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. 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.
Domain theorists will probably know him best for the compendium [1], then the red book [2]. From what I understand, he had been the driving force behind the two projects. He had also been the man behind the highly successful Domains workshop, of which the 12th and latest edition took place in Cork in 2015. 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. He could surprise you by telling you about K-theory, about sheaves, or about traces in C*-algebras.
First contact
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 in Novosibirsk (2007). You will not find my name there: I could not accept, being at the CSL 2007 conference in Edinburgh at the same time, and so I didn’t go to Novosibirsk.
That was totally mysterious to me, too. Imagine my situation at the time. I had never been to any Domains workshop, in fact I had almost not worked in domain theory at all. I was working in computer security, and I had worked in logic and automated deduction before that. 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]. I had also put a long set of notes in French, le Pavé, on my Web page. But nobody could know me as somebody doing domain theory at the time, really.
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 Pavé… fantastically enough, Klaus was fluent in French. He had spent a few years in Paris in the late 1960s, where he prepared a thèse d’État, which he defended in 1971.
He had other ties with France, too. 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. If you look at a map of Bagneux, there are some chances that you’ll see something called “École normale supérieure de Paris-Saclay” close by, below and on the right: that is where I work. From time to time, he would visit me, and we would spend an afternoon discussing science. I hold fond memories of these visits.
Anyway, as far as domain theory is concerned, Klaus was the person who discovered me. He could base his opinion only on [3] and on le Pavé, and despite that, my first domain theory talk was an invited talk at the next Domains workshop in Brighton. Imagine that: the first talk you give in a given area is an invited talk!
I am persuaded Klaus was incredible at discovering people. I am not saying that because he discovered me. Several years later, he told me about a brilliant young American researcher working in Japan, Matthew de Brecht. What Matthew had done—the discovery of quasi-Polish spaces—was so brilliant! Matthew was then an invited speaker at the Domains XI conference in Paris, and again Klaus is the person who suggested the name first.
The mathematics of Klaus
There is one thing that Klaus kept saying, that one should do mathematics “at the right level of generality“. That continues to strike me as exactly right.
I cannot say exactly what Klaus meant by that, and I’ll propose my own interpretation. Some mathematicians go overboard with generality, perhaps. I know some instances of that trend, and I am pretty sure I even know one example that Klaus had in mind. (Not me. Of course, I won’t tell.) Maybe I could get away with an analogy. It is said that Edwin Hewitt, who did many important things in topology and then went to explore other fields, once said “Right now, I don’t care a bit whether every beta-capsule of type delta is also a T-spot of the second kind.” Certainly, I’ve heard quite a number of talks in mathematics of that kind, only to be left wondering why and whither.
Anyway, the nice thing with Klaus’s motto is that it also applies to the other end of the spectrum. I am officially working in computer science, and there your aim is not generality but applications. Also, a paper should be as easy to read as possible, replete with examples, applications, illustrations, comparison with previous work, and what have you. (Otherwise, it will most likely be rejected.) 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. If there is no known non-trivial T-spot of the second kind, your paper will be rejected. 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. There are nice aspects to the situation. Notably, good computer science papers are a pleasure to read, and often a good introduction to the general field. The downside is that the mathematics is sometimes done at the least general level that is enough to cope with the problem. 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. There are exceptions, of course, but I tend to think that is rather faithful to the general picture.
So what does “the right level of generality” mean? One could give examples and counterexamples, but certainly no formal definition. Klaus and I sometimes disagreed, too. 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. Precisely, there is a dcpo, which Klaus called L(X) (with a calligraphic L) of all continuous maps from X to the set of extended non-negative reals (which I will just write as R+ in this post, although there should be a bar on top of it), and the map that sends f in L(X) to its integral with respect to a given continuous valuation is a Scott-continuous, linear map from L(X) to R+.
If you come from the computer scientific side of the world, or more precisely, from semantics, then you only need to care about X being a dcpo, and more probably a continuous dcpo with possibly some additional properties. I quickly realized that it was profitable to sit at a slightly higher level of generality, where X is allowed to be a topological space, possibly locally compact or stably compact for example. 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. By that I mean that some of the proofs suddenly become easier, or simply feasible, at the level of topological spaces.
Klaus felt that this was not the right level of generality (that was mine). L(X) is only one example of a d-cone, a notion he defined and organized with Gordon Plotkin, notably. Hence what is important is the space of linear Scott-continuous maps form a d-cone C to R+.
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 from (semi)topological cones to R+. That is clearly imitated from a similar theory—convex analysis—of topological vector spaces, convexity and linear continuous maps from topological vector spaces to R. And that is the right level of generality.
Naturally, why should you limit yourselves to R+? Cannot we replace it with something more general? I am sure Klaus thought about it, and dismissed it. You can do convex analysis in real vector spaces, but also in complex vector spaces, and also in more complicated situations. 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. (I am not claiming to be entirely successful.)
A final gem
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.
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].
A particularly useful theorem in that domain is due to Walter Roth, and here it is.
First, a cone is a set C together with two operations: addition, and scalar multiplication by non-negative reals. These two operations satisfy all the usual identities that you can express in vector spaces without taking opposites or multiplying by a negative number. The usual cones inside real vector spaces qualify. The important thing is that L(X) is a cone in Keimel and Roth’s sense, too, although L(X) does not even embed in a vector space. Indeed, in a vector space addition is cancellative, namely f+g=f+h implies g=h, but that is not true in L(X): take the constant function equal to infinity for f for example.
An ordered cone is a cone C with an ordering that makes both addition and scalar multiplication monotonic (in both arguments). Again, L(X) is an example of that notion. A map f from C to R+ is linear if f(x+y)=f(x)+f(y) and f(a.x)=a.f(x). If you replace = by ≤ in the first equality, then you obtain sublinear maps, and superlinear maps with ≥. Roth’s Sandwich Theorem says the following: let p, q, be two maps from C to R+ with q≤p, q superlinear, and p sublinear, and p or q monotonic; then there is a linear monotonic map f such that q≤f≤p. More generally, the claim holds without assuming q≤p or p or q monotonic, rather requiring that x≤y implies q(x)≤p(y).
Roth’s Sandwich Theorem is cognate with similar sandwich theorems in convex analysis, due to Hahn and Banach. However, its proof requires different methods. In a nutshell, the proof strategy is as follows. For simplicity, I will assume that q≤p and that both p and q are monotonic. Among the sublinear monotonic maps above q and below p, there is a minimal one p’, by Zorn’s Lemma. Among the superlinear monotonic maps above q and below p’, there is a maximal one q’, by Zorn’s Lemma again. Now do some computations involving superlinearity, sublinearity and the fact that x≤y implies q’(x)≤p’(y), and you should manage to show that q’=p’. Call that function f: this is the desired function.
Klaus observed that you could generalize that to lower semicontinuous (super, sub) linear functions from a semitopological cone C to R+. (Lower semicontinuous simply means continuous, understanding that the topology of R+ is the Scott topology.) 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.
A cone C is semitopological if and only if addition and scalar multiplication are separately continuous in each of their arguments, where R+ is equipped with its Scott topology. C is topological if they are jointly continuous. L(X) is always a semitopological cone in its Scott topology, but is only known to be topological if X is core-compact. Here is Klaus’ Theorem.
Theorem (Keimel, 2008, Theorem 8.2 in [4]). Let C be a semitopological cone, and q≤p, where q is superlinear and lower semicontinuous, and p is sublinear. Then there is a linear lower semicontinuous map f such that q≤f≤p.
Proof. We consider C as an ordered cone whose ordering is the specialization ordering of C seen as a topological space. We already have a minimal sublinear map p’ between q and p from the proof of Roth’s theorem, and we have seen that that was a linear monotonic map f such that f such that q≤f≤p. It remains to observe that f is necessarily lower semicontinuous.
Of couse not every monotonic map, even linear, is lower semicontinuous! Klaus instead makes clever use of the so-called lower semicontinuous envelope f’ of f. By definition, this is the (pointwise) supremum of all lower semicontinuous maps below f, and is itself lower semicontinuous. Klaus shows, by elementary arguments which I’ll let you discover by yourselves (Lemma 5.7, [4]) that the lower semicontinuous envelope f’ of a sublinear map is always sublinear.
Since f’ is the largest lower semicontinuous map below f, and q is lower semicontinuous and below f, q must be below f’, right? But f’ is sublinear, as we have seen, and is below f, which is minimal among the sublinear maps between q and p. Hence f’=f. And that shows that f is lower semicontinuous, since f’ is. ☐
I find that argument marvelous. Not just that, look at the assumptions of the theorem: we never had to assume any continuity property of p: only q must be lower semicontinuous!
I have used that in [6]. Lemma 3.16 there says that, given a topological space X (no assumption at all on X), for every superlinear continuous map F from L(X) to R+, the set s(F) of linear continuous maps G≥F is compact saturated in L(X) with its Scott topology. 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 whatsoever. 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, in Keimel’s Sandwich Theorem.
Thanks a lot, Klaus! That miracle would have been a good enough reason to thank you alone. I should have thanked you for all the rest, too.
- Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, and Dana S. Scott. A compendium of continuous lattices. Springer Science & Business Media, 2012.
- Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, and Dana S. Scott. Continuous Lattices and Domains. Encyclopedia of Mathematics and Its Applications, vol. 93, 2003. Cambridge University Press.
- . Extensions of Valuations. Mathematical Structures in Computer Science 15(2), pages 271-297, 2005.
- Klaus Keimel. Topological cones: functional analysis in a T0-setting. Semigroup forum 77(1), pages 109-142, 2008.
- Klaus Keimel and Walter Roth. Ordered cones and approximation. Springer Verlag Lecture Notes in Mathematics 1517, 1992.
- . Isomorphism theorems between models of mixed choice. Mathematical Structures in Computer Science, 2016.
Oh yes, the anecdote… Klaus apparently made some efforts to conceal the fact he had refereed paper [3] from me. He was right: anonymity is a key principle in the evaluation of scientific papers. 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. But I am naive, and the thought he might have been my referree did not occur to me. 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. When I received the letter, I thought I should properly thank the anonymous referee. 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. 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.
— Jean Goubault-Larrecq (November 21st, 2017)