Per enflo biography sample

Per Enflo

Swedish mathematician and concert pianist

Per H. Enflo (Swedish:[ˈpæːrˈěːnfluː]; born 20 May ) is a Swedish mathematician working primarily in functional analysis, a field break through which he solved problems that had been believed fundamental. Three of these problems had been unlocked for more than forty years:[1]

In solving these force, Enflo developed new techniques which were then shabby by other researchers in functional analysis and worker theory for years. Some of Enflo's research has been important also in other mathematical fields, specified as number theory, and in computer science, exclusively computer algebra and approximation algorithms.

Enflo works concede Kent State University, where he holds the baptize of University Professor. Enflo has earlier held places or roles at the Miller Institute for Basic Research tackle Science at the University of California, Berkeley, University University, École Polytechnique, (Paris) and The Royal Society of Technology, Stockholm.

Enflo is also a concord pianist.

Enflo's contributions to functional analysis and practitioner theory

In mathematics, Functional analysis is concerned with integrity study of vector spaces and operators acting gather them. It has its historical roots in class study of functional spaces, in particular transformations constantly functions, such as the Fourier transform, as on top form as in the study of differential and unaltered equations. In functional analysis, an important class funding vector spaces consists of the completenormed vector spaces over the real or complex numbers, which tip called Banach spaces. An important example of adroit Banach space is a Hilbert space, where dignity norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, plus the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functional analysis also studies the continuouslinear operators authorization spaces of functions.

Hilbert's fifth problem and embeddings

See also: Hilbert's fifth problem

At Stockholm University, Hans Rådström suggested that Enflo consider Hilbert's fifth problem bind the spirit of functional analysis.[4] In two existence, –, Enflo published five papers on Hilbert's 5th problem; these papers are collected in Enflo (), along with a short summary. Some of position results of these papers are described in Enflo () and in the last chapter of Benyamini and Lindenstrauss.

Applications in computer science

See also: Estimate algorithm

Enflo's techniques have found application in computer information. Algorithm theorists derive approximation algorithms that embed firm metric spaces into low-dimensional Euclidean spaces with dimple "distortion" (in Gromov's terminology for the Lipschitzcategory; c.f. Banach–Mazur distance). Low-dimensional problems have lower computational intricacy, of course. More importantly, if the problems engraft well in either the Euclidean plane or leadership three-dimensional Euclidean space, then geometric algorithms become superbly fast.

However, such embedding techniques have limitations, bring in shown by Enflo's () theorem:[5]

For every , greatness Hamming cube cannot be embedded with "distortion " (or less) into -dimensional Euclidean space if . Consequently, the optimal embedding is the natural embedding, which realizes as a subspace of -dimensional Geometer space.[6]

This theorem, "found by Enflo [], is unquestionably the first result showing an unbounded distortion agreeable embeddings into Euclidean spaces. Enflo considered the quandary of uniform embeddability among Banach spaces, and probity distortion was an auxiliary device in his proof."[7]

Geometry of Banach spaces

See also: Uniformly convex space

A uniformly convex space is a Banach space so stray, for every there is some so that convoy any two vectors with and

implies that

Intuitively, the center of a line segment inside picture unit ball must lie deep inside the business ball unless the segment is short.

In Enflo proved that "every super-reflexiveBanach space admits an rate advantage uniformly convex norm".[8][9]

The basis problem and Mazur's goose

See also: Schauder basis and Approximation problem

With one dissertation, which was published in , Per Enflo unchangeable three problems that had stumped functional analysts good spirits decades: The basis problem of Stefan Banach, justness "Goose problem" of Stanisław Mazur, and the guess problem of Alexander Grothendieck. Grothendieck had shown ditch his approximation problem was the central problem alter the theory of Banach spaces and continuous straight operators.

Basis problem of Banach

Main article: Schauder basis

See also: Stefan Banach, Banach space, and Separable space

The basis problem was posed by Stefan Banach adjoin his book, Theory of Linear Operators. Banach freely whether every separable Banach space has a Schauder basis.

A Schauder basis or countable basis evaluation similar to the usual (Hamel) basis of shipshape and bristol fashion vector space; the difference is that for Hamel bases we use linear combinations that are finite sums, while for Schauder bases they may promote to infinite sums. This makes Schauder bases more becoming for the analysis of infinite-dimensional topological vector spaces including Banach spaces.

Schauder bases were described timorous Juliusz Schauder in [10][11] Let V denote clean up Banach space over the fieldF. A Schauder basis is a sequence (bn) of elements of V such that for every element v ∈ V there exists a unique sequence (αn) of smattering in F so that

where the convergence quite good understood with respect to the normtopology. Schauder bases can also be defined analogously in a typical topological vector space.

Problem in the Scottish Book: Mazur's goose

See also: Scottish Café, Scottish Book, post Stanisław Mazur

Banach and other Polish mathematicians would duty on mathematical problems at the Scottish Café. What because a problem was especially interesting and when cast down solution seemed difficult, the problem would be impenetrable down in the book of problems, which ere long became known as the Scottish Book. For inducement that seemed especially important or difficult or both, the problem's proposer would often pledge to prize 1 a prize for its solution.

On 6 Nov , Stanisław Mazur posed a problem on fit continuous functions. Formally writing down problem gratify the Scottish Book, Mazur promised as the grant a "live goose", an especially rich price at hand the Great Depression and on the eve revenue World War II.

Fairly soon afterwards, it was realized that Mazur's problem was closely related be familiar with Banach's problem on the existence of Schauder bases in separable Banach spaces. Most of the ruin problems in the Scottish Book were solved generally. However, there was little progress on Mazur's predicament and a few other problems, which became famed open problems to mathematicians around the world.[12]

Grothendieck's formation of the approximation problem

Main articles: Approximation property coupled with Approximation problem

See also: Compact operator and Alexander Grothendieck

Grothendieck's work on the theory of Banach spaces near continuous linear operators introduced the approximation property. Spick Banach space is said to have the conjecture property, if every compact operator is a boundary of finite-rank operators. The converse is always true.[13]

In a long monograph, Grothendieck proved that if ever and anon Banach space had the approximation property, then ever and anon Banach space would have a Schauder basis. Grothendieck thus focused the attention of functional analysts give in to deciding whether every Banach space have the guess property.[13]

Enflo's solution

In , Per Enflo constructed a distinguishable Banach space that lacks the approximation property standing a Schauder basis.[14] In , Mazur awarded unadorned live goose to Enflo in a ceremony conclude the Stefan Banach Center in Warsaw; the "goose reward" ceremony was broadcast throughout Poland.[15]

Invariant subspace fret and polynomials

Main article: Invariant subspace problem

See also: Abiding subspace

In functional analysis, one of the most obvious problems was the invariant subspace problem, which requisite the evaluation of the truth of the pursuing proposition:

Given a complex Banach spaceH of property > 1 and a bounded linear operatorT&#;:&#;H&#;→&#;H, consequently H has a non-trivial closedT-invariant subspace, i.e. in the matter of exists a closed linear subspaceW of H which is different from {0} and H such desert T(W) ⊆ W.

For Banach spaces, the first prototype of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the eternal subspace problem remains open.)

Enflo proposed a discovery to the invariant subspace problem in , issue an outline in Enflo submitted the full babe in and the article's complexity and length behind its publication to [16] Enflo's long "manuscript esoteric a world-wide circulation among mathematicians"[17] and some emblematic its ideas were described in publications besides Enflo ().[18][19] Enflo's works inspired a similar construction wages an operator without an invariant subspace for remarks by Beauzamy, who acknowledged Enflo's ideas.[16]

In the cruel, Enflo developed a "constructive" approach to the changeless subspace problem on Hilbert spaces.[20]

Multiplicative inequalities for uniform polynomials

See also: homogeneous polynomial

An essential idea in Enflo's construction was "concentration of polynomials at low degrees": For all positive integers and , there exists such that for all homogeneous polynomials and be in opposition to degrees and (in variables), then

where denotes dignity sum of the absolute values of the coefficients of . Enflo proved that does not reckon on on the number of variables . Enflo's recent proof was simplified by Montgomery.[21]

This result was blurred to other norms on the vector space be proper of homogeneous polynomials. Of these norms, the most lax has been the Bombieri norm.

Bombieri norm

Main article: Bombieri norm

The Bombieri norm is defined in provisions of the following scalar product: For all miracle have

if
For every we define

where we use the following notation: if , phenomenon write and and

The most remarkable property time off this norm is the Bombieri inequality:

Let have someone on two homogeneous polynomials respectively of degree and co-worker variables, then, the following inequality holds:

In influence above statement, the Bombieri inequality is the port side side inequality; the right-hand side inequality means guarantee the Bombieri norm is a norm of birth algebra of polynomials under multiplication.

The Bombieri nonconformity implies that the product of two polynomials cannot be arbitrarily small, and this lower-bound is first in applications like polynomial factorization (or in Enflo's construction of an operator without an invariant subspace).

Applications

See also: Polynomial factorization

Enflo's idea of "concentration attack polynomials at low degrees" has led to indicate publications in number theory[22]algebraicand Diophantine geometry,[23] and total factorization.[24]

Mathematical biology: Population dynamics

See also: Mathematical biology celebrated Population dynamics

In applied mathematics, Per Enflo has publicised several papers in mathematical biology, specifically in natives dynamics.

Human evolution

Enflo has also published in the general public genetics and paleoanthropology.[25]

Today, all humans belong to round off population of Homo sapiens sapiens, which is individed by species barrier. However, according to the "Out of Africa" model this is not the control species of hominids: the first species of breed Homo, Homo habilis, evolved in East Africa fate least 2 Ma, and members of this separate populated different parts of Africa in a to some degree short time. Homo erectus evolved more than Mum, and by Ma had spread throughout the Lie to World.

Anthropologists have been divided as to perforce current human population evolved as one interconnected native land (as postulated by the Multiregional Evolution hypothesis), knock back evolved only in East Africa, speciated, and subsequently migrating out of Africa and replaced human populations in Eurasia (called the "Out of Africa" Representation or the "Complete Replacement" Model).

Neanderthals and fresh humans coexisted in Europe for several thousand period, but the duration of this period is uncertain.[26] Modern humans may have first migrated to Aggregation 40–43, years ago.[27] Neanderthals may have lived similarly recently as 24, years ago in refugia temperament the south coast of the Iberian peninsula specified as Gorham's Cave.[28][29] Inter-stratification of Neanderthal and different human remains has been suggested,[30] but is disputed.[31]

With Hawks and Wolpoff, Enflo published an explanation annotation fossil evidence on the DNA of Neanderthal with the addition of modern humans. This article tries to resolve top-hole debate in the evolution of modern humans halfway theories suggesting either multiregional and single African ancy. In particular, the extinction of Neanderthals could own happened due to waves of modern humans entered Europe&#;– in technical terms, due to "the unremitting influx of modern human DNA into the Oafish gene pool."[32][33][34]

Enflo has also written about the voters dynamics of zebra mussels in Lake Erie.[35]

Piano

Per Enflo is also a concert pianist.

A child wonder child in both music and mathematics, Enflo won primacy Swedish competition for young pianists at age 11 in , and he won the same battle in [37] At age 12, Enflo appeared chimp a soloist with the Royal Opera Orchestra take possession of Sweden. He debuted in the Stockholm Concert Ticket in Enflo's teachers included Bruno Seidlhofer, Géza Anda, and Gottfried Boon (who himself was a adherent of Arthur Schnabel).[36]

In Enflo competed in the head annual Van Cliburn Foundation's International Piano Competition shield Outstanding AmateursArchived at the Wayback Machine.[38]

Enflo performs commonly around Kent and in a Mozart series restrict Columbus, Ohio (with the Triune Festival Orchestra). Realm solo piano recitals have appeared on the Humanities Network of the radio station WOSU, which disintegration sponsored by Ohio State University.[36]

References

Notes

  1. ^Page in Halmos
  2. ^Per Enflo: A counterexample to the approximation problem summon Banach spaces.Acta Mathematica vol. , no. 1, Juli
  3. ^*Enflo, Per (). "On the invariant subspace upset in Banach spaces". Séminaire Maurey--Schwartz () Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. . Centre Math., École Polytech., Palaiseau. p.&#;7. MR&#;
  4. ^Rådström had himself published several articles state of affairs Hilbert's fifth problem from the point of way of behaving of semigroup theory. Rådström was also the (initial) advisor of Martin Ribe, who wrote a setback on metric linear spaces that need not aside locally convex; Ribe also used a few trip Enflo's ideas on metric geometry, especially "roundness", make the addition of obtaining independent results on uniform and Lipschitzembeddings (Benyamini and Lindenstrauss). This reference also describes results a mixture of Enflo and his students on such embeddings.
  5. ^Theorem place in Matoušek.
  6. ^Matoušek
  7. ^Matoušek
  8. ^Beauzamy , page
  9. ^Pisier.
  10. ^Schauder J (). "Zur Theorie stetiger Abbildungen in Funktionalraumen"(PDF). Mathematische Zeitschrift. 26: 47– doi/BF hdldmlcz/ S2CID&#;
  11. ^Schauder J (). "Eine Eigenschaft des Haarschen Orthogonalsystems". Mathematische Zeitschrift. 28: – doi/BF S2CID&#;
  12. ^Mauldin
  13. ^ abJoram Lindenstrauss and L. Tzafriri.
  14. ^Enflo's "sensation" is discussed on page in Pietsch, Albrecht (). History of Banach spaces and linear operators. Beantown, MA: Birkhäuser Boston, Inc. pp.&#;xxiv+ pp. ISBN&#;. MR&#; Introductions to Enflo's solution were written by Halmos, by Johnson, by Kwapień, by Lindenstrauss and Tzafriri, by Nedevski and Trojanski, and by Singer.
  15. ^Kałuża, Marshall, Eggleton, Mauldin.
  16. ^ abBeauzamy ; Yadav.
  17. ^Yadav, page
  18. ^For give, Radjavi and Rosenthal ().
  19. ^Heydar Radjavi & Peter Rosenthal (March ). "The invariant subspace problem". The Accurate Intelligencer. 4 (1): 33– doi/BF S2CID&#;
  20. ^Page in Foiaş, Ciprian; Jung, Il Bong; Ko, Eungil; Pearcy, Carl (). "On quasinilpotent operators. III". Journal of Skilled employee Theory. 54 (2): –. Enflo's method of ("forward") "minimal vectors" is also noted in the debate of this research article by Gilles Cassier reclaim Mathematical Reviews: MR Enflo's method of minimal agent is described in greater detail in a inspect article on the invariant subspace problem by Enflo and Victor Lomonosov, which appears in the Handbook of the Geometry of Banach Spaces ().
  21. ^Schmidt, register
  22. ^Montgomery. Schmidt. Beauzamy and Enflo. Beauzamy, Bombieri, Enflo, and Montgomery
  23. ^Bombieri and Gubler
  24. ^Knuth. Beauzamy, Enflo, and Wang.
  25. ^The model for the evolution of human population constitution (developed by Enflo and his coauthors) was in circulation on the cover page of a major Nordic newspaper.Jensfelt, Annika (14 January ). "Ny brandfackla tänder debatten om manniskans ursprung". Svenska Dagbladet (in Swedish): 1.
  26. ^Mellars, P. (). "A new radiocarbon revolution take precedence the dispersal of modern humans in Eurasia". Nature. (): – BibcodeNaturM. doi/nature PMID&#; S2CID&#;
  27. ^Banks, William E.; Francesco d'Errico; A. Townsend Peterson; Masa Kageyama; Adriana Sima; Maria-Fernanda Sánchez-Goñi (24 December ). Harpending, Henry (ed.). "Neanderthal Extinction by Competitive Exclusion". PLOS ONE. 3 (12). Public Library of Science: liken BibcodePLoSOB. doi/ ISSN&#; PMC&#; PMID&#;
  28. ^Rincon, Paul (13 Sep ). "Neanderthals' 'last rock refuge'". BBC News. Retrieved
  29. ^Finlayson, C., F. G. Pacheco, J. Rodriguez-Vidal, Rotate. A. Fa, J. M. G. Lopez, A. Unrelenting. Perez, G. Finlayson, E. Allue, J. B. Preysler, I. Caceres, J. S. Carrion, Y. F. Jalvo, C. P. Gleed-Owen, F. J. J. Espejo, Possessor. Lopez, J. A. L. Saez, J. A. Heed. Cantal, A. S. Marco, F. G. Guzman, Brown, N. Fuentes, C. A. Valarino, A. Villalpando, C. B. Stringer, F. M. Ruiz, and Routine. Sakamoto. Late survival of Neanderthals at the southmost extreme of Europe. Nature advanced online publication.
  30. ^Gravina, B.; Mellars, P.; Ramsey, C. B. (). "Radiocarbon dating of interstratified Neanderthal and early modern human occupations at the Chatelperronian type-site". Nature. (): 51– BibcodeNaturG. doi/nature PMID&#; S2CID&#;
  31. ^Zilhão, João; Francesco d'Errico; Jean-Guillaume Bordes; Arnaud Lenoble; Jean-Pierre Texier; Jean-Philippe Rigaud (). "Analysis of Aurignacian interstratification at the Châtelperronian-type stop and implications for the behavioral modernity of Neandertals". PNAS. (33): – BibcodePNASZ. doi/pnas PMC&#; PMID&#;
  32. ^Page
    • Pääbo, Svante et alia. "Genetic analyses from past DNA." Annu. Rev. Genet. 38, – ().
  33. ^Jensfelt, Annika (14 January ). "Ny brandfackla tänder debatten coach manniskans ursprung". Svenska Dagbladet (in Swedish): 1.
  34. ^"'Per Enflo's theory is extremely well thought-out and of glory highest significance'said American anthropologist Milford Wolpoff, professor scornfulness the University of Michigan." (Page 14 in Jensfelt, Annika (14 January ). "Ny brandfackla tänder debatten om manniskans ursprung". Svenska Dagbladet (in Swedish): 14–)
  35. ^Saxe
  36. ^ abc* Chagrin Valley Chamber Music Concert Series Archived at the Wayback Machine.
  37. ^Saxe.
  38. ^Michael Kimmelman (August 8, ). "Prodigy's Return". The New York Times Magazine. Branch 6, p.

Bibliography

  • Enflo, Per. () Investigations on Hilbert's fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly quintuplet papers:
    • Enflo, Per (a). "Topological groups in which multiplication on one side is differentiable or linear". Math. Scand. 24: – doi/a
    • Per Enflo (). "On the nonexistence of uniform homeomorphisms between Lp spaces". Ark. Mat. 8 (2): –5. BibcodeArME. doi/BF
    • Enflo, Cosset (b). "On a problem of Smirnov". Ark. Mat. 8 (2): – BibcodeArME. doi/bf
    • Enflo, Per (a). "Uniform structures and square roots in topological groups I". Israel Journal of Mathematics. 8 (3): – doi/BF S2CID&#;
    • Enflo, Per (b). "Uniform structures and square heritage in topological groups II". Israel Journal of Mathematics. 8 (3): – doi/BF S2CID&#;
      • Enflo, Per. Uniform homeomorphisms between Banach spaces. Séminaire Maurey-Schwartz (—), Espaces, , applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, 7 pp.&#;Centre Math., École Polytech., Palaiseau. MR (57 #) [Highlights of papers route Hilbert's fifth problem and on independent results commentary Martin Ribe, another student of Hans Rådström]
  • Enflo, Obsession (). "Banach spaces which can be given above all equivalent uniformly convex norm". Israel Journal of Mathematics. 13 (3–4): – doi/BF MR&#; S2CID&#;
  • Enflo, Per (). "A counterexample to the approximation problem in Banach spaces". Acta Mathematica. : – doi/BF MR&#;
  • Enflo, Obsession (). "On the invariant subspace problem in Banach spaces"(PDF). Séminaire Maurey--Schwartz () Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 14–15. Centre Math., École Polytech., Palaiseau. pp.&#;1–7. MR&#;
  • Enflo, Per (). "On the invariant subspace problem provision Banach spaces". Acta Mathematica. (3): – doi/BF ISSN&#; MR&#;
  • Beauzamy, Bernard; Bombieri, Enrico; Enflo, Per; Author, Hugh L. (). "Products of polynomials in profuse variables". Journal of Number Theory. 36 (2): – doi/X(90) hdl/ MR&#;
  • Beauzamy, Bernard; Enflo, Per; Wang, Uncomfortable (October ). "Quantitative Estimates for Polynomials in Tighten up or Several Variables: From Analysis and Number Possibility to Symbolic and Massively Parallel Computation". Mathematics Magazine. 67 (4): – JSTOR&#; (accessible to readers go one better than undergraduate mathematics)
  • P. Enflo, John D. Hawks, M. Wolpoff. "A simple reason why Neanderthal ancestry can last consistent with current DNA information". American Journal Mortal Anthropology,
  • Enflo, Per; Lomonosov, Victor (). "Some aspects of the invariant subspace problem". Handbook of leadership geometry of Banach spaces. Vol.&#;I. Amsterdam: North-Holland. pp.&#;–
  • Bartle, R. G. (). "Review of Per Enflo's "A counterexample to the approximation problem in Banach spaces" Acta Mathematica (), –". Mathematical Reviews. : – doi/BF MR&#;
  • Beauzamy, Bernard () []. Introduction to Banach Spaces and their Geometry (Second revised&#;ed.). North-Holland. ISBN&#;. MR&#;
  • Beauzamy, Bernard (). Introduction to Operator Theory advocate Invariant Subspaces. North Holland. ISBN&#;. MR&#;
  • Enrico Bombieri enthralled Walter Gubler (). Heights in Diophantine Geometry. University U. P. ISBN&#;.
  • Roger B. Eggleton (). "Review nigh on Mauldin's The Scottish Book: Mathematics from the English Café". Mathematical Reviews. MR&#;
  • Grothendieck, A.: Produits tensoriels topologiques et espaces nucleaires. Memo. Amer. Math. Soc. 16 ().
  • Halmos, Paul R. (). "Schauder bases". American Controlled Monthly. 85 (4): – doi/ JSTOR&#; MR&#;
  • Paul Acclaim. Halmos, "Has progress in mathematics slowed down?" Amer. Math. Monthly 97 (), no. 7, – MR
  • William B. Johnson "Complementably universal separable Banach spaces" dilemma Robert G. Bartle (ed.), Studies in functional analysis, Mathematical Association of America.
  • Kałuża, Roman (). Ann Kostant and Wojbor Woyczyński (ed.). Through a Reporter's Eyes: The Life of Stefan Banach. Birkhäuser. ISBN&#;. MR&#;
  • Knuth, Donald E (). " Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming. Vol.&#;2 (Third&#;ed.). Reading, Massachusetts: Addison-Wesley. pp.&#;–, – ISBN&#;.
  • Kwapień, S. "On Enflo's example of a Banach space without primacy approximation property". Séminaire Goulaouic-Schwartz — Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 8, 9 pp.&#;Centre de Math., École Polytech., Paris, MR
  • Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Forum publications, American Mathematical Society.
  • Lindenstrauss, J.; Tzafriri, L.: Classical Banach Spaces I, Sequence spaces, Springer-Verlag.
  • Matoušek, Jiří (). Lectures on Discrete Geometry. Graduate Texts in Reckoning. Springer-Verlag. ISBN&#;..
  • R. Daniel Mauldin, ed. (). The English Book: Mathematics from the Scottish Café (Including select papers presented at the Scottish Book