Geometrical Visions

1
Geometrical Visions

• The distinctive styles
• of Klein and Lie

2
Uses and Abuses of Style as an Explanatory Concept

• Style a vague and problematic notion
• Particularly problematic when extended to
  mathematical schools, research communities or
  national traditions (Duhem on German vs. French
  science)
• Or when used to discredit opponents
• (Bieberbachs use of racial stereotypes against
  Landau and others)

3
Types of Mathematical Creativity

• Hilbert as an algebraist, even when doing
  geometry
• Poincaré as a geometer, even when doing analysis
• Weyl commenting on Hilberts Zahlbericht
• Van der Waerden (algebraist) accounting for why
  Weyl (analyst) gave up Brouwers intuitionism

4
Different Views of Hilberts Work on Foundations
of Geometry

• Hans Freudenthal emphasized the modern elements
  how he broke the umbilical cord that connected
  geometry with investigations of the natural world
• Leo Corry emphasizes the empiricist elements that
  motivated Hilberts axiomatic approach to
  geometry but also his larger program for
  axiomatizing all exact sciences

5
Hilbert as a Classical Geometer

• Hilberts work can also be seen within the
  classical tradition of geometric problem solving
  (Pappus, Descartes)
• Greek tradition construction with straight edge
  and compass, conics (Knorr)
• Descartes more general instruments used to
  construct special types of algebraic curves (Bos)
• Hilbert, like Descartes, saw geometric problem
  solving as a paradigm for epistemology

6

• Methodological challenge to develop a systematic
  way to determine whether a well-posed geometrical
  problem can be solved with specified means
• Descartes showed that a problem which can be
  transformed into a quadratic equation can be
  solved by straight edge and compass
• 19th-century mathematicians used new methods to
  prove that trisecting an angle and doubling a
  cube could not be constructed using Euclidean
  tools

7
Impossibility Proofs

• Ferdinand Lindemann showed in 1882 that p is a
  transcendental number
• So even Descartes system of algebraic curves is
  insufficient for squaring the circle
• Hilbert regarded this as an important result, so
  he gave a new proof in 4 pages!
• He emphasized the importance of impossibility
  proofs in his famous Paris address on
  Mathematical Problems
• Not all problems are created equal he gave
  general criteria for those which are fruitful

8
Hilberts geometric vision

• Doing synthetic geometry with given constructive
  means corresponds to doing analytic geometry over
  a particular algebraic number field
• Solvability of a geometric problem is equivalent
  to deciding whether the corresponding algebraic
  equation has solutions in the field
• Paradigm for Hilberts 24th Paris problem to
  show that every well-posed mathematical problem
  has a definite answer (refutation of du Bois
  Reymonds Ignorabimus)

9
Hilberts Schlusswort aus derGrundlagen der
Geometrie
10

• Die vorstehende Abhandlung ist eine kritische
  Untersuchung der Prinzipien der Geometrie in
  dieser Untersuchung leitete uns der Grundsatz,
  eine jede sich darbietende Frage in der Weise zu
  erörtern, dass wir zugleich prüften, ob ihre
  Beantwortung auf einem vorgeschriebenen Wege mit
  gewissen eingeschränkten Hilfsmitteln möglich
  oder nicht möglich ist. Dieser Grundsatz scheint
  mir eine allgemeine und naturgemäße Vorschrift zu
  enthalten in der Tat wird, wenn wir bei unseren
  mathematischen Betrachtungen einem Probleme
  begegnen oder einen Satz vermuten, unser
  Erkenntnistrieb erst dann befriedigt, wenn uns
  entweder die völlige Lösung jenes Problem und der
  strenge Beweis dieses Satzes gelingt oder wenn
  der Grund für die Unmöglichkeit des Gelingens und
  damit zugleich die Notwendigkeit des Misslingens
  von uns klar erkannt worden ist.

11

• So spielt dann in der neueren Mathematik die
  Frage nach der Unmöglichkeit gewisser Lösungen
  oder Aufgaben eine hervorragende Rolle und das
  Bestreben, eine Frage solcher Art zu beantworten,
  war oftmals der Anlass zur Entdeckung neuer und
  fruchtbarer Forschungsgebiete. Wir erinnern nur
  an Abels Beweise für die Unmöglichkeit der
  Auflösung der Gleichungen fünften Grades durch
  Wurzelziehen, ferner an die Erkenntnis der
  Unbeweisbarkeit des Parallelaxioms und an
  Hermites und Lindemanns Sätze von der
  Unmöglichkeit, die Zahlen e und p auf
  algebraischem Wege zu konstruieren.

12

• Der Grundsatz, demzufolge man überall die
  Prinzipien der Möglichkeit der Beweise erläutern
  soll, hängt auch aufs Engste mit der Forderung
  der Reinheit der Beweismethoden zusammen, die
  von mehreren Mathematikern der neueren Zeit mit
  Nachdruck erhoben worden ist. Diese Forderung ist
  im Grunde nichts anderes als eine subjektive
  Fassung des hier befolgten Grundsatzes. In der
  Tat sucht die vorstehende geometrische
  Untersuchung allgemein darüber Aufschluss zu
  geben, welche Axiome, Voraussetzungen oder
  Hilfsmittel zu Beweise einer elementar-geometrisch
  en Wahrheit nötig sind, und es bleibt dann dem
  jedesmaligen Ermessen anheim gestellt, welche
  Beweismethode von dem gerade eingenommenen
  Standpunkte aus zu bevorzugen ist.

13
Klein and Lie as Creative Mathematicians

• Two full-blooded geometers

14
Kleins Universality

• Felix Klein was fascinated by questions of style
  and discussed it often in his lectures
• On a number of occasions he described Sophus
  Lies style as a geometer
• Geometry, for Klein, was essentially a
  springboard to a way of thinking about
  mathematics in general
• This is surely the most striking and also
  impressive feature in his research, which covered
  many parts of pure and applied mathematics

15
Felix Klein as a Young Admirer and Collaborator
of Lie

• Studied line geometry with Plücker in Bonn,
  1865-1868
• Protégé of Clebsch in Göttingen projective
  algebraic geometry
• Met Lie in Berlin, 1869
• Presented his work in Kummers seminar

16
Kleins first great discovery

• Lie was nowhere near as broad as Klein would
  become, but he was far deeper
• It is only a slight exaggeration to say that
  Klein discovered Lie
• During the early 1870s he was virtually the only
  one who had any understanding of Lies
  mathematics
• He described how Lie spent whole days living in
  the spaces he imagined

17
On Lies Relationship with Klein

• D. Rowe, Der Briefwechsel Sophus Lie Felix
  Klein, eine Einsicht in ihre persönlichen und
  wissenschaftlichen Beziehungen, NTM, 25 (1988)1,
  37-47.
• Sophus Lies Letters to Felix Klein,
• 1876-1898, ed. D. Rowe, to appear

18
Sophus Lie a Norwegian Hero
19

• Arild Stubhaugs Heroic Portrait of Lie, the
  Norwegian Patriot
• Interpretation of Lies Life as a Triumphant
  Struggle
• Story of Friends, Foes and Betrayal
• Subsidiary Theme French wisdom vs. German
  petty-mindedness

20
Sophus Lie, 1844-1899

• 1865-68 study at Univ. Christiania
• 1869-70 stipend to study in Berlin, Paris
• 1869-72 collaboration with Felix Klein
• 1872-86 Prof. in Christiania
• 1886-98 Leipzig
• 1898 return to Norway

21
On Lies Mathematics

• Hans Freudenthal, Marius Sophus Lie, Dictionary
  of Scientific Biography.
• Thomas Hawkins, Jacobi and the Birth of Lies
  Theory of Groups, Archive for History of Exact
  Sciences, 1991.

22
On Lies Early Work

• D. Rowe, The Early Geometrical Works of Felix
  Klein and Sophus Lie
• T. Hawkins, Line Geometry, Differential
  Equations, and the Birth of Lies Theory of
  Groups
• In The History of Modern Mathematics, vol. 1, ed.
  D. Rowe and J. McCleary, 1989.

23
Lies Early Career

• 1868-71 line and sphere geometry special
  contact transformations
• 1871-73 PDEs and line complexes general concept
  of contact transformations
• 1873-74 Lies vision for a Galois theory of
  differential equations

24
Lies Subsequent Career

• 1874-77 first work on continuous transformation
  groups classification of groups for line and
  plane
• 1877-82 return to geometry applications of
  group theory to differential geometry, in
  particular minimal surfaces
• 1882-85 group-theoretic investigations and
  differential invariants (with Friedrich Engel
  beginning 1884)

25
Lies Subsequent Career

• 1886 succeeds Klein as professor of geometry in
  Leipzig
• Continued collaboration with Engel on vol. 1 of
  Theorie der Transformationsgruppen
• 1889-90 Lie spends nine months at a sanatorium
  outside Hannover leaves without having fully
  recovered
• 1890-91 works on Riemann-Helmholtz space problem

26
On the History of Lie Theory

• Thomas Hawkins, Emergence of the Theory of Lie
  Groups. An Essay in the History of Mathematics,
  1869-1926, Springer 2000.
• Four Parts
• Sophus Lie, Wilhlem Killing,
• Élie Cartan, and Hermann Weyl

27
German line geometry andFrench sphere geometry

• 4-dimensional geometries derived from
  3-dimensional space

28
Julius Plücker and the Theory of Line Complexes

• Plücker took lines of space as elements of a
  4-dim geometry
• Algebraic equation of degree n leads to an
  nth-order line complex
• Locally, the lines through a point determine a
  cone of the nth degree
• Counterpart to French sphere geometry

29
Lie and Klein geometries based on free choice of
the space elements

• Line and sphere geometry were central examples
• Klein also studied spaces of line complexes in
  1860s, the space of cubic surfaces (1873), etc.
• In his Erlangen Program he emphasizes that the
  dimension of the geometry is insignificant, since
  one can always let the same group act on
  different spaces obtained by varying the space
  element, which may depend on an arbitrary number
  of coordinates

30
Kummer surfaces and their physical and
geometrical contexts
31
Kummer Surfaces

• Quartic surfaces with 16 double points (here all
  are real)
• Klein was the first to study these as the
  singularity surfaces that naturally arise for
  families of 2nd-degree line complexes

32
The Fresnel Wave Surface

• Kummers study of ray systems revealed that the
  Fresnel surface was a special type of Kummer
  surface
• It has 4 real and 12 complex double points

33
Lies Breakthrough, Summer 1870

• Line-to-sphere transformation
• Maps the principle tangent curves of one surface
  onto the lines of curvature of a second surface
• Lie applied this to show that the principle
  tangent curves of the Kummer surface were
  algebraic curves of degree 16
• Klein recognized that they were identical to
  curves he had obtained in his work on line
  geometry

34
Kleins Correspondence with Lie

• Used by Friedrich Engel in Band 7 of Lies
  Collected Works
• Fell into Hands of Ernst Hölder, son of Otto
  Hölder, who married one of Lies granddaughters
• Purchased by the Oslo University Library
• To be published by Springer in a German/English
  edition

35

• Kleins letters to Lie, 1870-1872
• Collaboration in Berlin and Paris, 1869-1870
• Klein had trouble following Lies ideas by 1871
• Lies visit in summer 1872 led to enriched
  version of Kleins Erlanger Programm

36
Kleins Style as a Geometer
37
Felix Klein as a Young Admirer of Riemann

• Came in Contact with Riemanns Ideas through
  Clebsch in Göttingen (1869-1872)
• Competed as self-appointed champion of Riemann
  with leading members of the Weierstrass school

38
Alfred Clebsch (1833-1872)

• Leading Southern German mathematician of the
  era
• Founder of Mathematische Annalen
• Klein was youngest member of the Clebsch School

39
Kleins Physical Mathematics

• Accounting for the Connection between singular
  points and the genus of a Riemann surface

40
Klein (borrowing from Maxwell) to Visualize
Harmonic Functions
41
Building complex functions on an abstract Riemann
surface

• Rather than introducing complex functions in the
  plane and then building Riemann surfaces over C,
  Klein began with a non-embedded surface of
  appropriate genus
• The harmonic functions were then introduced using
  current flows as before
• He visualized their behavior under deformations
  that affected the genus of the surface

42
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43
Klein on Visualizing Projective Riemann Surfaces

• Mathematische Annalen, 1873-76

44
Identifying Real and Imaginary Points on Real
Algebraic Curves

• Riemann and Clebsch had dealt with the genus of a
  curve as a fundamental birational invariant
• Klein wanted to find a satisfying topological
  interpretation of the genus which preserved the
  real points of the curve
• He did this by building a projective surface in
  3-space around an image of the real part of the
  curve in a plane

45
Carl Rodenbergs Modelsfor Cubic Surfaces
46
The Clebsch Model for a Diagonal Surface

• Klein studied cubics with Clebsch in Göttingen in
  1872
• Clebsch came up with this special case of a
  non-singular cubic where all 27 lines are real
• There are 10 Eckhard points where 3 of the 27
  lines meet

47
Klein on Constructing Models (1893)

• It may here be mentioned as a general rule,
  that in selecting a particular case for
  constructing a model the first prerequisite is
  regularity. By selecting a symmetrical form for
  the model, not only is the execution simplified,
  but what is of more importance, the model will be
  of such a character as to impress itself readily
  on the mind.

48
Klein on his Research on Cubics

• Instigated by this investigation of Clebsch, I
  turned to the general problem of determining all
  possible forms of cubic surfaces. I established
  the fact that by the principle of continuity all
  forms of real surfaces of the third order can be
  derived from the particular surface having four
  real conical points. . . .

49
A Cubic with 4 singular points

• Klein began by considering a cubic with 4
  singular points located in the vertices of a
  tetrahedron
• The 27 lines collapse into the 6 edges of the
  tetrahedron

50
Removing Singularities by Deformations

• Two basic types of deformations
• The first splits the surfaces at the singular
  points
• The second enlarges the surface around the
  singularity

51
Moving about in the Space of Cubic Surfaces

• The nonsingular cubics form a 19-dimensional
  manifold
• Those with a single conical point form an
  18-dimensional submanifold, and so on
• So starting with the special point in the
  15-dimensional submanifold with 4 singularities,
  Klein could move up step by step through the
  entire manifold to exhaust the classification

52
Vision behind this research

• What is of primary importance is the
  completeness of enumeration resulting from my
  point of view it would be of comparatively
  little value to derive any number of special
  forms if it cannot be proved that the method used
  exhausts the subject. Models of the typical cases
  of all the principal forms of cubic surfaces have
  since been constructed by Rodenberg for Brills
  collection.

53
Some Stylistic Elements in Lies Early Work
54
Scheffers editions of Lies lectures

• 1891-1896 Georg Scheffers writes three books
  based on Lies lectures
• 1) DEQs with known infinitesimal Transformations
  (1891)
• 2) Continuous Groups (1893)
• 3) Geometry of Contact Transformations (1896)

55
Solving Differential Equations

• According to Engel, Lie had already realized in
  1869 that an ordinary first-order DEQ
• can be reduced to quadratures if one can find a
  one-parameter family of transformations that
  leaves the DEQ invariant.

56

• By 1872 Lie saw that it was enough to have an
  infinitesimal transformation that generated the
  1-parameter group. Thus if the DEQ
• admits a known infinitesimal transformation
• in which, however, the individual integral curves
  do not remain invariant, then the DEQ has an
  integrating factor.

57

• The integrating factor
• then leads directly to a solution by quadrature
  in the form

58
Lies geometric interpretation of the integrating
factor
59
Lies Work on Tetrahedral Complexes

• A tetrahedral line complex consists of the lines
  in space that meet the four planes of a
  coordinate tetrahedron in a fixed cross ratio
• Such complexes were studied earlier by Theodor
  Reye and so were sometimes known as Reyesche
  Komplexe
• Lie generated such complexes by letting a
  3-parameter group act on a given line

60
Lie and Klein study W-Kurven

• Earliest jointly published work of Lie and Klein
  dealt with W-Kurven (W Wurf, an allusion to
  Staudts theory)
• Such curves in the plane are left invariant by a
  1-parameter subgroup of the projective group
  acting on the plane
• They work on W-Kurven and W-Flächen in space, but
  find this too complicated and tedious, so they
  never finish their manuscript

61
Lies interest in geometrical analysis

• Lie studied surfaces tangential to the
  infinitesimal cones determined by a tetrahedral
  complex, which leads to a first-order PDE of the
  form

62

• Lie used a special transformation to map this
  DEQ to a new one
• which was left invariant by the 3-parameter
  group of translations in the space (X,Y,Z). This
  enabled him to reduce the equation to one of the
  form
• which could be solved directly.

63

• This result soon led Lie to the following
  insights
• 1) PDEs that admit a
  commutative 3-parameter group can be reduced to
  the form
• 2) PDEs that admit a commutative 2-parameter
  group can be reduced to
• 3) PDEs that admit a 1-parameter group can be
  reduced to

64
Lies Theory of Contact Transformations

• Lie noticed that the transformations needed to
  carry out the above reductions were in all cases
  contact transformations.
• Earlier he had studied these intensively, in
  particular in connection with his line-to-sphere
  transformation.

65
Lies Surface Elements

• For a point (x,y,z) on a surface F given by z
  f(x,y), the equation for the tangent plane is
• For an infinitely small region, Lie associated
  to each point (x,y,z) of F the surface element
  with coordinates (x,y,z,p,q). All 5 coordinates
  are treated equally.

66

• The following local condition holds
• and describes the property that contiguous
  surface elements intersect. This Pfaffian
  relation must hold under an arbitrary contact
  transformation.
• Lie had no trouble extending these notions to
  n-dimensional space in order to deal with PDEs of
  the form

67

• Lie then (1872) defined a general contact
  transformation analytically as a mapping
• for which the condition
• remains invariant. He showed further that two
  first-order PDEs can be transformed to another by
  means of a contact transformation.

68
Lies Adaptation of Jacobis Theory

• In his Nova methodus Jacobi introduced the
  bracket operator
• within his theory of PDEs. This was a crucial
  tool for reducing a non-linear PDE to solving a
  system of linear PDEs.

69
Lies Notion of PDEs in Involution

• Lie interpreted the bracket operator
  geometrically, borrowing from Kleins notion of
  line complexes that lie in involution. He defined
  two functions
• to be in involution if

70
Lies First Results on Differential Invariants

• Lie showed that a system of m PDEs
• satisfying
• remains in involution after the application of a
  contact transformation.
• Such considerations led Lie to investigate the
  invariant theory of the group of all contact
  transformations.

71
On the Reception of Lies Work
72
Berlin Reactions to Lies Work

• Weierstrass considered Lies work so wobbly that
  it would have to be redone from the ground up
• Frobenius claimed Lies approach to differential
  equations represented a retrograde step compared
  with the elegant techniques Euler and Lagrange

73
Freudenthal on Lies failure to find an adequate
language

• Lie tried to adapt and express in a host of
  formulas, ideas which would have been better
  without them. . . . For by yielding to this
  urge, he rendered his theories obscure to the
  geometricians and failed to convince the
  analysts.
• The three volumes written by Engel had a
  distinctly function-theoretic touch

74
Where to look for Lies Vision

• According to his student Gerhard Kowalewski, Lie
  never referred to the volumes ghost-written by
  Engel but rather always cited his own papers
• This suggests that the true Lieto take up
  Kleins imageshould not be sought in the volumes
  produced with Engels assistance but rather in
  his own earlier papers and his lectures as edited
  by Georg Scheffers

75
Lies Break with Klein
76
Lies Preface from 1893

• Thanks those who helped pave his way
• Course with Sylow on Galois theory (1863)
• Clebsch, Cremona, Klein, Adolf Mayer, and
  especially Camille Jordan
• Darboux for promoting his geometrical work
• Picard, first to recognize importance of Lies
  group theory for analysis
• J. Tannery for sending students from ENS
• Engel and Scheffers for writing his books

77
Lie on Poincarés Support

• Lie expressed his gratitude to Poincaré for his
  interest in numerous applications of group
  theory. He was especially grateful that he
  Poincaré and later Picard stood with me in my
  fight over the foundations of geometry, whereas
  my opponents tried to ignore my works on this
  topic. (In the text one learns who these
  opponents were.)

78
Kleins Erlangen Program

• A supplement to Tom Hawkins, The Erlanger
  Programm of Felix Klein Reflections on its Place
  in the History of Mathematics, Historia
  Mathematica 11 (1984) 442-470

79
Kleins Lectures on Higher Geometry

• Circa 1890 Klein was returning to several topics
  in geometry he had pursued twenty years earlier
  in collaboration with Lie
• Corrado Segre had Gino Fano prepare an Italian
  translation of the Erlangen Program
• Soon afterward it appeared in French and English
  translations
• Klein wanted to republish it in German too, along
  with several of Lies earlier works

80
End of a Partnership

• Klein even wrote two drafts for an introductory
  essay on their collaboration during the period
  18691872
• Lie profoundly disagreed with Kleins portrayal
  of these events
• He also realized that his own subsequent research
  program had little to do with the Erlangen
  Program
• Lie felt under appreciated in Germany and from
  18891892 was severely depressed

81
Lie on Klein and the Erlangen Program from 1872

• Words that Scandalized the
• German Mathematical Community

82

• Felix Klein, to whom I communicated all of my
  ideas in the course of these years 1870-72,
  developed a similar point of view for
  discontinuous groups. In his Erlangen Program,
  where he reported on his and my ideas, he speaks
  beyond this of groups that are neither continuous
  nor discontinuous in my terminology, for example
  he speaks of the group of Cremona
  transformations. . .. That there is an essential
  difference between these types of groups and
  those I have named continuous groups, namely that
  my continuous groups can be defined by
  differential equations, whereas this is not the
  case for the former groups, evidently escaped him
  completely.

83

• Moreover, one finds hardly a trace of the all
  important concept of differential invariant in
  Kleins Program. Klein took no part in creating
  these concepts, which first make it possible to
  found a general theory of invariants, and it was
  only from me that he learned that every group
  defined by differential equations determines
  differential invariants that can be found by
  integration of complete systems.

84

• Lie felt compelled to clarify these matters
  because Kleins pupils and friends have
  continually represented the relationship between
  Kleins works and mine falsely, and also because
  some of Kleins remarks appended to the recently
  reissued Erlangen Program could easily be
  misconstrued.
• I am not a pupil of Klein, nor is the reverse
  the case, even though it perhaps comes closer to
  the truth. . . . I rate Kleins talent highly and
  will never forget the sympathy with which he
  followed my scientific efforts from the
  beginning, but I believe that he, for example,
  does not sufficiently distinguish between
  induction and proof, between the introduction of
  a concept and its utilization.

85
Seeking New Allies

• These remarks scandalized many within Kleins
  extensive network (Hilbert, Minkowski)
• But Lie also criticized several others by name,
  including Helmholtz, de Tilly, Lindemann, and
  Killing
• He also singled out several French mathematicians
  for praise