On Hilbert IVth Problem Marc Troyanov (EPFL) SJTU, June 21, 2019 - PowerPoint PPT Presentation

On Hilbert IVth Problem Marc Troyanov (EPFL) SJTU, June 21, 2019

On Hilbert IVth Abstract Problem Marc Troyanov (EPFL) In 1900, Hilbert formulated his famous list of 23 problems that greatly Introduction influenced mathematics throughout the 20th century. Ten among those Statement of the IVth problem problems where presented at the second International Congress of Historical context Mathematicians held in Paris in August 1900. Early results Busemann’s construction Hilbert’s fourth problem can be formulated (in a somewhat modern The Finsler Viewpoint language) as the problem of describing all geometries in a domain of the Euclidean space or projective space for which the straight lines have minimal length. Various specific meaning to the problem have been proposed and several partial solution have been given, notably by H. Hamel, H. Busemann, Pogorelov and Ambartzumian. In this talk, we will briefly present the historical context of Hilbert’s fourth problem, describe its relation with Hilbert and Finsler geometries and integral geometry. We will also describe some variant of the problem and classes of solution.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction Statement of the IVth problem Plan Historical context Early results ◮ Introduction: The Hilbert Problems. Busemann’s ◮ Historical context. construction The Finsler Viewpoint ◮ Early Results ◮ The Busemann Construction ◮ The Finsler Viewpoint ◮ Generalization ◮ References

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction Statement of the IVth problem Historical context Early results Busemann’s construction The Finsler Viewpoint Introduction: The Hilbert Problems.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction Statement of the IVth problem Historical context During the Second International Congress of Mathematicians held Early results in Paris in August 1900, David Hilbert delivered his famous lecture Busemann’s titled The future problems of mathematics . construction The Finsler Viewpoint In this lecture he presented 10 open problems, together with context and mathematical comments. He then published an extended list of 23 problems first in the Swiss journal L’Enesignement Math´ ematiques then in the Bulletin of the American Mathematical Society .

On Hilbert IVth Problem Marc Troyanov (EPFL) The Hilbert problems present a wide variety of research area and Introduction they greatly influenced mathematics during the 20th century. Here Statement of the IVth problem are some of the problems: Historical context Early results Problem I. The continuum hypothesis (P. Cohen proved the Busemann’s CH to be independent of ZFC in 1963). construction The Finsler Viewpoint Problem II. Consistency of the axioms of arithmetic (K.G¨ odel showed the consistency cannot be proved in 1931) Problem VIII. The Riemann hypothesis (yet unresolved). Problem X. Algorithmic solvability of Diophantine equations (answered negatively by Matiyasevich in 1970) Problem XIX. Are the solutions of regular problems in the calculus of variations always necessarily analytic? (solved positively by de Giorgi and Nash in 1957).

On Hilbert IVth Problem Marc Troyanov (EPFL) By contrast to some of the other problems, Hilbert Problem IV is Introduction written as a long discussion: Statement of the IVth problem Historical context Problem IV: Problem of the straight line as the shortest Early results distance between two points. Busemann’s construction The Finsler Viewpoint Another problem relating to the foundations of geometry is this: If from among the axioms necessary to establish ordinary Euclidean geometry, we exclude the axiom of parallels, or assume it as not satisfied, but retain all the other axioms, we obtain, as is well known, the geometry of Lobachevsky (hyperbolic geometry). We may therefore say that this is a geometry standing next to Euclidean geometry. If we require further that that axiom be not satisfied whereby, of three points on a straight line, one and only one lies between the other two, we obtain Riemann’s (elliptic) geometry, so that this geometry appears to be the next after Lobachevsky’s. [...]

On Hilbert IVth [...] The more general question now arises: Whether from other Problem Marc Troyanov suggestive standpoints geometries may not be devised which, with equal (EPFL) right, stand next to Euclidean geometry. Here I should like to direct Introduction your attention to a theorem which has, indeed, been employed by many Statement of the IVth authors as a definition of a straight line, viz., that the straight line is problem Historical context the shortest distance between two points. The essential content of this Early results statement reduces to the theorem of Euclid that in a triangle the sum of Busemann’s two sides is always greater than the third side – a theorem which, as is construction easily seen, deals solely with elementary concepts, i. e. , with such as are The Finsler Viewpoint derived directly from the axioms, and is therefore more accessible to logical investigation. Euclid proved this theorem, with the help of the theorem of the exterior angle, on the basis of the congruence theorems. Now it is readily shown that this theorem of Euclid cannot be proved solely on the basis of those congruence theorems which relate to the application of segments and angles, but that one of the theorems on the congruence of triangles is necessary. We are asking then, for a geometry in which all the axioms of ordinary Euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom.

On Hilbert IVth Problem Marc Troyanov One finds that such a geometry really exists and is no other than that (EPFL) which Minkowski constructed in his book, Geometrie der Zahlen , and Introduction made the basis of his arithmetical investigations. Minkowski’s is Statement of the IVth therefore also a geometry standing next to the ordinary Euclidean problem Historical context geometry; it is essentially characterized by the following stipulations: Early results 1. The points which are at equal distances from a fixed point O lie on a Busemann’s construction convex closed surface of the ordinary Euclidean space with O as a The Finsler Viewpoint center. 2. Two segments are said to be equal when one can be carried to the other by a translation of the ordinary Euclidean space. In Minkowski’s geometry the axiom of parallels also holds. By studying the theorem of the straight line as the shortest distance between two points, I arrived at a geometry in which the parallel axiom does not hold, while all other axioms of Minkowski’s geometry are satisfied. The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction Statement of the IVth problem Historical context Early results For this reason, and because I believe that the thorough investigation of Busemann’s construction the conditions for the validity of this theorem will throw a new light The Finsler Viewpoint upon the idea of distance, as well as upon other elementary ideas, e.g. , upon the idea of the plane, and the possibility of its definition by means of the idea of the straight line, the construction and systematic treatment of the geometries here possible seems to me desirable .

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction You may rightfully be confused by this text: What exactly is Statement of the IVth problem Hilbert talking about? What is the precise statement of the fourth Historical context problem? and why is it stated in such a complicate language? Early results Busemann’s construction To give a short answer, we can refer to Wikipedia, where the The Finsler Viewpoint problem is stated as “Construct all metrics where lines are geodesics”. Wikipedia also adds the following comment: “Too vague to be stated resolved or not” [sic.] On these points I dare say that Wikipedia is a bit too vague.

On Hilbert IVth Problem Marc Troyanov (EPFL) Introduction Statement of the IVth problem Historical context Early results Busemann’s construction The Finsler Viewpoint A brief historical context

On Hilbert IVth Problem Marc Troyanov (EPFL) To better grasp the problem it is useful to have in mind some of the key aspects of the history of Geometry in the XIXth century. Introduction Statement of the IVth problem Historical context In the 1820’s three major developments arised that greatly Early results influenced the development of Geometry. These are Busemann’s construction ◮ The revival of Projective Geometry by Jean-Victor Poncelet The Finsler Viewpoint and Charles Julien Brianchon (and later Karl von Staudt). ◮ The discovery of Non Euclidean Geometry by Nikolai Ivanovich Lobachevsky and J´ anos Bolyai. ◮ The work on Carl Friedriech Gauss on the differential geometry of surfaces (in particular the notion of intrinsic curvature). It will take several decades until the deep relations between these three sides of geometry will be clarified.