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| Per Enflo | |
|---|---|
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[[Image:Photograph (Kent State
University)|225px|alt=]] |
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| Born |
1944 Stockholm, Sweden |
| Residence | Kent, Ohio, United States |
| Fields | Functional
analysis Operator theory Analytic number theory |
| Institutions | University of
California, Berkeley Stanford University École Polytechnique, Paris The Royal Institute of Technology, Stockholm Kent State University |
| Alma mater | Stockholm University |
| Doctoral advisor | Hans Rådstrom |
| Doctoral students |
Nilson Bernardes Miguel Lacruz Jan-Ove Larsson Marie Lövblom Bruce Reznick Anthony Weston |
| Known for | Approximation problem Schauder basis Hilbert's fifth problem (infinite-dimensional) uniformly convex renorms of superreflexive Banach spaces embedding metric spaces (unbounded distortion of cube) "Concentration" of polynomials at low degree Invariant subspace problem |
| Influences | Joram
Lindenstrauss Laurent Schwartz |
| Influenced | Bernard Beauzamy |
| Notable awards | Mazur's "live goose" for solving "Scottish Book" Problem 153 |
Per Enflo is a mathematician who has solved fundamental problems in functional analysis. Three of these problems had been open for more than forty years:
In solving these problems, Enflo developed new techniques which were then used by other researchers in functional analysis and operator theory for years. Some of Enflo's research has been important also in other mathematical fields, such as number theory, and in computer science, especially computer algebra and approximation algorithms.
Per Enflo was born in Stockholm, Sweden in 1944. Enflo works at Kent State University, where he holds the title of University Professor. Enflo has earlier held positions at the University of California, Berkeley, Stanford University, École Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm.
Per Enflo is also a concert pianist.
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At Stockholm University, Hans Rådström suggested that Per Enflo consider Hilbert's fifth problem in the spirit of functional analysis.[3] In two years, 1969-1970, Enflo published five papers on Hilbert's fifth problem; these papers are collected in Enflo (1970), along with a short summary. Some of the results of the these papers are described in Enflo (1976) and in the last chapter of Benyamini and Lindenstrauss.
Enflo's techniques have found application in computer science. Algorithm theorists derive approximation algorithms that embed finite metric-spaces into low-dimensional Euclidean spaces with low "distortion" (in Gromov's terminology for the Lipschitz category; c.f. Banach-Mazur distance). Low-dimensional problems have lower computational complexity, of course. More importantly, if the problems embed well in either the Euclidean plane or the three-dimensional Euclidean space, then geometric algorithms become exceptionally fast.
However, such embedding techniques have limitations, as shown by Enflo's (1969) theorem:[4]
,
the Hamming cube Cm cannot be
embedded with "distortion D"
(or less) into 2m-dimensional Euclidean
space if 
.
Consequently, the optimal embedding is the natural embedding, which
realizes {0,1}m as
a subspace of m-dimensional
Euclidean space.[5]This theorem, "found by Enflo [1969], is probably the first result showing an unbounded distortion for embeddings into Euclidean spaces. Enflo considered the problem of uniform embeddability among Banach spaces, and the distortion was an auxiliary device in his proof."[6]
In 1972 Enflo proved that "every super-reflexive Banach space admits an equivalent uniformly convex norm".[7]
At the Scottish Café on 6 November 1936, Stanislaw Mazur posed the "basis problem" of determining whether every Banach space have a Schauder basis, with Mazur promising a "live goose" as a reward: In 1972, Mazur awarded a live goose to Enflo in a ceremony at the Stefan Banach Center in Warsaw; the "goose reward" ceremony was broadcast throughout Poland.
In fact, Enflo's construction also resolved another outstanding problem, the "Approximation Problem", which had been posed and worked on for years by Alexander Grothendieck (just before Grothendieck turned from functional analysis to algebraic geometry). In mathematics, a Banach space is said to have the approximation property, if every compact operator is a limit of finite rank operators. The converse is always true.
Every Hilbert space has the approximation property. Enflo's solution of the basis problem also provided an example of a Banach space that lacked the approximation property. The relation between the approximation problem and the basis problem had been clarified by Grothendieck.[8]
In functional analysis, one of the most prominent problems was the invariant subspace problem, which required the evaluation of the truth of the following proposition:
For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.)
Per Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987[9] Enflo's long "manuscript had a world-wide circulation among mathematicians"[10] and some of its ideas were described in publications besides Enflo (1976).[11] : Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.[9]
In the 1990s, Enflo developed other approaches to the invariant subspace problem, some of which are discussed in a survey article written by Enflo and Victor Lomonosov (2001).
An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers m and n, there exists C(m,n) > 0 such that for all homogeneous polynomials P and Q of degrees m and n (in k variables), then
,
where | P | denotes the sum of the absolute values of the coefficients of P. Enflo proved that C(m,n) does not depend on the number of variables k. Enflo's original proof was simplified by Montgomery.[12]
This result was generalized to other norms on the vector space of homogeneous polynomials. Of these norms, the most used has been the Bombieri norm.
The Bombieri
norm is defined in terms of the following scalar product: 
we have 
if 
we define 
,where we use use the following notation: if 
,
we write 
and 
and 
The most remarkable property of this norm is the Bombieri inequality:
Let P,Q be two homogeneous polynomials
respectively of degree 
and 
with N variables, then, the
following inequality holds:
In the above statement, the Bombieri inequality is the left-hand side inequality; the right-hand side inequality means that the Bombieri norm is a norm of the algebra of polynomials under multiplication.
The Bombieri inequality implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is fundamental in applications like polynomial factorization (or in Enflo's construction of an operator without an invariant subspace).
Enflo's idea of "concentration of polynomials at low degrees" has led to important publications in number theory[13] algebraic and Diophantine geometry[14], and polynomial factorization[15].
Per Enflo has also published articles in mathematical biology, on the DNA of Neanderthal and modern humans (with Hawks & Wolpoff) and on zebra mussels in the Great Lakes of the United States (with Heath).
Per Enflo is also a concert pianist.
A child prodigy in both music and mathematics, Enflo performed in a piano recital at age 8 and gave a solo recital at age 11. Enflo won the Swedish competition for young pianists at age 11 in 1956; he won the same competition in 1961.[16]
In 1999 Enflo competed in the first annual Van Cliburn Foundation’s International Piano Competition for Outstanding Amateurs.[17] Enflo performs regularly around Kent and in a Mozart series in Columbus, Ohio.
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