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W2171312000 abstract "Courtesy of Miami University (colour online) Nigel Kalton was born in Bromley, Kent, on 20 June 1946. He was the third and last child of Gordon Edelbert Kalton (1903–1971) and Stella Vickery (1911–1981), 12 years younger than his sister Pam (who died of cancer at the early age of 38) and ten years younger than his brother Graham. His paternal grandfather was Gordon Edelbert Kaltenbach (1879–1955), a photographic dealer living in Birmingham. The family changed its name to Kalton at the time of the First World War, when anti-German feeling was extreme, but never did it legally, and hence Nigel's birth certificate bears both names. Both Stella and Gordon Kalton had only limited schooling, although it seems that Nigel's father was good at performing numerical calculations. But Gordon Kalton left school at a young age to help in the family business, which consisted of photographic shops in major cities in England. He met his wife Stella when he was running the London shop, and they married in June 1932. The family business prospered before World War II; however, it was of course hard hit by shortages during and after the war, and the London shop was destroyed during the bombing of the capital. A few years after the war, Gordon Kalton gave up his stake in the business and relied thereafter on a modest income from stocks and shares. Nigel grew up in Bromley, on the outskirts of London, in a small, semi-detached house. His family did not have a telephone, a car, or a television until Nigel was a teenager. Gordon Kalton was a recluse, being extremely hard of hearing. He was also very frugal. However, he did attach a great deal of importance to his children's education. As a result, Nigel, and Graham before him, commuted by train to Dulwich College rather than attend a local grammar school. Dulwich, an extremely well-endowed school (founded in 1619) with extensive playing fields, good science laboratories, and good teachers, had become a magnet school for bright students. Both Graham and Nigel went on to university at a time when only about five per cent of school-leavers did so. They were in fact the first members of their family to obtain higher education. Their sister Pam resisted attempts to persuade her to apply to university. Graham went to the London School of Economics to study mathematical statistics, before engaging in an academic career which led him to a full professorship in Great Britain and then to a distinguished career, continuing to this day, in the United States of America. Nigel's path took him to Cambridge to study mathematics, as described later. When Nigel was very young, Pam and Graham were sometimes given the task of taking their baby brother in his pushchair for walks in the neighbourhood. On one such walk, when Nigel was about two years old, they taught him some lines based on Shakespeare's ‘As You Like It’, with a change made to reflect Gordon's taste for whisky. When they returned home Nigel recited to his father ‘Oh good old man how well in thee appears the ancient vintage of the antique world’. But it was Nigel's unusual mathematical abilities that most distinguished him at an early age. An example of this occurred while Graham was a graduate student at LSE but still living at home for financial reasons. He was working through Kendall and Stuart's two-volume The Advanced Theory of Statistics, and would leave one or other volume at home when going to London. Nigel would read the volume left behind. One evening, Nigel, then around 13 years old, showed Graham a neat, simple derivation of the Poisson distribution as a limiting case of the binomial distribution that he had devised, resulting from his reading of Kendall and Stuart. Shortly after, and while still at Dulwich, Nigel completed his first refereed paper published in the Cambridge Mathematical Gazette, ‘Quadratic forms that are perfect squares’ [1]. As a teenager, Nigel did not have much interest in the usual sports of the public school tradition such as rugby, football, or cricket, but he certainly was an intense chess player, and a member of the Dulwich College chess club, which came second in the nation in the Sunday Times competition in 1964. But then came the time to leave Dulwich College for Cambridge, the right place for an outstanding boy who explained once that ‘from a young age, I was very good at mental arithmetic, and somehow math was my subject. I never really thought to do anything else’. Nigel Kalton entered Trinity College, Cambridge, in 1964. More precisely, he chose to miss out the first year (Mathematical Tripos Part I) and moved directly into the second year. This daring move was certainly appropriate, since Nigel quickly impressed his fellow students by his problem-solving abilities, even in the Cambridge environment, where such talents are not unknown. Peter Goddard, former director of the Institute for Advanced Studies at Princeton, had entered Trinity College in 1963 and thus took the same examinations as Nigel between 1964 and 1966. According to his testimony: ‘Also marked was his ability to absorb new mathematics easily and quickly. He was cheerful and modest without being falsely so: he knew his abilities but he saw no need to base his whole persona on them; rather the reverse. His ability to absorb mathematics and solve difficult problems quickly suited him ideally to the formal Tripos examinations as they were then. Thus, unsurprisingly, he came first by a clear margin in the unofficial orders of merit for the Preliminary Examination (1965) and the Mathematical Tripos Part II (1966)’. It is plain that these mathematical and personal qualities remained Nigel's features throughout his life. Fortunately, his abilities were recognized and he was awarded the G. F. A. Osborn Prize, awarded to the most distinguished second-year mathematician at Trinity College, in 1965. Besides Peter Goddard, contemporary students of Nigel Kalton included Garth Dales, Alexander Davie, Peter Dixon, and Ian Stewart. In Cambridge, Nigel had also met Jennifer Bursey. Jennifer's family tree has been rooted in England for a millennium, since her ancestry goes back to a companion of William the Conqueror named Sirloin de Burcy. The author of these lines had the chance to escort Jenny and Nigel to the church of Dives-sur-Mer (Normandy), where the names of the known companions of William are carved on the wall, and to check that Sirloin de Burcy's name was there. Jenny and Nigel were married in 1969, and they had two children, Neil (born in 1973) and Helen (born in 1976); later there were four grandchildren. Even the most demanding studies allow for some amount of socializing. Most of Nigel's social life was devoted to chess, and a number of recorded games that he played at the time with Raymond Keene (who won the British Chess Championship in 1971) demonstrate that they were of comparable level. Nigel himself tied for the sixth place in the Eastbourne Open in 1966, he represented Cambridge University in the matches against Oxford University in 1967 and 1968, and he won the Major Open in Warwick in 1970. This qualified him for the British Championship in 1971 at Blackpool, won by Ray Keene, where Nigel scored 5/11 (one win, two losses, eight draws) and was ranked 26th among 36 participants, the very best players in the country. This result is impressive enough, given the fact that at that time he was a full-time mathematician. Nigel could have considered a professional career as a chess player, but mathematics remained his first passion. He retired from over-the-board play in 1976, although he played a number of games with the International Email Chess Group between 1993 and 1996. The course ‘Analysis 4’ is one that was offered to second-year students at Cambridge. This course was taught by D. G. H. ‘Ben’ Garling, and was based on a book by Jean Dieudonné called Treatise on Analysis. It probably contributed to Nigel's choice of functional analysis, a subject that was at the time attracting a large number of graduate students at Cambridge, about fifteen, according to Garth Dales’ testimony, with strong weekly seminars. Nigel became a student of Ben Garling in 1967, and wrote under his supervision the thesis entitled ‘Schauder decompositions in locally convex spaces’; this was approved on 11 November 1970. The thesis earned Nigel the Raleigh Prize from Cambridge University. Nigel's mathematical genealogy includes Ben Garling's advisor Frank Smithies, who had introduced functional analysis to Cambridge after studying the distribution theory of Laurent Schwartz. This new functional analysis was sometimes called ‘soft analysis’ as opposed to the ‘hard analysis’ of Hardy and Littlewood, although Frank Smithies was himself a student of Hardy. Á propos, it should be recalled that John Littlewood, the senior Wrangler at Cambridge in 1905, was still active during Nigel's student years, and they sometimes met, although to the best of my knowledge they never had a mathematical conversation together. Ben Garling spent the academic year 1969–1970 at Lehigh University in Bethlehem, Pennsylvania, and Nigel Kalton accompanied him there as a visiting lecturer. This first contact with an American Department made a very positive impression on Nigel. In his own words: ‘At Lehigh during a talk, people would chime in with no air about them. Questions were asked and discussed. No one tried to score off a speaker, and ultimately people had fun talking about mathematics ’. This would have looked relaxing after the fiercely competitive atmosphere of Cambridge, and the remainder of Nigel's career confirms that this new-world attitude had a lasting influence on his choices. Also in 1970, Nigel's first papers appeared (if we rule out his promising early bird from the 1966 Mathematical Gazette); this launched a powerful flow of publications which only death broke off. After returning from the United States, Nigel Kalton spent the year 1970–1971 as a Science Research Council Fellow at Warwick University in Coventry. One of his good friends and collaborators there was Robert Elliott, whose wife Ann had taught Nigel's future wife Jenny at high school in 1962, and the two families became close friends. In 1971, Nigel was appointed as a lecturer at the University College of Swansea, part of the University of Wales. He was to stay there for eight years, until 1979. The Kaltons lived there with two young children on a modest lecturer's salary, and Nigel had no access to funds to allow him to travel to conferences. As a consequence of this relative isolation, he focused in part on somewhat unfashionable topics such as non-locally convex spaces. In this pre-internet era, his reasoning was that the risk was less that such topics would suddenly experience a dramatic overhaul that he would not know of, which would mean his working on a problem that someone had already solved. Anyhow Nigel's talent was so great that his research record quickly became impressive, but despite very strong support from the Department, the University of Swansea failed to promote him to a Readership. Meanwhile, Nigel had been invited to the United States as a Visiting Associate Professor at the University of Illinois in Urbana-Champaign (by N. Tenney Peck, in 1977) and then at Michigan State University, East Lansing (by Joel H. Shapiro, in 1978). Several universities in the United States were interested in offering him a position, and the first to do so was the University of Missouri-Columbia through Dennis Sentilles; Nigel accepted this offer. When the Kaltons settled in Columbia in 1979, the place was still quite provincial, and during his first year there Nigel had the only NSF Grant of the Department. However, according to his testimony, he had ‘jumped at the chance of a job at Missouri-Columbia, because the conditions were so much better and allowed me to pursue my research without impediment’. Nigel was to spend the rest of his life in Columbia, which thanks to his influence would become a major centre in functional analysis, not to mention other fields which also benefited from the Department's rise. He clearly preferred the quiet surroundings provided by a midwest college town to the buzzing and steaming of large cities, and Columbia was a place where he could work in peace and welcome collaborators, such as the author of these lines, who was privileged to share five academic years with him between 1985 and 1997. Hence Nigel Kalton fully became an American faculty member. It should, however, be mentioned that he kept his British citizenship to the end, without ever bothering to seek an American passport. The University of Missouri at Columbia was prompt to realize what a catch Nigel Kalton was: several awards were bestowed upon him, such as the Chancellor's award for outstanding research in the physical and mathematical sciences in 1984 and the Weldon Springs presidential award for research and creativity in 1987. Nigel was named Houchins Professor of Mathematics in 1985 and became a Curator's Professor, the highest position that the University of Missouri can provide, in 1995. However, the Banach Medal awarded to him by the Polish Academy of Sciences in 2004 is surely his most prestigious award. Public recognition is definitively of importance, but maybe not as important as the freedom to ‘pursue research without impediment’. Nigel was left in peace by a wise Department, which valued his research as it deserved. He usually taught graduate courses in functional analysis, and his legendary finals were simply a gathering with the students over a beer in the nearby Heidelberg Pub. I attended some of these finals, where Nigel was sharing ideas and opinions with his students in his usual unassuming way. I suspect that some of these students remained unaware of the true stature of their professor. Fortunately, some of them understood who Nigel was. Adam Bowers, when a post-doctoral fellow in Columbia, attended the last course taught in 2009–2010 and the notes he took will be published shortly as a joint work by Nigel and himself [271]. Nigel Kalton worked extremely fast both to establish his results and to write them down. He typed his articles at high speed with no scrap paper around him, however complex the arguments were. He did not write much on paper, except for some explicit computations, and the whole work took place in his brain. I can actually testify to the quite amazing fact that he was able to solve highly non-trivial problems while talking about a completely different topic. And although he was entirely self-sufficient, he would listen to all those, students or colleagues, who approached him with sensible questions, and his attention would soon induce a drastic change in their mathematical landscape. Nigel seldom read articles or books, and would rather rebuild by himself what he needed in his work. It was indeed his way of saving time. This did not prevent him, however, from being very careful about references: he knew that work had been done by such and such, and he would quote the relevant articles. But the whole theory was in Nigel's mind anyhow. Let me call this a mystery for lack of a better word. Nigel also had a unique ability to use mathematical objects which are sometimes considered marginal, such as non-locally convex spaces or quasi-linear maps, not for their own sake, but as tools for showing spectacular results in main-stream analysis. His problem-solving power was famous and went much beyond answering open questions; he would build the proper framework in which the original problem was to be understood inside out with a collection of related results, and hence would prepare the ground for further work. He usually wrote down his theorems in the greatest generality, certainly not out of pedantry, but simply because his proofs reached this level and he trusted, maybe exceedingly, his reader's ability to find out what the applications were. As a rule, he submitted his articles to relatively modest journals and never attempted to publish in the most famous ones, although his work deserved the best. But it seems that his desire was to make things simple and be left in peace rather than to strive for fame. Vanity was foreign to him. Of course, Nigel Kalton accepted a number of invitations to various universities, including Paris, although that big city was not among his favourite places, and turned down a few, sometimes half-jokingly claiming that unfortunately he had to stay at home since his cats needed him. He attended scores of conferences in North America, Europe, and Australia. However, after being hired by the University of Missouri, he never stayed away from Columbia for more than a semester at a time. Nigel lived a quite well-regulated life there; he was not a morning person, and his routine was rather to work at home in the evening, and frequently well into the night. He would usually come to the Department around lunch time, happily explaining, for example to me, that all of yesterday's questions were solved, and much more. My contribution was to sit down and listen, but Nigel's generosity was such that this invariably resulted in a joint paper. Nigel's working power was impressive, but he also knew how to relax, if not to rest. Racquetball is a popular sport in Columbia and Nigel played it on a regular basis, in his usual rather competitive way. And although his mind was constantly in gear, he was also excellent company, a man of taste who knew how to enjoy good food and fine wine, and besides mathematics a man of culture with a definite interest in historical matters. Hence sharing time with such a friend was both pleasant and instructive. Above all, he was a family man, Jenny's husband for 41 years and a proud father and grandfather. Working under the supervision of a generous first-class mathematician is a PhD student's dream. A steady flow of students found their way to Columbia to turn this dream into reality, namely David Trautman (defense in 1983), Carolyn Eoff (1988), Camino Leranoz (1990), Beata Randrianantoanina (1993), Sik-Chung Tam (1994), Dan Cazacu (1997), Roman Vershynin (1999), Roman Shvidkoy (2002), Mark Hoffman (2003), Jakub Duda (2004), Pierre Portal (2004, joint supervision with Gilles Lancien from Besançon, France), Tamara Kucherenko (2005), Mikhail Ganichev (2006), Simon Cowell (2009). Daniel Fresen, who was Nigel's student in 2010, continued his PhD (defended in 2012) under the supervision of Alexander Koldobsky and Mark Rudelson. In my opinion, it would be fair to augment this list by the crowd of colleagues who benefited from Nigel Kalton's mathematical power, insight, and vision. Some among them gathered to celebrate Nigel's sixtieth birthday at the meeting organized in his honour by Beata and Narcisse Randrianantoanina in Oxford (Ohio). Nigel Kalton suffered a devastating stroke on Sunday, 29 August 2010. He passed away peacefully in his sleep two days later in University Hospital, Columbia, in the presence of his wife Jennifer and his children Neil and Helen. A gathering in his honour was organized on 1 October 2010 in Columbia, to which his family, friends, and colleagues were invited to honour his memory, and following Jenny's wish to celebrate his life. The Notices of the American Mathematical Society devoted an obituary article 〈11〉 to Nigel Kalton with Peter Casazza as Coordinating Editor. Fritz Gesztesy has set up a website to honour Nigel's memory and achievements; this contains, in particular (with the publishers’ permission), his publications (http://kaltonmemorial.missouri.edu). A selection of his articles, with for each one extensive comments by an expert of the field, edited by Fritz Gesztesy, Loukas Grafakos, Igor Verbitzky, and myself, is presently under completion and will be published by Birkhäuser under the title ‘Kalton Selecta’ 〈24〉. None of those who were privileged to know Nigel Kalton will ever forget him. But Nigel was an achiever who always tried and never gave up. He left us his spirited example and his inspiring mathematics, and his will clearly was that research should go on, no matter what. Commenting now on some of his works is a modest attempt to fulfill this wish. Before doing so, I should make it clear that to present every item of Nigel's formidable list of publications is way beyond my abilities, and I apologize to any reader who is unhappy about the lack of comments on his/her favourite among Nigel's theorems. I will simply choose some fields with which I am relatively familiar, and in which his influence is especially important. These selected items should, I hope, give some idea of the width, depth, and scope of Nigel Kalton's contribution to mathematics. Hahn–Banach theorems are cornerstones of functional analysis. But it turns out that non-locally convex spaces show up very naturally in many cases when there is no reason to ‘stop at p = 1 ’ and actually what happens when p < 1 provides precious information on the somewhat more classical locally convex setting. This is an invitation to visit what I suggest we call the Kalton zone: 0 ⩽ p < 1 . This terminology is amply justified by the fact that Nigel Kalton is the undisputed leader on non-locally convex analysis and its uses. ∥ x ∥ > 0 if x ≠ 0 ; ∥ α x ∥ = | α | ∥ x ∥ for all x ∈ X and α ∈ K ; ∥ x + y ∥ ⩽ C ( ∥ x ∥ + ∥ y ∥ ) for all ( x , y ) ∈ X 2 . Here C ⩾ 1 is the modulus of concavity of the quasi-norm. A locally bounded F-space is called a quasi-Banach space. We refer to [97] for an authoritative book on F-spaces. It is clear that the Hahn–Banach theorem is sensitive to local convexity assumptions, to the point where it leads to a characterization [31]: a quasi-Banach space X is locally convex (that is, it is a Banach space) if and only if every continuous linear functional defined on a closed subspace of X has an extension to a continuous linear functional on X. In other words, a quasi-Banach space is a Banach space if and only if the weak and quasi-norm topologies have the same closed subspaces. The proof of this theorem relies on the construction of Markushevich basic sequences (see [203, Propositon 3.4]), obtained by refining Mazur's classical argument. To be able to do this, one needs, however, a weaker topology, even if it is not ‘weak’ in the classical sense. A quasi-Banach space is minimal if it does not have any weaker Hausdorff vector topology. As suggested by Mazur's technique, a separable quasi-Banach space is minimal exactly when it contains no basic sequence. On the other hand, it is shown in [39] that an F-space satisfies the restricted Hahn–Banach extension property (that is, if L is an infinite-dimensional, closed subspace and 0 ≠ x ∈ L , then there exists an infinite-dimensional, closed subspace M of L such that x ∉ M ) if and only if every infinite-dimensional, closed subspace contains a basic sequence. It turns out that quite general assumptions, such as the existence of an equivalent pluri-subharmonic quasi-norm, force the existence of basic sequences. This applies to subspaces of L p for 0 < p < 1 and more generally to all natural quasi-Banach spaces, where ‘natural’ means ‘subspace of a lattice-convex quasi-Banach lattice’. Along these lines, we refer to the articles [200, 216] devoted to quasi-Banach sequence spaces such as l p ( l 1 ) and l 1 ( l p ) (with 0 < p < 1 ), which have a unique unconditional basis up to permutation. However, minimal quasi-Banach spaces do exist, and the following is shown in [151]: there is a quasi-Banach space M which contains a one-dimensional subspace E such that every infinite-dimensional, closed subspace Y of M contains E. In particular, M contains no basic sequence. Indeed, a basic sequence would provide a decreasing sequence of infinite-dimensional, closed subspaces with intersection equal to { 0 } , and this cannot be the case if they all contain E. Thus M is minimal. The reader may find it amusing to think of that space as a book: it has many pages but they all meet on the one-dimensional binding. The proof of this theorem is the culmination of several works, which we now outline. Suppose that X and Y are quasi-Banach spaces. Then Y is an extension of X by E if Y / E ≃ X ; when E is one-dimensional this extension is said to be Minimal. The reader should be warned that the word ‘minimal’ is used here with two different meanings, and to prevent confusion we shall use an initial capital letter to denote extensions by a one-dimensional space E. A Minimal extension is usually not a minimal quasi-Banach space, but the above theorem asserts that this may happen. A Minimal extension is said to be trivial if it splits, that is, if Y ≃ X ⊕ E . The case where X is actually a Banach space is important, and indeed Kalton [55], Ribe 〈51〉, and Roberts 〈53〉 independently constructed non-trivial Minimal extensions of X = ℓ 1 , thus solving negatively the three-space problem for local convex spaces. A minimal quasi-Banach space M is a rather strange object, since every one-to-one continuous linear map from M into a Hausdorff topological vector space is actually an isomorphism onto its range. However, existing examples are ‘non-isotropic’, in the sense that they contain a distinguished line, namely the orthogonal complement of the dual space. It in not known whether an even stranger example exists which would exhibit this behaviour everywhere, that is, is there a quasi-Banach space which contains no infinite-dimensional, proper closed subspace? Note that an algebraic complement of E in Kalton's space M is a quasi-normed space with the Hahn–Banach extension property that is not locally convex, and hence the characterization from [31] requires completeness. The Ribe space, for instance, is a non-trivial Minimal extension of ℓ 1 . However, there exist infinite-dimensional quasi-Banach spaces X which are such that every Minimal extension of X is trivial. Such spaces are called K−spaces in [59], and it is shown in [55] that, for 0 < p < 1 , the spaces ℓ p and L p are K-spaces, from which it follows, in particular, that L p / E is not isomorphic to L p [59], whenever E is a one-dimensional subspace of L p . Some Banach spaces are K-spaces: it is shown in [84] that every quotient space of an L ∞ -space is a K-space, and in [55] that a Banach space with non-trivial type is a K-space. In fact, Kalton conjectured that a Banach space is a K-space exactly when its dual space has non-trivial cotype. Minimal extensions of L ∞ -spaces are trivial, in other words, all quasi-linear maps on ‘cubes’ are close to linear ones. This creates a link between this field and the Maharam problem, explored by Nigel Kalton and Jim Roberts, who showed, in particular, that the existence of a control measure is equivalent to the uniform exhaustivity of the given sub-measure. The Maharam problem has been solved negatively in 〈59〉; this shows, in particular, that the Kalton–Roberts theorem is optimal. For sake of brevity, we simply state the following from [84]. Theorem 5.1.There is a universal constant K such that, if Σ is an algebra of subsets of some set Ω and φ : Σ → R is a set function such that The best value of K belongs to the interval [ 3 / 2 , 45 ] , but its precise value seems to be unknown. Let us conclude this section with Nigel's investigations on the fundamental theorem of calculus for functions which take values in F-spaces. He shows in [73] that, if X is an F-space with trivial dual and x ∈ X , then there exists an X-valued dyadic martingale ( u n ) with u 0 = x which converges uniformly to 0. It follows that, under these assumptions on X, there exists a non-constant Lipschitz function from [ 0 , 1 ] to X whose derivative vanishes identically [73]. This result is used in [157], where Nigel, answering a question of Popov 〈47〉, shows that, if X is a quasi-Banach space with trivial dual, every continuous function from [ 0 , 1 ] to X has a primitive. This result contrasts with 〈2〉, where it is proved that, if X is a non-locally convex quasi-Banach space with separating dual, then there is an X-valued continuous function which fails to have a primitive. An important feature of Nigel's contribution to non-linear geometry concerns embeddings of special metric graphs into Banach spaces and ‘concentration results’ when the target space satisfies certain properties. Such investigations were motivated, in particular, by attempts to attack the Novikov conjecture by relating the geometry of groups with coarse embeddings into Hilbert spaces or super-reflexive spaces 〈34, 67〉. Such embedding results are found in [240], where the Kalton–Randrianarivony graphs G k ( M ) (increasing sequences of integers of length k equipped with a weighted Hamming distance) are used to show that the space l p 1 ⊕ l p 2 ⊕ ⋯ ⊕ l p n is determined by its nets (in particular, by its uniform structure) provided that 1 < p i < ∞ for all i. Nigel also used these graphs (equipped this time with the graph distance, where two sequences are adjacent if they interlace) to show that, if c 0 coarsely embeds into a Banach space X, then one of the iterated duals is non-separable, and, in particular, X is not reflexive [229], although c 0 embeds uniformly and coarsely into a Banach space with the Schur property [213]. On the other hand, any stable metric space (where ‘stable’ means that the order of limits can be permuted in lim k lim n d ( x k , y n ) whenever both limits exist) can be coarsely embedded into a reflexive space [229]. The line of thought that was opened in [229] leads to the property denoted Q by Kalton, a necessary condition for coarse embeddability into a reflexive space that could possibly be sufficient as well. The interlacing distance w" @default.
W2171312000 created "2016-06-24" @default.
W2171312000 creator A5074278380 @default.
W2171312000 date "2014-11-15" @default.
W2171312000 modified "2023-10-18" @default.
W2171312000 title "Obituary: Nigel John Kalton, 1946-2010" @default.
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