Per Enflo - Math or Music? - Yes, Math AND Music!
This work was done in the years 1964-70. It is a study of an infinite-dimensional version of Hilbert's fifth problem for topological groups - together with a study on uniform homeomorphisms between Banach spaces. Notions of non-linear type were introduced. They turned out to be useful also in other areas of Mathematics and in Computer Science. The long-standing problem whether "Enflo type" and "Rademacher type" coincide was recently (2020) solved affirmatively >>
In 1946 I.Segal showed that groups modeled on Euclidean Space with x→xy differentiable for fixed y, are Lie groups. It was at that time the strongest result on Hilbert's fifth problem, in the sense that it was the weakest assumption on the group operations to get a Lie group. By a very different method I showed that this result generalizes to local groups modeled on Banach Spaces, if one assumes local uniform continuity of (x,y)→xy. This assumption is trivially satisfied in the finite-dimensional case and it cannot be removed in the infinite-dimensional case. If one removes differentiability conditions on the group operations and replaces them with weaker conditions - Lipschitz or Hölder or just uniform continuity - one is led to non-linear geometric problems in Banach Spaces. In the 1960:s, there was no theory developed in this direction and so I started by considering commutative groups modeled on Banach Spaces. And until today, there is no theory of non-commutative groups modeled on Banach Spaces, without differentiability assumptions on the group operations.
For a commutative group modeled (locally) on a Banach Space, Hilbert's fifth problem would ask if the group is (locally) isomorphic to the additive group of the Banach Space. A first step in proving that is to establish a linear structure in the group. And a first step in establishing a linear structure is to solve the equation y = x2 in the group. I managed to do that under some Hölder condition on the group operation, by developing a notion of non-linear type in Banach Spaces. And, somewhat later, I had to develop a stronger non-linear notion to solve an embedding problem of Smirnov. Both these notions have turned out to be useful also in other areas of Mathematics and in Computer Science.
Once one has estabished that a commutative group modeled on a Banach Space is in itself a linear space, one has to decide whether this linear space is isomorphic to the Banach Space it is modeled on. I managed to show that, under a local Lipschitz condition on the group operations, and if the group is modeled on a reflexive Banach Space and if locally x2 =0 implies x=0, then the group is locally the additive group of some Banach Space (0 is assumed to be the unit element of the group). And so, one is finally led to the problem: If a Banach Space is Lipschitz equivalent to a reflexive Banach, is it linearly isomorphic to that Banach Space? This problem is still open. In 1969 I managed to prove: If a Banach Space is uniformly homeomorphic to a Hilbert Space, it is linearly isomorphic to the Hilbert Space (and the conclusion holds also if they are just locally Lipschitz equivalent). At that time, it was the first result of that type. It has been referred to as a "pioneering result by Enflo". Since then, much research has been done in that direction.
An infinite-dimensional version of Hilbert's fifth problem can be seen as a particular case of a very general question : To what extent do topological notions and results in finite dimensions carry over to Lipschitz (or Hölder or uniform continuity) notions and results in infinite dimensions?
Investigations on Hilbert's fifth problem for non locally compact groups.
Dissertation, Stockholm University, 1970
Topological groups in which multiplication on one side is differentiable or linear. Math. Scand. 24 (1969) 195-207
Uniform structures and square roots in topological groups. Israel J. Math. 8 (1970) 230-272
On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat. 8 (1969) 103-105 [note: Roundness is introduced which in the 1980:s developed into Enflo type]
On a problem of Smirnov. Ark. Math. 8 (1969) 107-10
(Unpublished) L1 and l1 are not uniformly homeomorphic. The proof of this result uses that the set of metrical midpoints between two points is much larger in L1 than in l1. This midpoint argument has later been used and taken much further by other researchers.
This work was done in September 1971. The main result is that the notions uniformly convex, uniformly smooth, uniformly non-square and super-reflexive are all isomorphically equivalent.
The problem whether uniformly convex and uniformly smooth are isomorphically equivalent had been posed 30 years earlier, in 1941, by M.Day. Since the concepts uniformly convex and uniformly smooth had become increasingly important in Functional Analysis, it had become increasingly important to know whether they are isomorphically equivalent. Before me, R.C.James had solved half the problem. I learned about his result, soon after arriving in Berkeley in the end of August 1971. I felt that I had a chance to solve the other half of the problem - with the general strategies that I had developed in my isolation in Sweden. For me it was an unusually short time to spend on a problem - less than a month.
Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math., 13 (1972), 281-288
This work started in 1967-68 and was completed in 1972. The slow progress and the many obstacles on the way is described in my Autobiography. The two problems - the Basis problem and the Approximation problem - had been open for more than 40 years, and many mathematicians had tried to solve them. I learned about them as a student. They were the big problems in Functional Analysis, that everybody wanted to solve.
AMS had its 75 year celebration in 1991 and for that Professor P.Halmos wrote a report on "the progress of mathematics in the last 75 years". He reported on 22 mathematical discoveries and my solution of the Basis problem was one of them. And on IBM:s big poster over the History of Mathematics over the last 1000 years, the solution of the Basis problem is one of the less than 50 discoveries from the last century that is mentioned.
In 1936 Professor S.Mazur promised a live goose for the solution of a problem, that turned out to be equivalent to the Approximation problem. I received the goose at a ceremony in Warsaw, in December 1972; the "goose reward" ceremony was broadcast throughout Poland.
A Banach space with basis constant greater than 1. Ark Mat. 11 (1973) 103-107
A counterexample to the approximation problem in Banach spaces, Acta Mathematica (1973) 130, Number 1, 309-317
The Invariant Subspace Problem is also one of the fundamental problems from the early days of Functional analysis. J. von Neumann showed in the 1930s that compact operators on Hilbert space have invariant subspaces and in 1973, V.Lomonosov showed that operators on Banach spaces which commute with an operator that commutes with a compact operator (obs. 2 steps) have invariant subspaces.
My work shows that there are operators without invariant subspaces on some Banach spaces. For reflexive Banach spaces, especially Hilbert space, the problem remains open. My work required new analytical techniques, especially new types of results for products of polynomials. This work started in 1970 and was completed with a first manuscript in 1975. The complexity of the paper - the proof is 100 pages long - made the published paper appear only in 1987, 6 years after its submission in 1981. In K.Saxe's book "Beginning Functional Analysis" it is referred to as "arguably, his most famous mathematical contribution thus far".
On the invariant subspace problem in Banach spaces, Seminaire Maurey-Schwartz (1975-76)
On the invariant subspace problem in Banach spaces; Mittag-Leffler report (1980), Acta Math. Vol. 158, (submitted 1981, appeared 1987): 213-313
Some results concerning Lp(μ)-spaces (with H.P. Rosenthal), J. Funct. Anal. 14(1973), 325-348
On the "Three Space Problem" (with J. Lindenstrauss and G. Pisier) Math. Scand. 36 (1975), 199-210
On the structure of separable lp spaces (1 < p < oo) (with D. Alspach and E. Odell) Studia Math., 60 (1977), 79-90
Subspaces of L1 containing L1. (with T. Starbird) Studia Math. 65. (1979), 203-225
Some results and open questions on spaceability in function spaces. (with V. Gurariy and J. Seoane-Sepúlveda) Transactions of the American Mathematical Society. 366 (2014), 611-625 >>pdf
These results are mostly from the late 80:s and early 90:s. The concept "minimal point" is an outgrowth of my efforts on the Basis problem. It turns out to be a special case of the concept Pareto optimal point in Economics. It turned out to also be a fruitful concept in several areas of non-linear theory in Banach spaces. The first paper contains Fixed point results for contractive maps and an Approximation result for Pareto optimal poins in Economics.
The other papers contain characterizations of contractive subsets of lp -spaces, and - considerably harder - the same characterizations in Lp-spaces. Special cases of these results have been published as results in Approximation theory. The problem whether "optimal" and "contractive" are equivalent for uniformly convex spaces remains open.
Theoremes de point fixe et d'approximation (with B. Beauzamy); Ark. Mat. Vol. 23 no. 1 (1985), 19-34
Contractive projections on lp-spaces (with W. J. Davis) London Math.Soc. Lecture Notes Series 137 (1989), 151–161
Contractive projections onto subsets of L1(0,1), Analysis at Urbana, Vol. I, London Math. Soc. Lecture Note Ser. 137 (1989), 162–184
Contractive projections onto subsets of Lp-spaces, Lecture Notes in. Pure and Applied Mathematics 136 (1992), 79–94
Minimal points and contractive projections. Constructive Theory of Functions, Marin Drinov Publishing House (2006), 98-103
Estimations de produits de polynomes, (with B. Beauzamy) J. of Number Theory, 21-3 (1985), pp. 390-412
Products of polynomials in many variables (with B. Beauzamy, E. Bombieri and H. Montgomery) J. of Number Theory, Vol. 36 No. 2 (1990) 219-245
The largest coefficient in products of polynomials, Function Spaces, Lecture Notes in Pure and Appl. Math, 136, (1992) pp. 97-105
Polynomials in many variables: real vs complex norms (with R. Aron and B. Beauzamy), J. Approx. Theory 74 (1993) 181–198
Some Problems in the Interface between Number Theory, Harmonic Analysis and Geometry of Euclidean Space, Quaestiones Mathematicae 18 (1995), 309-323
Exponential numbers of linear operators in normed spaces, (with V. Gurariy, V.I. Lomonosov, Yu.I. Lyubich), Linear Algebra and its Applications, 219, (1995) pp. 225-260
On Montgomery's conjecture and the distribution of Dirichlet sums, (with V. Gurariy and J. Seoane-Sepúlveda), Journal of Functional Analysis 267, (2014) 1241-1255 pdf >>
Quantitative and qualitative estimates on the norm of products of polynomials, (with G. Araújo, G.A. Muñoz-Fernández, et al.) Israel J. Math. 236, (2020) 727–745 link>
Extremal vectors and invariant subspaces (with Shamim Ansari) Trans. Amer. Math. Soc. 350 (1998), 539-558
Extremal vectors for a class of linear operators, Functional Analysis and Economic Theory, Springer-Verlag (1998), 61-64
Some results on extremal vectors and invariant subspaces (with Terje Hõim) Proc. Amer. Math. Soc.131 (2003), 379-387
Extremal vectors and rectifiability (with Aderaw Fenta) Quaestiones Mathematicae - vol 34(1) (2011), 119-123
On Sparse Languages L such that LL = Sigma. (with Andrew Granville, Jeffrey Shallit, Sheng Yu ) Discrete Applied Mathematics 52(3) (1994), 275-285 pdf >>
Stable basis families and complexity lower bounds (with Meera Sitharam) Electronic Colloquium on Computational Complexity (ECCC) 3(49): (1996) pdf >>
WORKS IN MATHEMATICS
11. Mathematical Biology/Ecology
12. Population Genetics. Human Evolution
I think that mathematics can successfully be used in many more areas of science than is done today - to help us better understand the world around us. The three areas below, that I have worked in, represent three different ways to apply mathematics - or mathematical thinking.
In the Acoustics work (number 10.) the mathematical equations were already there. The challenge was to find as precise and explicit solutions as possible.
The Mathematical Biology/Ecology work (number 11.) was initially to study, whether changes in the composition of algae in Lake Erie after the zebra mussel invasion, could be understood as consequences of basic ecological phenomena: Different species competing for the same nutrients, and the predator-prey relation. In the big picture, there are equations for these basic phenomena - systems of equations of Lotka-Volterra type. Thus it was fairly clear what type of mathematical equations one should look for. One of the biggest challenges was to find realistic parameter values in these equations.
My Human Evolution/Population Genetics work (number 12.) was initially motivated by the challenge to understand an apparent contradiction between fossil data and DNA data concerning the Neandertals. It was not clear that mathematics would play any role in the solution of this problem, but it actually did. Mathematics certainly played a role in the evaluation of different scenarios, that I studied. The second project is a study of how much of Human Evolution can be understood as a consequence of the reproduction strategy of humans. No mathematics is involved.
The background of the first paper is that perturbations at the vertex of an acoustic sawtooth wave in a slightly dispersive medium (i.e. water) have been observed. By solving a variant of Korteweg-de Vries-Burgers' equation, we showed that such perturbations will happen as a consequence of small bubbles in the medium.
In the second paper we gave an exact solution formula of the sound field from a point source over a large surface with two different materials (the "land-sea" problem). The formula is complex and in the years 1982-86 we worked it out to be easily interpreted near the surface. Our formula gives information on how the surface waves depend on the impedances and information on the farfield solution. Before our work, such formulas had only been given for large homogeneous surfaces.
Perturbations at the vertex of an acoustic sawtooth wave in a slightly dispersive medium (with B. Enflo), Report/Institut Mittag-Leffler, 99-0178853-6 ; 1982:1 Stockholm
An exact solution formula for sound wave propagation from a point source over a surface with an impedance discontinuity (with B. Enflo), J. of Acoustical Soc. of America, Vol. 82 (1987) 2123-2134
We worked out (joint work with A.Spalsbury, R.Heath and myself) numerical solutions of Lotka-Volterra systems with 3 equations - involving zebra mussels, "edible" algae and "inedible" algae and, as a parameter, phosphorus loading of Lake Erie.
The model suggests the following: 1) Loading more phosphsorus into the lake will have a negative effect on the composition of algae. 2) The system will have oscillations, and one can expect more oscillations nearshore than offshore, in the quantity of zebra mussels as well as in the quantity of the different types of algae . 3) If algae had been grazed by a short-lived aggressive species (e.g. Daphnia) rather than by zebra mussels, one would have more oscillations in the quantity of algae than have been observed.
Refugia for edible algae - larger offshore than nearshore - stabilize the system. In fact, the modeling suggests the following general ecological principle: Introducing refugia for prey, will change an oscillating ecological system into an ecological system which is in a stable equilibrium state. So, also outside of the refugium, the oscillations will stop.
Modeling the effects of nutrient concentrations on community production and ecosystem stability: Framework for a Great Lakes Model. (with R.T.Heath, R.Sturtevant and D.Shoup). In Great Lakes Modeling Summit--Focus on Lake Erie. L.Tulen and J.dePinto, Eds., published by Int. Joint Commission, Windsor, ONT., ISBN 1-894280-17-2 (2000), 37-50
Long-term Plankton Community Effects of Zebra Mussels: Mathematical Analysis of a Simple Predator-prey Model With and Without Refugia. (with R.T. Heath). 48th Annual Conference of the International Association for Great Lakes Research, Abstracts, Ann Arbor, 2005, Great lakes ecosystem forecasting : improving understanding and prediction. (2005) p 53
Manuscript: A Model to Study the Effects of the Zebra Mussel and the Effects of Different Levels of Phosphorus Loading in Lake Erie (with R.T.Heath and A.Spalsbury) (2005)
In the 1990:s there was an apparent contradiction between fossil data and DNA data concerning the Neandertals. Some fossil data indicated that there had been interbreeding between Neandertals and Early Moderns. On the other hand, all Neandertal DNA seemed to have disappeared in today's human population. I showed that the concept of "regions, where the populations do not reproduce themselves" could be used to give a simple explanation of the loss of Neandertal genes, even under a scenario of total and random interbreeding between Neandertals and Early Moderns. This model also predicted that one might find a fossil with modern anatomy and mitochondrial DNA very different from ours. Such a find was done in Australia in the end of the year 2000.
The low genetic variation in today's human population has been attributed to a bottleneck in the past in the human population. The concept of "regions where the populations do not reproduce themselves" gives a possible explanation of this low genetic variation, without assuming any bottleneck.
Background and Annika Jensfelt's
A simple reason, why Neandertal ancestry can be consistent with current DNA information Abstract of talk extracted from Program [American Journal of Physical Anthropology, 114 p. 62] Presentation at the Yearly Meeting of American Association of Physical Anthropologists, Kansas City, Missouri, March 28 to March 31, 2001, web publ.>>
Reproduction strategy as a driving force for Human Evolution, Presentation for a Panel discussion on the topic "How unique, in fact, is the animal man? (Hur unik är djuret människa, egentligen?)" Göteborg Book Fair, Gothenburg, Sweden, 2008, web publ.>>
Reproductively disadvantageous regions and archaic humans (with Gustavo A. Muñoz-Fernández, Juan B. Seoane-Sepúlveda) arXiv.org Submitted on 23 May 2017,
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