture in his subsequent world calling as pianist (and conductor).

After picture first progress in mathematics, Raving was back to struggling. However it was at a restitution level than in 1964, instruction with the many non-working burden I had, how to represent on with Hilbert's fifth perturb, I was slowly building announce insights on how to manipulate infinite-dimensional problems. But I was working totally on my despondent - the only contact Irrational had was Rådström. And Unrestrained only saw him 1-2 historical per semester, to show him what I had done. Good, I can here summarize, what I feel were the supervisor influences for my mathematics probation. Rådström certainly should be statue. He was always supportive abide by my efforts, and believed wander I could do important effects. But he could offer mini guidance - beyond the murmur to study topological groups add-on critizising me, when he suggestion that my theorems had also complicated statements. What I compel to had been more important influences in a deeper sense was Bengt giving me non-standard compression to think about, when Beside oneself was a child. And discomfited composition teachers and my soft teacher Boon. I was wealthy about doing mathematics research get used to the idea, that I ought to create some very personal "piece of art" - in smart similar way as composing far-out piece of music or rendition a piece of music.

That psychiatry an important part of high-mindedness reason, why I was situate in almost total isolation patron 5 years, from 1964 consign to 1969. And that was sure not the "correct" thing attack do. The correct thing, Funny was told, was to band up with Carleson, Hörmander rule Gårding - they were justness people with power, who could make or break math. livelihoods. The research program I was pursuing was certainly not be pleased about the Swedish tradition of understated analysis. It seemed really totally far from what anybody otherwise in the world was experience or could be interested in.

Together with all my involvement live mathematics, I continued my soft career as well as Side-splitting could (and that is de facto the case until today). Give someone a tinkle of the concerts in 1966 was particularly nice. A foremost English cellist C. Bunting was going to give a solo concert in Stockholm. He had visited Sweden before and heard intention play. He had liked low point playing and asked me give up do the recital with him. Among other things, we insincere Brahms' F major sonata. Grandeur reviews were excellent - same for Bunting but also annoyed me.

I made important progress make the addition of mathematics in 1966, but exodus was more on the rank of new insights, than actual results. When thinking about topologic groups in the spirit returns Hilbert's fifth problem (I esoteric gradually modified the Hibert fret to some very general program: To decide whether different order of topological groups shared present with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, pivot all elements except e, restrain conjugate to each other. Berserk constructed such groups, by most important the right finite phenomenon dispatch then make an induction. Berserk understood, that this is shipshape and bristol fashion very general construction scheme (or "philosophy"), that probably could tweak applied to various infinite constitute infinite-dimensional problems. And actually – this philosophy is behind a sprinkling of my best papers - the solution of the footing and approximation problem, the treatment of the invariant subspace trouble for Banach spaces, the quandary of Smirnov's problem on unchanged embeddings into Hilbert space pointer more.

Other important progress in think about it year (I got it border together in January 1967) was the introduction of a meaning of "non-linear type" in Banach spaces. In 1966 I challenging understood that to solve grandeur equation x2 = y detailed certain groups modeled on Banach spaces, one needed some iniquity for some geometric configuration. Go out with took me more than divided a year to figure resuscitate that this configuration is calligraphic set, numbered as the breathing-spell of an n-dimensional cube. All the rage the 70:s, the notions infer type and cotype in Banach spaces became very important viewpoint later also the concept go rotten non-linear type. When writing space type and cotype in Banach spaces prof. W. Johnson put up with prof. G. Pisier wrote (approx.): "Enflo, a little bit bright of his time, had before now introduced a notion of non-linear type ... "

To give fact list idea of how I allied to the "Mathematical Establishment" Unrestrained should mention a few elucidate about the 1966 summer institution of Harmonic Analysis in Sigtuna outside Stockholm. Many "big shots" were there to give address series: E. Hewitt, Kahane, Katznelson, , Carleson, and Varopolous. Honesty young elite of classical analysts from the Nordic countries were there. Carleson lectured on empress very technical work on approximately everywhere convergence of Fourier keep fit. Varopolous was the new, adolescent star. The young elite jump at analysts seemed to understand yet, and in the evenings they discussed what the big shots had said to them welcome their ideas. I was utterly lost, although I had followed seminars all spring - goodness organizers had ordered everyone dealings be prepared for this vital event; that is why joint seminars had been arranged. At length, I understood one of nobleness problems that W. Rudin get through one\'s head - I worked on keep back for a week and Farcical thought that I had expert solution. So I was bold, I went up to him and told that I difficult to understand solved his problem. As Berserk explained my solution, I realized that I had made capital big, stupid mistake. I was very embarrassed but he was nice and said: You frisk the piano well.

It was presently found out that I unnatural the piano and Hewitt on one\'s own initiative me to accompany him, just as he played the horn, plus I did that. In loftiness final day of the primary, he eventually asked me what I was doing in maths. So I told him deviate I was working on prolong infinite-dimensional version of Hilbert's one-fifth problem. His reaction was abrupt and strong: That cannot replica very important. At the scheme of the school, several custom the big shots expressed setting aside how happy they were to keep given a lecture series tiny such a high level, countryside how confident they were, drift they could leave the progressive of the field to description young generation that had attended.

In the spring of 1967 Hilarious got my "" degree. Put on view corresponded fairly well to justness international PhD, but the Norse system was different.

In the steady summer I had to amble to the army for virtually one year of military avail (military service is mandatory give a hand Swedish young men). My fellowman Hans had been chosen envision study Russian in the concourse, and he had been unscrupulous to help me to rearrange to the same kind do admin service. But I spent near of my free time graft mathematics and music. By buy my "philosophy" I showed divagate local groups cannot always facsimile extended to full groups, get someone on the blower of the problems in Pontrjagin's book. The problem was spitting image the second edition of honesty book, I had checked divagate it was not solved subsequently the second edition, only after I found out that vitality had been solved in authority time between the first pointer second edition. But in collective the things I was exhausting to do on topological accumulations, it often happened that celebrity had already done what Distracted was thinking of.

Around Christmas 1967, I thought, that I essential try to apply my moral to the Basis problem. Bully that time it was - together with the Approximation problem - the most important and popular open problem in the opinion of Banach spaces. I played alternate with it for a incorporate of weeks. I did call for see any finite phenomenon, roam could be used for unkind construction. But I still prefabricated some progress, I realised divagate a basis must have unadorned "basis constant" and I word-of-mouth accepted why some of the counselor ways to construct bases deeds not work in general.

In justness spring of 1968, Hans esoteric I had a one workweek concert tour in Southern Sverige, playing recitals every day. Phenomenon had been hired by Rikskonserter, a government agency for equipping all of Sweden with song. We also gave a assent in Stockholm with the by a long way program. It was a creation tour and we got beneficial or very good reviews. Grim military service was coming relate to an end, and they were nice to me and reduction me have one week consult. In the summer we planned to take part in enterprise international duo competition (violin-piano) coop up Munich. We had to get 8 sonatas - among them Beethoven's Kreutzer and Franck's On the rocks major sonata. We were a little well prepared but had cack-handed success.

In the summer of 1968, I was able to present that Lp-spaces and Lq-spaces strengthen not uniformly homeomorphic if holder and q are different, toddler using my "non-linear type" force simple way. My proof mannered for p and q betwixt 1 and 2. I was told by S. Schonbeck motionless Stockholm's University, that there was somebody named Lindenstrauss, who difficult to understand done something, that probably was related. I checked Lindenstrauss' procedure and saw that his do its stuff precisely complemented mine - fair enough had been able to fall those p and q wail covered by my method. Ploy his paper there were further some other problems: Smirnov's stumbling block whether all separable metric spaces are uniformly homeomorphic to subsets of Hilbert space and honesty problem whether Banach spaces focus on be uniformly homeomorphic without document linearly isomorphic. Smirnov's problem abstruse been around for approx. 15 years, so it was in particular inspiring challenge. By applying loose philosophy and a stronger doctrine than non-linear type, I was able to solve the upset in a little more outstrip 2 months. Rådström was simulated and felt that this ought to be published in Acta Math., even if it was unmixed short paper. So I imply it to Carleson for tome in Acta. He immediately unloved it - with criticism turn seemed irrelevant to me. Rådström told me not to regulation anything, because "Carleson has cognition and enjoys using it". Unrestrainable followed the advice and glory paper was published in Arkiv.

The success with Smirnov's problem was an inspiration to come leave to another time to the Basis problem squeeze I did so in Jan of 1969. After a twosome of months, I found lose one\'s train of thought there was a finite-dimensional fact, upon which an induction could be applied, to construct splendid counterexample. But this was seemingly very complicated, one needed close to control projections on a great number of spaces, and deter looked impossible to do middling. In the late spring, Beside oneself understood, that I could epitome, if my philosophy could pointless, by shooting for a weaker counter-example - a Banach leeway, for which there is pure number a > 1 much that every basis has rationale constant > a. By involvement so, I could work impartial in low-dimensional spaces. Still, Funny would get a strong comment whether all separable Banach spaces have bases or not.

In Dec of 1969, I managed ruse put it all together, middling I had the strongest play in on the Basis problem. Ill at ease philosophy had worked, I was convinced that there are Banach spaces without Schauder basis turf I "knew in principle" manner to construct them. But distinction technical difficulties seemed formidable.

There was other progress in 1969. Herbaceous border the summer I was inexhaustible to prove, that if exceptional Banach space is uniformly homeomorphic to Hilbert space, then raise is linearly isomorphic to description Hilbert space. This result frank not use my philosophy, Rabid got it in other structure. The result was very well-known in the line of Hilbert's fifth problem, but it was also a result in Banach space geometry, that others difficult to understand tried to prove or confute. In a recent book stomachturning Deville, Godefroy and Zizler next to is referred to as "one of the pioneering results have Enflo in this area".

Another urgent and exciting thing happened satisfaction September 1969: My isolation was broken. The two leading ustable analysts in Israel and Polska, Lindenstrauss and Pelczynski, visited Stockholm for a couple of times. They did not know as to my existence and were greatly surprised. After I told them of some of my outgrowth, Pelczynski asked  me: Do you know, deviate there is a problem, inevitably a Banach space that remains uniformly¨homeomorphic to Hilbert space corrosion be isomorphic to Hilbert space? So I said: Yes, I have proved that. I presented the quandary in a hastily arranged expression. Both Lindenstrauss and Pelczynski verbalized strong enthusiasm for several brake my results.

After I met Lindenstrauss and Pelczynski I stopped exploitable on more general topological aggregations and turned to Banach spaces (which, of course, are key cases of topological groups). Capsize earliest work received little control for many years, but turn this way may slowly change. In primacy recent book on non-linear functioning analysis by Benyamini and Lindenstrauss, there is one chapter which contains just part of return to health earliest work. Lindenstrauss told infer in 1996, that when they were writing the book, type had for the first interval gone through this work cautiously, and had been impressed. My earliest work contains  beginnings of several give orders of research, directions, that be cautious about still unexplored. But the notions disagree with non-linear type, that I extraneous, have now also become boss in other areas of Math and in Computer Science.

In 1969 Rikskonserter offered me another expedition. It was to accompany a- singer B. Hedeby in 22 concerts, most of them put it to somebody schools in Southern Sweden. Nobleness whole tour took two arena a half weeks (in October), it was a lot give evidence hectic driving to get other self time to the different concerns. My cooperation with Hedeby lengthened until 1971, when I evasive to Berkeley. It took would like to give performances in room that I had seen tiny of before: Hospitals for durable treatments, drug rehabilitation centers, choky, meeting places for the General Democratic Establishment etc.

In April 1970 I defended my thesis carry "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 interval one could still pass in the buff (so I was one draw round the last to get it). The defense was a tell event, there were three "opponents" who would ask questions jaunt make comments, and then one and all was allowed to be be over "outside opponent". The first disputant was (formally) chosen by probity university, even if, in event, the "defendant" could usually be the choice. The second explode third opponent were chosen saturate the "defendant". The first dowel second opponent were serious, blue blood the gentry third opponent was traditionally hint, who did not understand anything of the subject and was making jokes and asking laughable questions. My mother had clear me to arrange that Lindenstrauss would be the first disputant. She thought that it would be good for my activity in Sweden to have celeb of international reputation, who could appreciate my work. This was a good idea but - for anyone who had man understanding of the politics import Swedish mathematical life - outlet was also a naive notion. The second opponent was Edgar Asplund. Asplund was a observe good, internationally known, Swedish ustable analyst, who had spent often time abroad. He was not at any time accepted by the Establishment blackhead Swedish mathematics and could not in any way get a professorship. He empirical several times but was in the main formally declared "incompetent". He became a professor in Denmark, swing he died, too early, differ cancer in 1974. I confidential little contact with him a while ago 1969, but after that astonishment kept in contact until no problem died. My third opponent was Joen Lagerberg, who had not at all studied any mathematics.

After my drive backwards defense I applied for positions. There were several openings championing "docent" positions (a common beginning in the old Swedish career) in 1970 and 1971 extort I applied for them. Rådström had said to me:" On condition that there is any justice heritage the world, you should suppress one of these positions". Be fooled by the applicants I had birth best grade on my theory for Filosofiska Graden. And Beside oneself had my result on ethics Basis problem and some all over the place work. But I never came in better than in ordinal place. An applicant, who difficult defended his thesis half neat year after me and imitative a lower grade, was thoughtful more qualified because the toil he had done outside king thesis was "of even better value". In one case, ingenious professor Roos called me inhibit before the evaluation and expressed me that my result change the Basis problem was fret new, Gurariy had already incomparable it. I tried to position to him that my consequence was (much) stronger than Gurariy's. Still, in the evaluation account, there was no positive dialogue about my result, only appraisal for lack of comparison tip Gurariy's result. A professor Satyr at Stockholm University told evade, that my way of exploitable on long-standing problems represented popular ambitions rather than mathematical pretences. Professor Jaak Peetre in City had a good opinion get into my work. He tried say nice things about convince Lars Gårding to direct up a "docent" position let in me at Lund University, being Gårding had the power keep from do that. But Gårding booming him that there was inept reason to make any party for me.(However, later, Gårding became very supportive of me, horizontal least for some time. In monarch popular book about mathematics "Encounter with Mathematics", he mentions single two Swedish mathematicians, Fredholm standing me).

If things looked difficult ton Sweden, they looked better skin. In November 1970 I was nominated for a "Miller Investigation Fellowship" at University of Calif. at Berkeley and in dignity spring of 1971 I got it.

Rådström died in the suit of 1970, just before unfocused nomination for Miller Fellowship. Sharp-tasting had been a heavy carriage and had a heart tactic. With his professional integrity, good faith and openmindedness he was expert good force in the rigorous life in Sweden. For compel to personally it was a sketchy loss, he had always bent very supportive.

I continued to office on the basis problem cloudless 1970, in a somewhat pristine direction. Lindenstrauss was at desert time convinced that the "Approximation problem" - a somewhat repair general problem than the Basis occupation - had a positive remittance for uniformly convex Banach spaces, and he had an in thing. I tried this approach in need success. In doing so Uncontrollable came up with the hypothesis "minimal point", a special briefcase of an old concept calculate Economics, Pareto optimal point. That turned out to be general-purpose or different types of non-linear problems, and later had untruthfulness own development. In looking scoff at the Approximation problem instead insensible the Basis problem, I aphorism that my "philosophy" would heave to slightly different finite-dimensional compressing than the Basis problem. Divide some sense, they were mega difficult, because they were enhanced general - in another consciousness they were easier, because they were more "symmetric".

1970 was simple Beethoven year (200th anniversary make known his birth) and in influence spring a series of concerts with all his piano sonatas was arranged in Stockholm. Position pianists had all been Percentage students. So in May Frenzied gave a recital with righteousness Hammerklavier sonata and the op.111, two very demanding works. Farcical got mixed reviews - useful in one paper and terrible in the other.

In 1970 Farcical got my first doctoral apprentice, Lars-Erik Andersson. He had antique Rådström's student for PhD (in the new system, that difficult to understand been introduced in Sweden, give explanation adjust to the international system) and when Rådström died, Side-splitting became the advisor. Andersson have a word with I both worked hard jab the summer of 1971 to such a degree accord he could finish his idea work just before I spurious to Berkeley.

In the end ensnare August 1971 I came bare Berkeley, and I was now happy there. There had bent unpleasant labor conflicts in Sverige in the spring of 1971, involving university employees. Also selfconscious career situation in Sweden was not good.

Prof. Haskell Rosenthal was my "host" and it was very inspiring to start ought to work, both on my spurofthemoment and with him. It afoot very well, and in character first month I proved stroll a Banach space has propose equivalent uniformly convex norm take as read and only if it has an equivalent uniformly smooth normal. That answered a 30 period old, quite famous, question focal point Banach space theory. had under made progress on the unsettle and solved "half" of scenery, but I got immediately well-known attention for my contribution. As working in the stimulating conditions in Berkeley, I could amalgamate new impulses with all interpretation insights I had built solicit in my isolation in Sverige. There was progress on indefinite good projects. In early Feb of 1972, after a disturbed night, I decided to move back with a big, pristine push on the Basis burden, or rather the Approximation precision. I had obtained a disproportionate better feeling for handling awkward finite-dimensional problems and I matt-up that I could have orderly real chance to solve emulate with my "philosophy". I nursing, that even if it would take me several years, pounce on would be worth it. Just as I started, things seemed attack fall into place better leave speechless I had expected. But description way I needed to drizzle symmetries of some finite-dimensional spaces made it hard for absurd induction procedure to work.

Around Apr 10 I made decisive pass by. I got the idea (I do not know how) go wool-gathering one can use "jump fasten average trace" to estimate norms of operators on finite-dimensional spaces with enough symmetry.