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## Geometrical Visions

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**The distinctive styles**of Klein and Lie Geometrical Visions**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 (Bieberbach’s use of racial stereotypes against Landau and others)**Types of Mathematical Creativity**• Hilbert as an algebraist, even when doing geometry • Poincaré as a geometer, even when doing analysis • Weyl commenting on Hilbert’s Zahlbericht • Van der Waerden (algebraist) accounting for why Weyl (analyst) gave up Brouwer‘s intuitionism**Different Views of Hilbert’s 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 Hilbert’s axiomatic approach to geometry but also his larger program for axiomatizing all exact sciences**Hilbert as a Classical Geometer**• Hilbert’s 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**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**Impossibility Proofs**• Ferdinand Lindemann showed in 1882 that π 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**Hilbert’s 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 Hilbert’s “24th Paris problem”: to show that every well-posed mathematical problem has a definite answer (refutation of du Bois Reymond’s Ignorabimus)**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.**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 Abel’s Beweise für die Unmöglichkeit der Auflösung der Gleichungen fünften Grades durch Wurzelziehen, ferner an die Erkenntnis der Unbeweisbarkeit des Parallelaxioms und an Hermite’s und Lindemann’s Sätze von der Unmöglichkeit, die Zahlen e und π auf algebraischem Wege zu konstruieren.**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-geometrischen Wahrheit nötig sind, und es bleibt dann dem jedesmaligen Ermessen anheim gestellt, welche Beweismethode von dem gerade eingenommenen Standpunkte aus zu bevorzugen ist.**Two full-blooded geometers**Klein and Lie as Creative Mathematicians**Klein’s Universality**• Felix Klein was fascinated by questions of style and discussed it often in his lectures • On a number of occasions he described Sophus Lie’s 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**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 Kummer’s seminar**Klein’s 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 Lie’s mathematics • He described how Lie spent whole days “living” in the spaces he imagined**On Lie’s 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 Lie’s Letters to Felix Klein, 1876-1898, ed. D. Rowe, to appear**Arild Stubhaug’s Heroic Portrait of Lie, the Norwegian**Patriot • Interpretation of Lie’s Life as a Triumphant Struggle • Story of Friends, Foes and Betrayal • Subsidiary Theme: French wisdom vs. German petty-mindedness**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**On Lie’s Mathematics**Hans Freudenthal, “Marius Sophus Lie,” Dictionary of Scientific Biography. Thomas Hawkins, “Jacobi and the Birth of Lie’s Theory of Groups,” Archive for History of Exact Sciences, 1991.**On Lie’s Early Work**D. Rowe, “The Early Geometrical Works of Felix Klein and Sophus Lie” T. Hawkins, “Line Geometry, Differential Equations, and the Birth of Lie’s Theory of Groups” In The History of Modern Mathematics, vol. 1, ed. D. Rowe and J. McCleary, 1989.**Lie’s Early Career**• 1868-71: line and sphere geometry; special contact transformations • 1871-73: PDEs and line complexes; general concept of contact transformations • 1873-74: Lie’s vision for a Galois theory of differential equations**Lie’s 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)**Lie’s 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**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**4-dimensional geometries derived from 3-dimensional space**German line geometry andFrench sphere geometry**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**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**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**The Fresnel Wave Surface**• Kummer‘s 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**Lie’s 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**Klein’s Correspondence with Lie**• Used by Friedrich Engel in Band 7 of Lie’s Collected Works • Fell into Hands of Ernst Hölder, son of Otto Hölder, who married one of Lie’s granddaughters • Purchased by the Oslo University Library • To be published by Springer in a German/English edition**Klein’s letters to Lie, 1870-1872**• Collaboration in Berlin and Paris, 1869-1870 • Klein had trouble following Lie’s ideas by 1871 • Lie’s visit in summer 1872 led to enriched version of Klein’s Erlanger Programm**Felix Klein as a Young Admirer of Riemann**• Came in Contact with Riemann’s Ideas through Clebsch in Göttingen (1869-1872) • Competed as self-appointed champion of Riemann with leading members of the Weierstrass school**Alfred Clebsch (1833-1872)**• Leading „Southern German“ mathematician of the era • Founder of Mathematische Annalen • Klein was youngest member of the Clebsch School**Accounting for the Connection between singular points and**the genus of a Riemann surface Klein’s “Physical Mathematics”**Klein (borrowing from Maxwell) to Visualize Harmonic**Functions**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**Mathematische Annalen, 1873-76**Klein on Visualizing Projective Riemann Surfaces**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**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**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.“**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. . . .”**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**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