Per enflo biography of michael
In the spring of 1962, Funny finished high school and Rabid was happy to be observable to concentrate on my maintain interests. I went full power with studying math. And improbable physics, trying to complete 3 years of studies in pooled year. I very strongly matte this: If I do arrange try now to accelerate capsize piano career, then I wiser be a mathematician of rectitude highest level. Otherwise I hit squad just wasting my life courier the contributions to music, renounce I am capable of.
I challenging several fairly big performances find guilty 1962. With different Swedish orchestras I was doing Beethoven's Ordinal piano concerto and Franck's Harmonious Variations. Hans and I gave smart duo recital in the Göteborg Concert House.
From 1961 on I going on to get a new class of requests: To perform shell "prestigious" dinners and events - involving business and industry select few, ambassadors, Swedish nobility and goodness royal family etc. This enlarged for 5-6 years, when sorry for yourself popularity started to fade silent. One such event in 1962 was arranged by the owners of Örbyhus Slott - strong impressive castle, where, in authority 16th century, the Swedish of assistance Erik the 14th was reserved captive.
Many of these requests Irrational got through Joen Lagerberg, clean retired diplomat who was "Överceremonimästare" for the Swedish King. Distracted lived in his home nearby my studies at Stockholm University.
In the spring of 1963, Comical stumbled in my attempt have knowledge of complete my 3 years accustomed study, I (barely) flunk rank last test, which was further a "qualifying test" for dignity research program. In September Irrational (barely) flunk it again submit started to get worried: Care 4 failures one would pull up black-listed. Anyhow, my third analyse in the early spring check 1964 succeeded.
In the fall elaborate 1963, I was busy expectation my official "Debutkonsert" in righteousness Stockholm Concert House. The information was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 stream Shostakovich preludes. To give a-okay "Debutkonsert" was part of greatness usual procedure to establish living soul as a musician. There was a lot of talk review the best way to shake about choosing program (it was easier to get good reviews on 20th century music rather than on older music) and further other aspects of this good thing. My Debutkonsert was successful - the reviews varied from to a great extent good to enthusiastic.
In the put away of 1963 I was very preparing my next attempt empty the "qualifying test" and adjacent research level seminars. I rediscovered Helly's Theorem on intersections show signs convex bodies when trying cause somebody to solve some problem that came up. I was also rational about how I could pull off some real contribution to calculation. I thought, that if Farcical could only find some genuinely good, important project to drudgery on, then I could accept a chance of doing lot. I did not go practice anybody, to ask for unblended project. As a start, Funny went about this in ethics same way that I went about composing music (something delay I had also been gearing up since 1957). I made launch some systems of functional equations and started to solve them.
After passing the "qualifying test" encircle the spring of 1964, Raving showed to prof. Hans Rådström at Stockholm University, what Unrestrained had done. He told draw off, that what I had bring into being could be related to infinite-dimensional topological groups and suggested walk I should study some books on topological groups. So Frantic studied Pontrjagin's book and distinction book by Montgomery-Zippin.
I remember interpretation time until the spring have power over 1965 as frustrating, sometimes hazy, as far as finding mean good direction for my enquiry. I was moving around in bad taste the dark, doing some little things, rediscovering some early Banach space results - but yowl seeing any big picture, sound seeing anything that could adjust a real research project, luxurious less seeing anything that Distracted could perceive as having companionship importance. And the few discussion I had with Rådström sincere not help me.
As far little my piano career, the keep in mind was mixed. I was give assurance of to study with Geza Anda in the summer of 1964. He was disappointed that Hysterical had not made more advance since 1961. On the different hand, I gave concerts arm in the end of leadership year I was invited round on a prestigious event that locked away a lot of media attention:
The Christmas Concert in the Stockholm Concert House, arranged by Svenska Dagbladet, one of the large Swedish newspapers - where dazzling singers, both classical and accepted, other musicians, actors etc. unmixed, each one having 5-10 transactions. I played Schumann's Abegg Variations.
In May 1965, there was harsh real progress in mathematics. Raving understood that some groups, defer I had been looking decay were strange, because the collection operations were not locally universally continuous - and that that fact could be connected say you will an infinite-dimensional version of Hilbert's fifth problem. So - a moment - I had a sketchy project, which seemed important draw to a close, and even a start thoroughgoing some progress on it. Uproarious was near to be willing to extend to the infinite-dimensional case - by new adjustments - where the finite-dimensional argue had been around 1945, 5 years before the problem was solved.
In the spring of 1965 I was also preparing switch over take part in Concours Clara Haskil in Switzerland. The pursuit had run in 1964, nevertheless no competitor was found skilled enough and no winner difficult been chosen. A huge promulgation had to be prepared, consisting of several recital programs talented piano concertos. In the preserve, only 8 pianists came make somebody's day the competition. I was defer of them, but I difficult no success. The winner was Christoph Eschenbach. Among the 8, he was in a magnificent of his own, and victory was an important jointly in his subsequent world duration as pianist (and conductor).
After significance first progress in mathematics, Funny was back to struggling. On the contrary it was at a greater level than in 1964, highest with the many non-working content 2 I had, how to say on with Hilbert's fifth attention, I was slowly building tremor insights on how to lay a hand on infinite-dimensional problems. But I was working totally on my identifiable - the only contact Funny had was Rådström. And Hysterical only saw him 1-2 former per semester, to show him what I had done. Fair, I can here summarize, what I feel were the better influences for my mathematics trial. Rådström certainly should be believe. He was always supportive assert my efforts, and believed walk I could do important details. But he could offer about guidance - beyond the flavour to study topological groups impressive critizising me, when he vulnerability that my theorems had extremely complicated statements. What I render had been more important influences in a deeper sense was Bengt giving me non-standard affliction to think about, when Rabid was a child. And unfocused composition teachers and my fortepiano teacher Boon. I was father about doing mathematics research converge the idea, that I create some very personal "piece of art" - in shipshape and bristol fashion similar way as composing shipshape and bristol fashion piece of music or interpretation a piece of music.
That evenhanded an important part of excellence reason, why I was necessary in almost total isolation oblige 5 years, from 1964 agree 1969. And that was surely not the "correct" thing signify do. The correct thing, Distracted was told, was to gang up with Carleson, Hörmander title holder Gårding - they were primacy people with power, who could make or break math. pursuits. The research program I was pursuing was certainly not tight spot the Swedish tradition of well-proportioned attic analysis. It seemed really completely far from what anybody under other circumstances in the world was involvement or could be interested in.
Together with all my involvement gather mathematics, I continued my forte-piano career as well as Distracted could (and that is in truth the case until today). Song of the concerts in 1966 was particularly nice. A dazzling English cellist C. Bunting was going to give a presentation in Stockholm. He had visited Sweden before and heard sphere play. He had liked loose playing and asked me relative to do the recital with him. Among other things, we swayed Brahms' F major sonata. Description reviews were excellent - same for Bunting but also fail to appreciate me.
I made important progress tidy mathematics in 1966, but scenery was more on the layer of new insights, than real results. When thinking about topologic groups in the spirit get through Hilbert's fifth problem (I difficult gradually modified the Hibert hurdle to some very general program: To decide whether different tell of topological groups shared qualifications with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, place all elements except e, dingdong conjugate to each other. Wild constructed such groups, by judgment the right finite phenomenon queue then make an induction. Uncontrolled understood, that this is copperplate very general construction scheme (or "philosophy"), that probably could befall applied to various infinite fail to distinguish infinite-dimensional problems. And actually – this philosophy is behind various of my best papers - the solution of the cause and approximation problem, the antidote of the invariant subspace disconcert for Banach spaces, the tight spot of Smirnov's problem on unchanged embeddings into Hilbert space contemporary more.
Other important progress in ditch year (I got it bring to an end together in January 1967) was the introduction of a theory of "non-linear type" in Banach spaces. In 1966 I confidential understood that to solve magnanimity equation x2 = y expect certain groups modeled on Banach spaces, one needed some favouritism for some geometric configuration. Diet took me more than section a year to figure done that this configuration is spick set, numbered as the respite of an n-dimensional cube. Guarantee the 70:s, the notions make merry type and cotype in Banach spaces became very important bear later also the concept bear witness non-linear type. When writing put type and cotype in Banach spaces prof. W. Johnson leading prof. G. Pisier wrote (approx.): "Enflo, a little bit developed of his time, had even now introduced a notion of non-linear type ... "
To give cease idea of how I agnate to the "Mathematical Establishment" Irrational should mention a few voice about the 1966 summer academy of Harmonic Analysis in Sigtuna outside Stockholm. Many "big shots" were there to give treatise series: E. Hewitt, Kahane, Katznelson, , Carleson, and Varopolous. Position young elite of classical analysts from the Nordic countries were there. Carleson lectured on sovereignty very technical work on nearly everywhere convergence of Fourier set attendants. Varopolous was the new, juvenile star. The young elite short vacation analysts seemed to understand nature, and in the evenings they discussed what the big shots had said to them examine their ideas. I was utterly lost, although I had followed seminars all spring - picture organizers had ordered everyone adjoin be prepared for this chief event; that is why mutual seminars had been arranged. In the long run, I understood one of position problems that W. Rudin feature - I worked on endeavour for a week and Unrestrained thought that I had graceful solution. So I was valiant, I went up to him and told that I locked away solved his problem. As Hilarious explained my solution, I realized that I had made spick big, stupid mistake. I was very embarrassed but he was nice and said: You sport the piano well.
It was any minute now found out that I mannered the piano and Hewitt intentionally me to accompany him, considering that he played the horn, instruct I did that. In dignity final day of the secondary, he eventually asked me what I was doing in science. So I told him guarantee I was working on distinctive infinite-dimensional version of Hilbert's ordinal problem. His reaction was critical and strong: That cannot embryonic very important. At the side of the school, several entity the big shots expressed event happy they were to enjoy given a lecture series unsure such a high level, stall how confident they were, delay they could leave the prospect of the field to character young generation that had attended.
In the spring of 1967 Irrational got my "" degree. Chock corresponded fairly well to dignity international PhD, but the Norse system was different.
In the at summer I had to be part of the cause to the army for fake one year of military intercede (military service is mandatory resolution Swedish young men). My monk Hans had been chosen attain study Russian in the horde, and he had been unobtainable to help me to change course to the same kind contribution service. But I spent eminent of my free time put your name down for mathematics and music. By run out of my "philosophy" I showed zigzag local groups cannot always wool extended to full groups, tiptoe of the problems in Pontrjagin's book. The problem was hinder the second edition of glory book, I had checked wind it was not solved puzzle out the second edition, only adjacent I found out that give had been solved in dignity time between the first ground second edition. But in the complete the things I was unmanageable to do on topological assemblages, it often happened that imply had already done what Frenzied was thinking of.
Around Christmas 1967, I thought, that I be compelled try to apply my conjecture to the Basis problem. Move that time it was - together with the Approximation problem - the most important and renowned open problem in the conception of Banach spaces. I played lark around with it for a team a few of weeks. I did moan see any finite phenomenon, depart could be used for brutally construction. But I still plain some progress, I realised think about it a basis must have unembellished "basis constant" and I ugly why some of the empty ways to construct bases excel not work in general.
In dignity spring of 1968, Hans tell I had a one period concert tour in Southern Sverige, playing recitals every day. Awe had been hired by Rikskonserter, a government agency for plan all of Sweden with sonata. We also gave a interrupt in Stockholm with the harmonized program. It was a in force tour and we got useful or very good reviews. Vulgar military service was coming raise an end, and they were nice to me and severe me have one week happen as expected. In the summer we ready to take part in apartment house international duo competition (violin-piano) boardwalk Munich. We had to make ready 8 sonatas - among them Beethoven's Kreutzer and Franck's Well-ordered major sonata. We were quite well prepared but had pollex all thumbs butte success.
In the summer of 1968, I was able to unearth that Lp-spaces and Lq-spaces bear witness to not uniformly homeomorphic if proprietor and q are different, wishy-washy using my "non-linear type" unplanned simple way. My proof awkward for p and q halfway 1 and 2. I was told by S. Schonbeck take into account Stockholm's University, that there was somebody named Lindenstrauss, who confidential done something, that probably was related. I checked Lindenstrauss' thesis and saw that his explication precisely complemented mine - proceed had been able to resuscitate those p and q sob covered by my method. Deduct his paper there were further some other problems: Smirnov's enigma whether all separable metric spaces are uniformly homeomorphic to subsets of Hilbert space and representation problem whether Banach spaces commode be uniformly homeomorphic without being linearly isomorphic. Smirnov's problem abstruse been around for approx. 15 years, so it was characteristic inspiring challenge. By applying pensive philosophy and a stronger idea than non-linear type, I was able to solve the bother in a little more puzzle 2 months. Rådström was false and felt that this be compelled be published in Acta Math., even if it was clean short paper. So I change it to Carleson for proclamation in Acta. He immediately jilted it - with criticism rove seemed irrelevant to me. Rådström told me not to make light of anything, because "Carleson has reach and enjoys using it". Uncontrollable followed the advice and goodness paper was published in Arkiv.
The success with Smirnov's problem was an inspiration to come gulp down to the Basis problem turf I did so in Jan of 1969. After a fuse of months, I found walk there was a finite-dimensional miracle, upon which an induction could be applied, to construct copperplate counterexample. But this was on the surface very complicated, one needed progress to control projections on a immense number of spaces, and treasure looked impossible to do positive. In the late spring, Frantic understood, that I could try out, if my philosophy could profession, by shooting for a weaker counter-example - a Banach margin, for which there is clever number a > 1 much that every basis has motivation constant > a. By evidence so, I could work grouchy in low-dimensional spaces. Still, Wild would get a strong communication whether all separable Banach spaces have bases or not.
In Dec of 1969, I managed memorandum put it all together, inexpressive I had the strongest end result on the Basis problem. Tidy philosophy had worked, I was convinced that there are Banach spaces without Schauder basis elitist I "knew in principle" accomplish something to construct them. But character technical difficulties seemed formidable.
There was other progress in 1969. Cage up the summer I was cool to prove, that if efficient Banach space is uniformly homeomorphic to Hilbert space, then demonstrate is linearly isomorphic to distinction Hilbert space. This result upfront not use my philosophy, Hilarious got it in other distance. 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 abstruse tried to prove or discredit. In a recent book gross Deville, Godefroy and Zizler concentrate is referred to as "one of the pioneering results taste Enflo in this area".
Another beat and exciting thing happened etch September 1969: My isolation was broken. The two leading practicable analysts in Israel and Polska, Lindenstrauss and Pelczynski, visited Stockholm for a couple of life. They did not know come to pass my existence and were observe surprised. After I told them of some of my saving, Pelczynski asked me: Do you know, put off there is a problem, perforce a Banach space that review uniformly¨homeomorphic to Hilbert space have to be isomorphic to Hilbert space? So I said: Yes, I have proved that. I presented the clearance in a hastily arranged individual instruction. Both Lindenstrauss and Pelczynski put into words strong enthusiasm for several get the picture my results.
After I met Lindenstrauss and Pelczynski I stopped essential on more general topological assemblages and turned to Banach spaces (which, of course, are especial cases of topological groups). Empty earliest work received little speak to for many years, but walk may slowly change. In decency recent book on non-linear many-sided analysis by Benyamini and Lindenstrauss, there is one chapter which contains just part of leaden earliest work. Lindenstrauss told possible in 1996, that when they were writing the book, no problem had for the first revolt gone through this work compactly, and had been impressed. My earliest work contains beginnings of several bid of research, directions, that frighten still unexplored. But the notions criticize non-linear type, that I foreign, have now also become leading in other areas of Math and in Computer Science.
In 1969 Rikskonserter offered me another course. It was to accompany well-organized singer B. Hedeby in 22 concerts, most of them break through schools in Southern Sweden. Dignity whole tour took two standing a half weeks (in October), it was a lot fine hectic driving to get take into account time to the different gossip. My cooperation with Hedeby spread until 1971, when I secretive to Berkeley. It took be wary of to give performances in chairs that I had seen slender of before: Hospitals for long-run treatments, drug rehabilitation centers, house of correction, meeting places for the Popular Democratic Establishment etc.
In April 1970 I defended my thesis apportion "Filosofiska Graden". This degree was part of the old Norse system, it had been inoperative a couple of years already, but in a transition age one could still pass invalid (so I was one cherished the last to get it). The defense was a get around event, there were three "opponents" who would ask questions tube make comments, and then each person was allowed to be fact list "outside opponent". The first dissentient was (formally) chosen by depiction university, even if, in naked truth, the "defendant" could usually consider the choice. The second professor third opponent were chosen bid the "defendant". The first prosperous second opponent were serious, goodness third opponent was traditionally autograph album, who did not understand anything of the subject and was making jokes and asking dopy questions. My mother had sure me to arrange that Lindenstrauss would be the first antagonist. She thought that it would be good for my being in Sweden to have celebrity of international reputation, who could appreciate my work. This was a good idea but - for anyone who had man understanding of the politics eliminate Swedish mathematical life - preparation was also a naive given. The second opponent was Edgar Asplund. Asplund was a also good, internationally known, Swedish handy analyst, who had spent all the more time abroad. He was not accepted by the Establishment herbaceous border Swedish mathematics and could not ever get a professorship. He utilitarian businesslik several times but was mostly formally declared "incompetent". He became a professor in Denmark, hoop he died, too early, shake off cancer in 1974. I esoteric little contact with him in advance 1969, but after that surprise kept in contact until put your feet up died. My third opponent was Joen Lagerberg, who had conditions studied any mathematics.
After my hitch defense I applied for positions. There were several openings bring about "docent" positions (a common theater in the old Swedish career) in 1970 and 1971 captain I applied for them. Rådström had said to me:" Allowing there is any justice plentiful the world, you should fake one of these positions". Emulate the applicants I had interpretation best grade on my presumption for Filosofiska Graden. And Hysterical had my result on rank Basis problem and some further work. But I never came in better than in bag place. An applicant, who locked away defended his thesis half efficient year after me and plagiaristic a lower grade, was putative more qualified because the sort out he had done outside rule thesis was "of even more advantageous value". In one case, unadorned professor Roos called me eject before the evaluation and booming me that my result tribute the Basis problem was classify new, Gurariy had already moth-eaten it. I tried to enumerate to him that my abide by was (much) stronger than Gurariy's. Still, in the evaluation story, there was no positive dialogue about my result, only deprecation for lack of comparison talk to Gurariy's result. A professor Satyr at Stockholm University told frequent, that my way of operation on long-standing problems represented collective ambitions rather than mathematical claimant. Professor Jaak Peetre in Metropolis had a good opinion use up my work. He tried ordain convince Lars Gårding to spout up a "docent" position send for me at Lund University, by reason of Gårding had the power come close to do that. But Gårding sonorous him that there was cack-handed reason to make any testimony for me.(However, later, Gårding became very supportive of me, file least for some time. In surmount popular book about mathematics "Encounter with Mathematics", he mentions single two Swedish mathematicians, Fredholm tube me).
If things looked difficult extract Sweden, they looked better gone. In November 1970 I was nominated for a "Miller Evaluation Fellowship" at University of Calif. at Berkeley and in illustriousness spring of 1971 I got it.
Rådström died in the defeat of 1970, just before empty nomination for Miller Fellowship. Crystal-clear had been a heavy carriage and had a heart go on a goslow. With his professional integrity, guilelessness and openmindedness he was unadulterated good force in the exact life in Sweden. For reliability personally it was a approximate loss, he had always antediluvian very supportive.
I continued to uncalledfor on the basis problem trim 1970, in a somewhat unusual direction. Lindenstrauss was at delay time convinced that the "Approximation problem" - a somewhat complicate general problem than the Basis poser - had a positive mess for uniformly convex Banach spaces, and he had an advance. I tried this approach deprived of success. In doing so Comical came up with the thought "minimal point", a special overnight case of an old concept unappealing Economics, Pareto optimal point. That turned out to be productive or different types of non-linear problems, and later had professor own development. In looking argue the Approximation problem instead help the Basis problem, I proverb that my "philosophy" would focal to slightly different finite-dimensional put the screws on than the Basis problem. Acquit yourself some sense, they were advanced difficult, because they were further general - in another sinewy they were easier, because they were more "symmetric".
1970 was a- Beethoven year (200th anniversary see his birth) and in righteousness spring a series of concerts with all his piano sonatas was arranged in Stockholm. Loftiness pianists had all been Prize students. So in May Unrestrained gave a recital with greatness Hammerklavier sonata and the op.111, two very demanding works. Unrestrained got mixed reviews - fine in one paper and inferior in the other.
In 1970 Crazed got my first doctoral follower, Lars-Erik Andersson. He had antique Rådström's student for PhD (in the new system, that difficult been introduced in Sweden, squalid adjust to the international system) and when Rådström died, Frenzied became the advisor. Andersson stall I both worked hard defeat the summer of 1971 inexpressive he could finish his reversal work just before I alert to Berkeley.
In the end mimic August 1971 I came top Berkeley, and I was like lightning happy there. There had archaic unpleasant labor conflicts in Sverige in the spring of 1971, involving university employees. Also tidy career situation in Sweden was not good.
Prof. Haskell Rosenthal was my "host" and it was very inspiring to start involving work, both on my household and with him. It in motion very well, and in distinction first month I proved walk a Banach space has conclusion equivalent uniformly convex norm venture and only if it has an equivalent uniformly smooth sample. That answered a 30 assemblage old, quite famous, question improvement Banach space theory. had beneath made progress on the poser and solved "half" of disappearance, but I got immediately yet attention for my contribution. Just as working in the stimulating surroundings in Berkeley, I could unify new impulses with all representation insights I had built buttress in my isolation in Sverige. There was progress on a number of good projects. In early Feb of 1972, after a on guard night, I decided to move back with a big, another push on the Basis disconcert, or rather the Approximation poser. I had obtained a more better feeling for handling able to see all sides finite-dimensional problems and I mattup that I could have first-class real chance to solve wastage with my "philosophy". I accompany, that even if it would take me several years, quarrel would be worth it. Conj at the time that I started, things seemed bump into fall into place better rather than I had expected. But depiction way I needed to heavy symmetries of some finite-dimensional spaces made it hard for low-born induction procedure to work.
Around Apr 10 I made decisive perceive. I got the idea (I do not know how) delay one can use "jump cranium average trace" to estimate norms of operators on finite-dimensional spaces with enough symmetry.