{"id":1320,"date":"2017-11-21T19:37:23","date_gmt":"2017-11-21T18:37:23","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1320"},"modified":"2022-05-17T09:38:26","modified_gmt":"2022-05-17T07:38:26","slug":"in-memoriam-klaus-keimel","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1320","title":{"rendered":"In memoriam: Klaus Keimel"},"content":{"rendered":"

Klaus Keimel<\/a> passed away on Saturday, November 18th, 2017. This is sad news.<\/p>\n

I would like to pay homage to him, and I am not sure I will manage to do so in the way he deserves.\u00a0 I cannot retrace everything he has done, but I will try to provide a personal perspective on the man, and on some of the science.<\/p>\n

Domain theorists will probably know him best for the compendium [1], then the red book [2].\u00a0 From what I understand, he had been the driving force behind the two projects.\u00a0 He had also been the man behind the highly successful Domains workshop, of which the 12th and latest edition took place in Cork<\/a> in 2015.\u00a0 His talents were not confined to domain theory, and he had been active in various other fields of mathematics, too, in analysis and algebra for example.\u00a0 He could surprise you by telling you about K-theory, about sheaves, or about traces in C*-algebras.<\/p>\n

First contact<\/h2>\n

My personal experience with him started, or so I thought at least, was when he invited me as an invited speaker to the Domains VIII workshop<\/a> in Novosibirsk (2007).\u00a0 You will not find my name there: I could not accept, being at the CSL 2007<\/a> conference in Edinburgh at the same time, and so I didn’t go to Novosibirsk.<\/p>\n

That was totally mysterious to me, too.\u00a0 Imagine my situation at the time.\u00a0 I had never been to any Domains workshop, in fact I had almost not worked in domain theory at all.\u00a0 I was working in computer security, and I had worked in logic and automated deduction before that.\u00a0 I had only started to be really interested in domain theory in 2005, less than two years before, but I only had one minor publication in the field [3].\u00a0 I had also put a long set of notes in French, le Pav\u00e9<\/a><\/em>, on my Web page<\/a>.\u00a0 But nobody could know me as somebody doing domain theory at the time, really.<\/p>\n

As I learned later, Klaus was one of the referees of the paper [3] (how do I know? read the anecdote at the end of this post), and he was aware of le Pav\u00e9<\/em>… fantastically enough, Klaus was fluent in French.\u00a0 He had spent a few years in Paris in the late 1960s, where he prepared a th\u00e8se d’\u00c9tat<\/em>, which he defended in 1971.<\/p>\n

He had other ties with France, too.\u00a0 Although he had a position in Darmstadt, he also had a house in Bagneux, right south of Paris, where he spent some of his holidays with his family.\u00a0 If you look at a map of Bagneux<\/a>, there are some chances that you’ll see something called “\u00c9cole normale sup\u00e9rieure de Paris-Saclay” close by, below and on the right: that is where I work.\u00a0 From time to time, he would visit me, and we would spend an afternoon discussing science.\u00a0 I hold fond memories of these visits.<\/p>\n

Anyway, as far as domain theory is concerned, Klaus was the person who discovered me.\u00a0 He could base his opinion only on [3] and on le Pav\u00e9, and despite that, my first domain theory talk was an invited talk at the next Domains workshop in Brighton<\/a>.\u00a0 Imagine that: the first talk you give in a given area is an invited<\/em> talk!<\/p>\n

I am persuaded Klaus was incredible at discovering people.\u00a0 I am not saying that because he discovered me.\u00a0 Several years later, he told me about a brilliant young American researcher working in Japan, Matthew de Brecht.\u00a0 What Matthew had done\u2014the discovery of quasi-Polish spaces<\/a>\u2014was so brilliant!\u00a0 Matthew was then an invited speaker at the Domains XI conference in Paris, and again Klaus is the person who suggested the name first.<\/p>\n

The mathematics of Klaus<\/h2>\n

There is one thing that Klaus kept saying, that one should do mathematics “at the right level of generality<\/em>“.\u00a0 That continues to strike me as exactly right.<\/p>\n

I cannot say exactly what Klaus meant by that, and I’ll propose my own interpretation.\u00a0 Some mathematicians go overboard with generality, perhaps.\u00a0 I know some instances of that trend, and I am pretty sure I even know one example that Klaus had in mind.\u00a0 (Not me.\u00a0 Of course, I won’t tell.)\u00a0 Maybe I could get away with an analogy.\u00a0 It is said that Edwin Hewitt, who did many important things in topology and then went to explore other fields, once said<\/a> “Right now, I don’t care a bit whether every beta-capsule of type delta is also a T-spot of the second kind.”\u00a0 Certainly, I’ve heard quite a number of talks in mathematics of that kind, only to be left wondering why and whither.<\/p>\n

Anyway, the nice thing with Klaus’s motto is that it also applies to the other end of the spectrum.\u00a0 I am officially working in computer science, and there your aim is not generality but applications.\u00a0 Also, a paper should be as easy to read as possible, replete with examples, applications, illustrations, comparison with previous work, and what have you.\u00a0 (Otherwise, it will most likely be rejected.)\u00a0 It does not matter that you can extend the theory of beta-capsules of type delta to the more general case of T-spots of the second kind.\u00a0 If there is no known non-trivial T-spot of the second kind, your paper will be rejected.\u00a0 If there are, but you cannot explain why they are crucial to making rocket-control software bug-free or something at least as serious, then your paper will be rejected.\u00a0 There are nice aspects to the situation.\u00a0 Notably, good computer science papers are a pleasure to read, and often a good introduction to the general field.\u00a0 The downside is that the mathematics is sometimes done at the least general level that is enough to cope with the problem.\u00a0 So you’ll find dozens of papers which each propose to make a slightly more general contribution than the previous one, each time with a motivation coming from the real world, but no completely general theory.\u00a0 There are exceptions, of course, but I tend to think that is rather faithful to the general picture.<\/p>\n

So what does “the right level of generality” mean?\u00a0 One could give examples and counterexamples, but certainly no formal definition.\u00a0 Klaus and I sometimes disagreed, too.\u00a0 For example, just like he was, I was interested in the theory of continuous valuations (a notion close to measures), and I thought that you had to see them through the lens of the linear continuous functionals they generate through the process known as integration, but he thought one should reason at a slightly more abstract level, that of cones.\u00a0 Precisely, there is a dcpo, which Klaus called L<\/em>(X<\/em>) (with a calligraphic L<\/em>) of all continuous maps from X<\/em> to the set of extended non-negative reals (which I will just write as R+<\/sub><\/strong> in this post, although there should be a bar on top of it), and the map that sends f<\/em> in L<\/em>(X<\/em>) to its integral with respect to a given continuous valuation is a Scott-continuous, linear map from L<\/em>(X<\/em>) to R+<\/sub><\/strong>.<\/p>\n

If you come from the computer scientific side of the world, or more precisely, from semantics, then you only need to care about X<\/em> being a dcpo, and more probably a continuous dcpo with possibly some additional properties.\u00a0 I quickly realized that it was profitable to sit at a slightly higher level of generality, where X<\/em> is allowed to be a topological space, possibly locally compact or stably compact for example.\u00a0 Other people had realized before me that going from pure domain theory to topology gave a bit of elbow room, if not of fresh air.\u00a0 By that I mean that some of the proofs suddenly become easier, or simply feasible, at the level of topological spaces.<\/p>\n

Klaus felt that this was not the right level of generality (that was mine).\u00a0 L<\/em>(X<\/em>) is only one example of a d-cone<\/em>, a notion he defined and organized with Gordon Plotkin, notably.\u00a0 Hence what is important is the space of linear Scott-continuous maps form a d-cone C<\/em> to R+<\/sub><\/strong>.<\/p>\n

However, d-cones are not as general as one would like, and Klaus later extended those to topological and semitopological cones [4], and produced a theory of convexity, separation theorems, etc., based on linear continuous maps\u00a0 from (semi)topological cones to R+<\/sub><\/strong>.\u00a0 That is clearly imitated from a similar theory\u2014convex analysis\u2014of topological vector spaces, convexity and linear continuous maps from topological vector spaces to R<\/strong>.\u00a0 And that is the right level of generality.<\/p>\n

Naturally, why should you limit yourselves to R+<\/sub><\/strong>?\u00a0 Cannot we replace it with something more general?\u00a0 I am sure Klaus thought about it, and dismissed it.\u00a0 You can do convex analysis in real vector spaces, but also in complex vector spaces, and also in more complicated situations.\u00a0 But what do you gain with the added generality in cones? Is it worth the price to be paid if that incurs extra complications, notably? That is, I think, the real question you should ask yourself if you, like me, aim at heeding by Klaus’s principle of right generality.\u00a0 (I am not claiming to be entirely successful.)<\/p>\n

A final gem<\/h2>\n

Since I have mentioned (semi)topological cones, I would like to mention a particular theorem due to Klaus in that area, which struck me as particularly clever when I discovered it.<\/p>\n

Klaus had been interested in convex analysis, in particular in those strange extensions of convex analysis to so-called ordered cones, for a long time [5].<\/p>\n

A particularly useful theorem in that domain is due to Walter Roth, and here it is.<\/p>\n

First, a cone<\/em> is a set\u00a0C<\/em> together with two operations: addition, and scalar multiplication by non-negative<\/em> reals.\u00a0 These two operations satisfy all the usual identities that you can express in vector spaces without taking opposites or multiplying by a negative number.\u00a0 The usual cones inside real vector spaces qualify.\u00a0 The important thing is that L<\/em>(X<\/em>) is a cone in Keimel and Roth’s sense, too, although L<\/em>(X<\/em>) does not even embed in a vector space.\u00a0 Indeed, in a vector space addition is cancellative, namely f<\/em>+g<\/em>=f<\/em>+h<\/em> implies g<\/em>=h<\/em>, but that is not true in L<\/em>(X<\/em>): take the constant function equal to infinity for f<\/em> for example.<\/p>\n

An ordered<\/em> cone is a cone C<\/em> with an ordering that makes both addition and scalar multiplication monotonic (in both arguments).\u00a0 Again, L<\/em>(X<\/em>) is an example of that notion.\u00a0 A map f<\/em> from C<\/em> to\u00a0R+<\/sub><\/strong> is\u00a0linear<\/em> if\u00a0f<\/em>(x<\/em>+y<\/em>)=f<\/em>(x<\/em>)+f<\/em>(y<\/em>) and\u00a0f<\/em>(a.x<\/em>)=a<\/em>.f<\/em>(x<\/em>).\u00a0 If you replace = by \u2264 in the first equality, then you obtain\u00a0sublinear maps<\/em>, and\u00a0superlinear maps<\/em> with \u2265. Roth’s Sandwich Theorem says the following: let p<\/em>, q<\/em>, be two maps from C<\/em> to\u00a0R+<\/sub><\/strong> with q<\/em>\u2264p<\/em>, q<\/em> superlinear, and p<\/em> sublinear, and p<\/em> or q<\/em> monotonic; then there is a linear monotonic map f<\/em> such that q<\/em>\u2264f<\/em>\u2264p<\/em>.\u00a0 More generally, the claim holds without assuming\u00a0q<\/em>\u2264p<\/em> or p<\/em> or q<\/em> monotonic, rather requiring that x<\/em>\u2264y<\/em> implies\u00a0q<\/em>(x<\/em>)\u2264p<\/em>(y<\/em>).<\/p>\n

Roth’s Sandwich Theorem is cognate with similar sandwich theorems in convex analysis, due to Hahn and Banach.\u00a0 However, its proof requires different methods.\u00a0 In a nutshell, the proof strategy is as follows.\u00a0 For simplicity, I will assume that q<\/em>\u2264p<\/em> and that both p<\/em> and q<\/em> are monotonic.\u00a0 Among the sublinear monotonic maps above q<\/em> and below p<\/em>, there is a minimal one p’<\/em>, by Zorn’s Lemma.\u00a0 Among the superlinear monotonic maps above q<\/em> and below p’<\/em>, there is a maximal one q’<\/em>, by Zorn’s Lemma again.\u00a0 Now do some computations involving superlinearity, sublinearity and the fact that x<\/em>\u2264y<\/em> implies\u00a0q’<\/em>(x<\/em>)\u2264p’<\/em>(y<\/em>), and you should manage to show that q’<\/em>=p’<\/em>.\u00a0 Call that function f<\/em>: this is the desired function.<\/p>\n

Klaus observed that you could generalize that to lower semicontinuous<\/em> (super, sub) linear functions from a semitopological cone C<\/em> to R+<\/sub><\/strong>.\u00a0 (Lower semicontinuous simply means continuous, understanding that the topology of\u00a0R+<\/sub><\/strong> is the Scott topology.)\u00a0 In his place, I would probably have tried to redo Roth’s proof in a topological setting, but Klaus observed that the (semi)topological case reduced to the order-theoretic case.<\/p>\n

A cone C<\/em> is semitopological<\/em> if and only if addition and scalar multiplication are separately<\/em> continuous in each of their arguments, where\u00a0R+<\/sub><\/strong> is equipped with its Scott topology.\u00a0 C<\/em> is topological<\/em> if they are jointly<\/em> continuous.\u00a0 L<\/em>(X<\/em>) is always a semitopological cone in its Scott topology, but is only known to be topological if X<\/em> is core-compact.\u00a0 Here is Klaus’ Theorem.<\/p>\n

Theorem (Keimel, 2008, Theorem 8.2 in [4]).<\/strong>\u00a0 Let C<\/em> be a semitopological cone, and q<\/em>\u2264p<\/em>, where q<\/em> is superlinear and lower semicontinuous, and\u00a0p<\/em> is sublinear.\u00a0 Then there is a linear lower semicontinuous map f<\/em> such that q<\/em>\u2264f<\/em>\u2264p<\/em>.<\/p>\n

Proof.<\/em> We consider C<\/em> as an ordered cone whose ordering is the specialization ordering of C<\/em> seen as a topological space.\u00a0 We already have a minimal sublinear map p’<\/em> between q<\/em> and p<\/em> from the proof of Roth’s theorem, and we have seen that that was a linear monotonic map f<\/em> such that f<\/em> such that q<\/em>\u2264f<\/em>\u2264p<\/em>.\u00a0 It remains to observe that f<\/em> is necessarily lower semicontinuous.<\/p>\n

Of couse not every monotonic map, even linear, is lower semicontinuous!\u00a0 Klaus instead makes clever use of the so-called lower semicontinuous envelope f’<\/em> of f<\/em>.\u00a0 By definition, this is the (pointwise) supremum of all lower semicontinuous maps below f<\/em>, and is itself lower semicontinuous.\u00a0 Klaus shows, by elementary arguments which I’ll let you discover by yourselves (Lemma 5.7, [4]) that the lower semicontinuous envelope f’<\/em> of a sublinear map is always sublinear.<\/p>\n

Since f’<\/em> is the largest lower semicontinuous map below f<\/em>, and q<\/em> is lower semicontinuous and below f<\/em>, q<\/em> must be below f’<\/em>, right?\u00a0 But f’<\/em> is sublinear, as we have seen, and is below f<\/em>, which is minimal among the sublinear maps between q<\/em> and p<\/em>.\u00a0 Hence f’<\/em>=f<\/em>.\u00a0 And that shows that f<\/em> is lower semicontinuous, since f’<\/em> is.\u00a0 \u2610<\/p>\n

I find that argument marvelous.\u00a0 Not just that, look at the assumptions of the theorem: we never had to assume any continuity property of p<\/em>: only q<\/em> must be lower semicontinuous!<\/p>\n

I have used that in [6].\u00a0 Lemma 3.16 there says that, given a topological space X<\/em> (no assumption at all on X<\/em>), for every superlinear continuous map F<\/em> from L<\/em>(X<\/em>) to R+<\/sub><\/strong>, the set s<\/em>(F<\/em>) of linear continuous maps G<\/em>\u2265F<\/em> is compact saturated in L<\/em>(X<\/em>) with its Scott topology.\u00a0 That is one of the steps which eventually allowed me to show the existence of certain retractions, and then of certain isomorphisms, without any assumption on X<\/em> whatsoever.\u00a0 The key of the miracle is the fact that you do not need to assume anything apart sublinearity, and certainly not lower semicontinuity, from p,<\/em> in Keimel’s Sandwich Theorem.<\/p>\n

Thanks a lot, Klaus!\u00a0 That miracle would have been a good enough reason to thank you alone.\u00a0 I should have thanked you for all the rest, too.<\/p>\n

    \n
  1. Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, and Dana S. Scott. A compendium of continuous lattices<\/a>.\u00a0 Springer Science & Business Media, 2012.<\/li>\n
  2. Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, and Dana S. Scott. Continuous Lattices and Domains<\/a>. Encyclopedia of Mathematics and Its Applications, vol. 93, 2003. Cambridge University Press.<\/li>\n
  3. J. Goubault-Larrecq<\/a><\/span><\/span>.\u00a0 Extensions of Valuations<\/a><\/span>.\u00a0 Mathematical Structures in Computer Science<\/span>\u00a015(2), pages\u00a0271-297<\/a>, 2005<\/span>.<\/li>\n
  4. Klaus Keimel.\u00a0 Topological cones: functional analysis in a T0<\/sub>-setting<\/a>.\u00a0 Semigroup forum 77(1), pages 109-142, 2008.<\/li>\n
  5. Klaus Keimel and Walter Roth.\u00a0 Ordered cones and approximation<\/a>.\u00a0 Springer Verlag Lecture Notes in Mathematics 1517, 1992.<\/li>\n
  6. J. Goubault-Larrecq<\/a><\/span><\/span>.\u00a0 Isomorphism theorems between models of mixed choice<\/a><\/span>.\u00a0 Mathematical Structures in Computer Science<\/span><\/a>, 2016<\/span>.<\/li>\n<\/ol>\n

    Oh yes, the anecdote…\u00a0 Klaus apparently made some efforts to conceal the fact he had refereed paper [3] from me.\u00a0 He was right: anonymity is a key principle in the evaluation of scientific papers.\u00a0 In retrospect, that he was one of my referees should have been obvious: he had cited one paper he had written with Jimmie Lawson on measure extensions for T0 spaces (which is an incredibly good paper, by the way), the paper had not been in print yet, and the only way to get it was from Klaus’s web page.\u00a0 But I am naive, and the thought he might have been my referree did not occur to me.\u00a0 Then I received an anonymous letter, posted from some place in the Netherlands (note: not from Darmstadt!) with an offprint of that paper in it and nothing else.\u00a0 When I received the letter, I thought I should properly thank the anonymous referee.\u00a0 The only proper way to do that is, of course, to ask the editor of the journal with whom you’ve been in touch to transmit your thanks to the anonymous referee.\u00a0 That editor, who is a very respectable person, thought I had guessed who the referee was for a long time (as I said, that was starting to be pretty obvious), and merely told me to thank Klaus directly.<\/p>\n

    \u2014 Jean Goubault-Larrecq<\/a>\u00a0(November 21st, 2017)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

    Klaus Keimel passed away on Saturday, November 18th, 2017. This is sad news. I would like to pay homage to him, and I am not sure I will manage to do so in the way he deserves.\u00a0 I cannot retrace … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-1320","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1320","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1320"}],"version-history":[{"count":16,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1320\/revisions"}],"predecessor-version":[{"id":5351,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1320\/revisions\/5351"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}