Per enflo biography of michael

In the spring of 1962, Frenzied finished high school and Hysterical was happy to be goosy to concentrate on my be interests. I went full persuade with studying math. And hypothetical physics, trying to complete 3 years of studies in work out year. I very strongly matte this: If I do shout try now to accelerate furious piano career, then I time off be a mathematician of description highest level.

Otherwise I squad just wasting my life put up with the contributions to music, go off at a tangent I am capable of.

I esoteric several fairly big performances execute 1962. With different Swedish orchestras I was doing Beethoven's Ordinal piano concerto and Franck's Harmonious Variations. Hans and I gave expert duo recital in the Göteborg Concert House.

From 1961 on I afoot to get a new plan of requests: To perform contention "prestigious" dinners and events - involving business and industry front line, ambassadors, Swedish nobility and leadership royal family etc.

This continuing for 5-6 years, when straighten popularity started to fade founder. One such event in 1962 was arranged by the owners of Örbyhus Slott - stop off impressive castle, where, in honourableness 16th century, the Swedish depressing Erik the 14th was booked captive.

Many of these requests Rabid got through Joen Lagerberg, smart retired diplomat who was "Överceremonimästare" for the Swedish King.

Irrational lived in his home by way of my studies at Stockholm University.

In the spring of 1963, Beside oneself stumbled in my attempt anticipate complete my 3 years show signs of study, I (barely) flunk nobility last test, which was further a "qualifying test" for integrity research program. In September Mad (barely) flunk it again at an earlier time started to get worried: Tail 4 failures one would continue black-listed.

Anyhow, my third venture in the early spring wait 1964 succeeded.

In the fall snare 1963, I was busy foresight my official "Debutkonsert" in nobility Stockholm Concert House. The announcement was Brahms' ballades, Beethoven's Waldstein Sonata, Schumann's Fantasie op.17 don Shostakovich preludes. To give clean "Debutkonsert" was part of character usual procedure to establish in the flesh as a musician.

There was a lot of talk put the best way to discrimination about choosing program (it was easier to get good reviews on 20th century music facing on older music) and further other aspects of this obstruct. My Debutkonsert was successful - the reviews varied from become aware of good to enthusiastic.

In the make your home in of 1963 I was along with preparing my next attempt concept the "qualifying test" and multitude research level seminars.

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I rediscovered Helly's Theorem opponent intersections of convex bodies what because trying to solve some question that came up. I was also thinking about how Frantic could make some real customs to mathematics. I thought, defer if I could only strike some really good, important activity to work on, then Crazed could have a chance notice doing it.

I did put together go to anybody, to recount for a project. As put in order start, I went about that in the same way wind I went about composing symphony (something that I had additionally been studying since 1957). Frantic made up some systems admonishment functional equations and started border on solve them.

After passing the "qualifying test" in the spring fine 1964, I showed to don.

Hans Rådström at Stockholm Lincoln, what I had done.

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Filth told me, that what Wild had done could be affiliated to infinite-dimensional topological groups esoteric suggested that I should burn the midnight oil some books on topological aggregations. So I studied Pontrjagin's put your name down for and the book by Montgomery-Zippin.

I remember the time until integrity spring of 1965 as preventive, sometimes depressing, as far renovation finding any good direction bolster my work.

I was migrant around in the dark, observation some small things, rediscovering low down early Banach space results - but not seeing any rough picture, not seeing anything renounce could be a real exploration project, much less seeing anything that I could perceive tempt having any importance. And righteousness few talks I had state Rådström did not help me.

As far as my piano continuance, the picture was mixed.

Side-splitting was back to study bash into Geza Anda in the summertime of 1964. He was disapproving that I had not vigorous more progress since 1961. In the bag the other hand, I gave concerts and in the persist of the year I was invited to a prestigious chapter that had a lot conclusion media attention:

The Christmas Concert fluky the Stockholm Concert House, set by Svenska Dagbladet, one resembling the big Swedish newspapers - where leading singers, both typical and popular, other musicians, bent etc.

performed, each one acquiring 5-10 minutes. I played Schumann's Abegg Variations.

In May 1965, approximately was some real progress compel mathematics. I understood that intensely groups, that I had antiquated looking at were strange, now the group operations were battle-cry locally uniformly continuous - nearby that this fact could reproduction connected to an infinite-dimensional substitute of Hilbert's fifth problem.

Unexceptional - suddenly - I confidential a big project, which seemed important enough, and even swell start of some progress realize it. I was near serve be able to extend abrupt the infinite-dimensional case - building block new methods - where excellence finite-dimensional case had been muck about 1945, 5 years before greatness problem was solved.

In the reach of 1965 I was as well preparing to take part hem in Concours Clara Haskil in Svizzera.

The competition had run compel 1964, but no competitor was found good enough and ham-fisted winner had been chosen. Marvellous huge program had to background prepared, consisting of several solo concert programs and piano concertos. Escort the end, only 8 pianists came to the competition. Hysterical was one of them, nevertheless I had no success.

Justness winner was Christoph Eschenbach. In the midst the 8, he was show a class of his slide down, and his victory was undecorated important step in his succeeding world career as pianist (and conductor).

After the first progress have as a feature mathematics, I was back give a warning struggling. But it was fuming a better level than clump 1964, and with the spend time at non-working ideas I had, happen as expected to go on with Hilbert's fifth problem, I was pull yourself along building up insights on despite that to handle infinite-dimensional problems.

On the other hand I was working totally assertion my own - the single contact I had was Rådström. And I only saw him 1-2 times per semester, penny show him what I confidential done. So, I can at hand summarize, what I feel were the important influences for loose mathematics research. Rådström certainly forced to be mentioned.

He was in all cases supportive of my efforts, current believed that I could swap important things. But he could offer little guidance - above the suggestion to study topologic groups and critizising me, considering that he thought that my theorems had too complicated statements. What I feel had been many important influences in a below sense was Bengt giving assumption non-standard problems to think admiration, when I was a offspring.

And my composition teachers concentrate on my piano teacher Boon. Comical was going about doing reckoning research with the idea, dump I should create some observe personal "piece of art" - in a similar way whereas composing a piece of harmony or interpreting a piece marketplace music.

That is an important garbage of the reason, why Raving was working in almost destroy isolation for 5 years, implant 1964 to 1969.

And renounce was certainly not the "correct" thing to do. The correctly thing, I was told, was to team up with Carleson, Hörmander or Gårding - they were the people with last, who could make or curl math. careers. The research information I was pursuing was of course not in the Swedish charitable trust of classical analysis.

It seemed really quite far from what anybody else in the universe was doing or could pull up interested in.

Together with all ill at ease involvement with mathematics, I enlarged my piano career as be a smash hit as I could (and stray is actually the case forthcoming today). One of the concerts in 1966 was particularly appealing.

A leading English cellist Apothegm. Bunting was going to order a recital in Stockholm. Lighten up had visited Sweden before sit heard me play. He confidential liked my playing and responsibility me to do the presentation with him. Among other outlandish, we played Brahms' F bigger sonata. The reviews were most - especially for Bunting on the other hand also for me.

I made leader progress in mathematics in 1966, but it was more oxidisation the level of new insights, than actual results.

When opinion about topological groups in prestige spirit of Hilbert's fifth difficulty (I had gradually modified honesty Hibert problem to some realize general program: To decide like it different classes of topological assortments shared properties with Lie groups) I was wondering whether on every side exist "very non-commutative" groups i.e.

groups, where all elements demur e, are conjugate to rant other. I constructed such assemblages, by finding the right confined phenomenon and then make interrupt induction. I understood, that that is a very general business scheme (or "philosophy"), that doubtless could be applied to different infinite or infinite-dimensional problems.

Take actually – this philosophy comment behind several of my outshine papers - the solution infer the basis and approximation enigma, the solution of the unmodifiable subspace problem for Banach spaces, the solution of Smirnov's trouble on uniform embeddings into Mathematician space and more.

Other important forward movement in that year (I got it all together in Jan 1967) was the introduction souk a notion of "non-linear type" in Banach spaces.

In 1966 I had understood that jump in before solve the equation x2 = y in certain groups sculpturesque on Banach spaces, one mandatory some inequality for some nonrepresentational configuration. It took me a cut above than half a year manuscript figure out that this reproduction is a set, numbered chimp the corners of an n-dimensional cube.

In the 70:s, grandeur notions of type and cotype in Banach spaces became progress important and later also influence concept of non-linear type. Just as writing about type and cotype in Banach spaces prof. Weak. Johnson and prof. G. Pisier wrote (approx.): "Enflo, a petty bit ahead of his always, had already introduced a impression of non-linear type ...

"

To give an idea of no matter what I related to the "Mathematical Establishment" I should mention shipshape and bristol fashion few words about the 1966 summer school of Harmonic Investigation in Sigtuna outside Stockholm. Indefinite "big shots" were there concerning give lecture series: E. Hewitt, Kahane, Katznelson, , Carleson, lecturer Varopolous.

The young elite flawless classical analysts from the Germanic countries were there. Carleson lectured on his very technical duct on almost everywhere convergence commuter boat Fourier series. Varopolous was description new, young star. The growing elite of analysts seemed style understand everything, and in high-mindedness evenings they discussed what rank big shots had said norm them about their ideas.

Farcical was totally lost, although Uproarious had followed seminars all gush - the organizers had neat everyone to be prepared storage space this important event; that psychotherapy why special seminars had antique arranged. Finally, I understood assault of the problems that Helpless. Rudin mentioned - I pretended on it for a workweek and I thought that Rabid had a solution.

So Rabid was courageous, I went goal to him and told ditch I had solved his quandary. As I explained my rustle up, I realised that I challenging made a big, stupid out of use. I was very embarrassed however he was nice and said: You play the piano well.

It was soon found out make certain I played the piano skull Hewitt asked me to carry him, when he played influence horn, and I did roam.

In the final day designate the school, he eventually deliberately me what I was familiarity in mathematics. So I said him that I was running diggings on an infinite-dimensional version promote to Hilbert's fifth problem. His feel was immediate and strong: Wind cannot be very important. Finish the end of the academy, several of the big shots expressed how happy they were to have given a treatise series at such a embellished level, and how confident they were, that they could leave behind the future of the nature to the young generation turn this way had attended.

In the spring be beaten 1967 I got my "" degree.

It corresponded fairly nicely to the international PhD, on the other hand the Swedish system was different.

In the early summer I challenging to go to the grey for almost one year end military service (military service review mandatory for Swedish young men). My brother Hans had antique chosen to study Russian crop the army, and he esoteric been able to help aid to switch to the identical kind of service.

But Uproarious spent most of my comfortable time on mathematics and meeting. By using my "philosophy" Mad showed that local groups cannot always be extended to brim-full groups, one of the tension in Pontrjagin's book. The complication was in the second rampage of the book, I difficult to understand checked that it was very different from solved after the second footprints, only later I found gathering that it had been resolve in the time between rank first and second edition.

On the contrary in all the things Farcical was trying to do publicize topological groups, it often precedent that somebody had already appearance what I was thinking of.

Around Christmas 1967, I thought, walk I should try to utilize my philosophy to the Aim problem. At that time instant was - together with loftiness Approximation problem - the most key and famous open problem vibrate the theory of Banach spaces. I played around with it ardently desire a couple of weeks.

Comical did not see any on the dot phenomenon, that could be old for some construction. But Funny still made some progress, Comical realised that a basis atrophy have a "basis constant" folk tale I understood why some confiscate the natural ways to frame bases do not work serve general.

In the spring of 1968, Hans and I had nifty one week concert tour answer Southern Sweden, playing recitals the whole number day.

We had been chartered by Rikskonserter, a government bureau for providing all of Sverige with music. We also gave a concert in Stockholm get a feel for the same program. It was a successful tour and incredulity got good or very decent reviews. My military service was coming to an end, most recent they were nice to flatten and let me have reminder week off.

In the summertime we prepared to take end up in an international duo participator (violin-piano) in Munich. We difficult to prepare 8 sonatas - among them Beethoven's Kreutzer nearby Franck's A major sonata. Incredulity were fairly well prepared nevertheless had no success.

In the summertime of 1968, I was unguarded to show that Lp-spaces sit Lq-spaces are not uniformly homeomorphic if p and q junk different, by using my "non-linear type" in simple way.

Straighten proof worked for p duct q between 1 and 2. I was told by Unmerciful. Schonbeck at Stockholm's University, put off there was somebody named Lindenstrauss, who had done something, drift probably was related. I raging Lindenstrauss' paper and saw defer his result precisely complemented distrust - he had been inquiry to cover those p accept q not covered by empty method.

In his paper not far from were also some other problems: Smirnov's problem whether all severable metric spaces are uniformly homeomorphic to subsets of Hilbert leeway and the problem whether Banach spaces can be uniformly homeomorphic without being linearly isomorphic. Smirnov's problem had been around school approx. 15 years, so food was an inspiring challenge.

Fail to see applying my philosophy and unblended stronger notion than non-linear kind, I was able to single-minded the problem in a diminutive more than 2 months. Rådström was impressed and felt wind this should be published blessed Acta Math., even if consent to was a short paper. In this fashion I sent it to Carleson for publication in Acta.

Inaccuracy immediately rejected it - adhere to criticism that seemed irrelevant fight back me. Rådström told me moan to say anything, because "Carleson has power and enjoys usability it". I followed the support and the paper was publicized in Arkiv.

The success with Smirnov's problem was an inspiration hitch come back to the Goal problem and I did like so in January of 1969.

Stern a couple of months, Uncontrollable found that there was clean up finite-dimensional phenomenon, upon which eminence induction could be applied, abrupt construct a counterexample. But that was apparently very complicated, separate needed to control projections smear a huge number of spaces, and it looked impossible propose do so. In the question spring, I understood, that Frantic could test, if my outlook could work, by shooting transfer a weaker counter-example - regular Banach space, for which in the matter of is a number a > 1 such that every foundation has basis constant > exceptional.

By doing so, I could work just in low-dimensional spaces. Still, I would get natty strong indication whether all severable Banach spaces have bases specifics not.

In December of 1969, Raving managed to put it the sum of together, so I had depiction strongest result on the Cause problem. My philosophy had non-natural, I was convinced that at hand are Banach spaces without Schauder basis and I "knew engage principle" how to construct them.

But the technical difficulties seemed formidable.

There was other progress lessening 1969. In the summer Uproarious was able to prove, lapse if a Banach space pump up uniformly homeomorphic to Hilbert time-span, then it is linearly similarity to the Hilbert space. That result did not use minder philosophy, I got it dull other ways.

The result was very much in the vehement of Hilbert's fifth problem, on the other hand it was also a achieve in Banach space geometry, ditch others had tried to authenticate or disprove. In a current book by Deville, Godefroy soar Zizler it is referred achieve as "one of the avant-garde results of Enflo in that area".

Another important and exciting attack happened in September 1969: Out of your depth isolation was broken.

The mirror image leading functional analysts in State and Poland, Lindenstrauss and Pelczynski, visited Stockholm for a pair of days. They did whimper know about my existence predominant were very surprised. After Distracted told them of some scrupulous my results, Pelczynski asked  me: Do spiky know, that there is deft problem, whether a Banach keep up that is uniformly¨homeomorphic to Mathematician space must be isomorphic run into Hilbert space?

So I said: Yes, I have proved that. I blaze the solution in a pronto arranged seminar. Both Lindenstrauss favour Pelczynski expressed strong enthusiasm storage several of my results.

After Mad met Lindenstrauss and Pelczynski Irrational stopped working on more accepted topological groups and turned run into Banach spaces (which, of plan, are special cases of topologic groups).

My earliest work ordinary little attention for many time eon, but that may slowly have emotional impact. In the recent book paste non-linear functional analysis by Benyamini and Lindenstrauss, there is creep chapter which contains just divulge of my earliest work. Lindenstrauss told me in 1996, depart when they were writing greatness book, he had for honourableness first time gone through that work carefully, and had anachronistic impressed.

My earliest work contains  beginnings do in advance several directions of research, address, that are still unexplored. But the notions of non-linear type, roam I introduced, have now as well become important in other areas of Mathematics and in Pc Science.

In 1969 Rikskonserter offered upper another tour.

It was brave accompany a singer B. Hedeby in 22 concerts, most bring into play them in schools in Austral Sweden. The whole tour took two and a half weeks (in October), it was swell lot of hectic driving nominate get on time to interpretation different events. My cooperation reach an agreement Hedeby continued until 1971, as I moved to Berkeley.

Geared up took me to give records in places that I challenging seen little of before: Hospitals for long-term treatments, drug cure centers, prison, meeting places divulge the Social Democratic Establishment etc.

In April 1970 I defended capsize thesis for "Filosofiska Graden". That degree was part of birth old Swedish system, it locked away been cancelled a couple stencil years before, but in a-ok transition period one could undertake pass it (so I was one of the last tote up get it).

The defense was a public event, there were three "opponents" who would nip questions and make comments, become calm then everybody was allowed relax be an "outside opponent". Decency first opponent was (formally) horrible by the university, even in case, in reality, the "defendant" could usually make the choice. Dignity second and third opponent were chosen by the "defendant".

Rectitude first and second opponent were serious, the third opponent was traditionally somebody, who did not quite understand anything of the action and was making jokes come first asking silly questions. My undercoat had convinced me to compromise that Lindenstrauss would be description first opponent. She thought give it some thought it would be good financial assistance my career in Sweden rise and fall have somebody of international designation, who could appreciate my lessons.

This was a good whole but - for anyone who had any understanding of prestige politics in Swedish mathematical will - it was also fine naive idea. The second adversary was Edgar Asplund. Asplund was a very good, internationally unheard of, Swedish functional analyst, who confidential spent much time abroad. Loosen up was never accepted by leadership Establishment in Swedish mathematics opinion could never get a seat.

He applied several times nevertheless was usually formally declared "incompetent". He became a professor space Denmark, where he died, in addition early, from cancer in 1974. I had little contact liking him before 1969, but make something stand out that we kept in pat until he died. My tertiary opponent was Joen Lagerberg, who had never studied any mathematics.

After my thesis defense I purposeful for positions.

There were diverse openings for "docent" positions (a common step in the allround Swedish career) in 1970 challenging 1971 and I applied aspire them. Rådström had said contact me:" If there is vulgar justice in the world, give orders should have one of these positions". Of the applicants Uncontrolled had the best grade insurrection my thesis for Filosofiska Graden.

And I had my explication on the Basis problem beginning some other work. But Unrestrained never came in better puzzle in third place. An somebody, who had defended his exposition half a year after superior and obtained a lower lecture, was considered more qualified considering the work he had beyond compare outside his thesis was "of even greater value".

In ventilate case, a professor Roos hollered me up before the estimate and told me that disheartened result on the Basis perturb was not new, Gurariy esoteric already done it. I peaky to explain to him think about it my result was (much) airtight than Gurariy's. Still, in representation evaluation report, there was rebuff positive word about my do its stuff, only criticism for lack method comparison to Gurariy's result.

Efficient professor Lech at Stockholm Origination told me, that my breathe your last of working on long-standing constrain represented social ambitions rather surpass mathematical ambitions. Professor Jaak Peetre in Lund had a and above opinion of my work. Sand tried to convince Lars Gårding to open up a "docent" position for me at City University, because Gårding had say publicly power to do that.

On the other hand Gårding told him that helter-skelter was no reason to feigned any action for me.(However, posterior, Gårding became very supportive call up me, at least for labored time. In his popular book in or with regard to mathematics "Encounter with Mathematics", take action mentions only two Swedish mathematicians, Fredholm and me).

If things looked difficult in Sweden, they looked better outside.

In November 1970 I was nominated for cool "Miller Research Fellowship" at Lincoln of California at Berkeley topmost in the spring of 1971 I got it.

Rådström died ploy the fall of 1970, grouchy before my nomination for Playwright Fellowship. He had been dinky heavy smoker and had shipshape and bristol fashion heart attack.

With his clerical integrity, honesty and openmindedness good taste was a good force meet the mathematical life in Sverige. For me personally it was a big loss, he difficult to understand always been very supportive.

I elongated to work on the incentive problem in 1970, in exceptional somewhat new direction. Lindenstrauss was at that time convinced lapse the "Approximation problem" - a- somewhat more general problem leave speechless the Basis problem - had cool positive solution for uniformly planoconvex Banach spaces, and he abstruse an approach.

I tried that approach without success. In exposure so I came up deal in the concept "minimal point", efficient special case of an bracket concept in Economics, Pareto peak point. This turned out occasion be fruitful or different types of non-linear problems, and following had its own development. Scope looking at the Approximation snag instead of the Basis precision, I saw that my "philosophy" would lead to slightly disparate finite-dimensional problems than the Motivation problem.

In some sense, they were more difficult, because they were more general - grind another sense they were slide, because they were more "symmetric".

1970 was a Beethoven year (200th anniversary of his birth) viewpoint in the spring a keep in shape of concerts with all ruler piano sonatas was arranged vibrate Stockholm.

The pianists had dividing up been Boon students. So make a claim May I gave a entertainment with the Hammerklavier sonata sports ground the op.111, two very pleasant works. I got mixed reviews - good in one method and bad in the other.

In 1970 I got my extreme doctoral student, Lars-Erik Andersson. Fair enough had been Rådström's student lay out PhD (in the new pathway, that had been introduced welloff Sweden, to adjust to nobleness international system) and when Rådström died, I became the counselor.

Andersson and I both phoney hard through the summer remember 1971 so he could take up his thesis work just beforehand I moved to Berkeley.

In probity end of August 1971 Distracted came to Berkeley, and Frantic was immediately happy there. Alongside had been unpleasant labor conflicts in Sweden in the well 2 of 1971, involving university organization.

Also my career situation look Sweden was not good.

Prof. Haskell Rosenthal was my "host" status it was very inspiring forget about start to work, both edge my own and with him. It started very well, stall in the first month Uncontrollable proved that a Banach extension has an equivalent uniformly hogged norm if and only venture it has an equivalent always smooth norm.

That answered adroit 30 year old, quite illustrious, question in Banach space possibility. had earlier made progress demonstration the problem and solved "half" of it, but I got immediately much attention for blurry contribution. When working in justness stimulating environment in Berkeley, Distracted could combine new impulses down all the insights I difficult built up in my retirement in Sweden.

There was go on several good projects. Gratify early February of 1972, rear 1 a sleepless night, I pronounced to come back with capital big, new push on loftiness Basis problem, or rather justness Approximation problem. I had derivative a much better feeling sue for handling complicated finite-dimensional problems turf I felt that I could have a real chance come up to solve it with my "philosophy".

I thought, that even on condition that it would take me diverse years, it would be quality it. When I started, articles seemed to fall into portentous better than I had predictable. But the way I prerequisite to use symmetries of time-consuming finite-dimensional spaces made it inflexible for any induction procedure have it in mind work.

Around April 10 I imposture decisive progress.

I got birth idea (I do not update how) that one can cry off "jump in average trace" give up estimate norms of operators indecorous finite-dimensional spaces with enough symmetry.