1. Johann Carl Friedrich Gauss 1777-1855

2.  1795-1798 –Gauss is a student of the University of Göttingen  1799 - Assistant Professor of the University of Braunschweig.  1806 - On the recommendation of Alexander von Humboldt Gauss appointed professor at Göttingen and director of the Gottingen Observatory.  1810 - Gauss received the award of the Paris Academy of Sciences and the Gold Medal of the Royal Society.  1824 - Foreign Member of the St. Petersburg Academy of Sciences.  Gauss died February 23, 1855 in Göttingen. Johann Carl Friedrich Gauss 1777-1855 Gauss's teacher were Johann Christian Martin Bartels and Abraham Gotthel Kästner

3. Summer of 1801 - «Disquisitiones Arithmeticae» (algebra, theory of number) 1799 - Doctoral dissertation ("New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors (i.e., polynomials) of the first or second degree") Astronomy (When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted). 1809 - Theory of motion of the celestial bodies moving in conic sections around the Sun Gauss showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. Mathematical Analysis, The theory of complex Gaussian integers, Gauss's potential theory, Probability theory and statistics: The method of least squares, the study of the normal distribution.

4. In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the Kingdom of Hanover, linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions. In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840 Gauss gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics)

6. János Bolyai 1802 - 1860 Farkas Wolfgang Bolyai 1775 - 1856 By 20 June 1831 the Appendix had been published («Tentamen»)

7. Nikolai Ivanovich Lobachevsky 1792-1856

8. 23.02.1826

10. Alexandert Vasiljevich Vasiljev (1853-1929) Eugenio Beltrami (1835-1900) Klein’s modelPoincaré’s model

11. Georg Friedrich Bernhard Riemann 1826- 1866 Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts.... The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented. «(On the hypotheses that lie at the foundations of geometry)» (10 июня 1854 г.)

12. Giovanni Girolamo Saccheri 1667 - 1733

13. Felix Christian Klein 1849 - 1925 1865 -1868 Klein entered the University of Bonn and studied mathematics and physics 1870 Klein was in Paris when Bismarck, the Prussian chancellor, published a provocative message aimed at infuriating the French government. France declared war on Prussia on the 19 th of July and Klein felt he could no longer remain in Paris and returned 1872 professor at Erlangen, 1875 professor at the Technische Hochschule at Munich 1876 The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities 1880 -1886 professor at Leipzig 1888 professor at the University of Göttingen Беркович Е. Феликс Клейн и его команда http://berkovich- zametki.com/2008/Starina/Nomer6/Berkovich1.phphttp://berkovich- zametki.com/2008/Starina/Nomer6/Berkovich1.php

14. Розов Н.Х. Феликс Клейн и его эрлангенская программа // Математическое просвещение, 1999. Сер. 3, в.3 – С. 49-55 http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mp&paperid=40&o ption_lang=eng Erlangen program Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form of Galois theory.

15. Hilbert's axioms 5 groups of axioms: Incidence Order Congruence Parallels Continuity Three primitive terms: point; line; plane; Three primitive relations: Betweenness, a ternary relation linking points; Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes;binary relations Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅. line segmentsangles

16. David Hilbert (1862-1943) Kenigsberg Heidelberg

17. Kenigsberg Heinrich Martin Georg Friedrich Weber 1842 - 1913 Ferdinand von Lindemann 1852 - 1939 Adolf Hurwitz 1859 - 1919  Doctor of Science  The Habilitation is the extra post-doctoral qualification needed to lecture at a German university.  Privatdozent  extraordinary professor  ordinary professor

18. D.Hilbert in Paris H.Poincaré Ch.Hermité E.Picard C.Jordan

19. Kenigsberg

20. Paul Albert Gordan 1837 - 1912 An invariant is something that is left unchanged by some class of functions. In particular, invariant theory studied quantities which were associated with polynomial equations and which were left invariant under transformations of the variables. Паршин А. Н. Давид Гильберт и теория инвариантов // Историко- математические исследования. — М.: Наука, 1975. — № 20. — С. 171—197  Ясная и легко понимаемая («так как, в то время как ясное и простое привлекает, сложное отталкивает»).  Трудная (чтобы нас привлекать») и в то же время не полностью недоступная («чтобы не сделать безнадёжными наши усилия»).  Важная («путеводная звезда на извилистых тропах к сокрытым истинам»).

21. Gottingen

22. Hermann Minkowski 1864 - 1909 * Abraham–Minkowski controversy * Brunn–Minkowski theorem * Hasse–Minkowski theorem * Minkowski addition Minkowski–Bouligand dimension * Minkowski diagram * Minkowski functional * Minkowski inequality * Minkowski problem * Minkowski's question mark function * Minkowski space * Minkowski–Steiner formula Minkowski's theorem in geometry of numbers * Smith–Minkowski–Siegel mass formula

23.  invariant theory (1885-1893)  theory of algebraic numbers (1893- 1898)  Foundations of Geometry (1898-1902)  Dirichlet principle and adjacent problems of the calculus of variations and partial differential equations (1900-1906),  theory of integral equations (1900-1910)  solution of Waring's problem in number theory (1908-1909)  fundamentals of mathematical physics (1910-1922)  logical foundations of mathematics (1922-1939)

24. Mathematical club of Gottingen

26. International Congress of Mathematicians in Paris on 8 August 1900

27. Participants and Sections France - 90 persons Germany - 25 persons. USA - 17 persons. Italy - 15 persons Belgium - 13 persons. Russia - 9 persons Austria, Switzerland - 8 persons England, Sweden - 7 persons Denmark, South America - 4 persons Netherlands, Spain, Romania - 2 persons Turkey, Greece, Norway, Canada, Japan, Mexico 1) arithmetic and algebra (chairman D.Hubert, Secretary of E.Cartan) 2) analysis (P. Painlevé, J.Hadamard) 3) geometry (E.Darboux? B. Nivenglovsky) 4) mechanics and mathematical physics (J.Larmo, T.Levi-Civita) 5) the history and bibliography of mathematics (Prince Roland Bonaparte, M. Okan) 6) teaching and methodology of mathematics (M.Cantor, Sh.Lesa). Vito Volterra Gesta Mittag-Leffler

28. 1. The cardinality of the continuum, including well-ordering. 2. The consistency of the axioms of arithmetic. 3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes. 4. The straight line as shortest connection between two points. 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining a group. 6. The axioms of physics. 7. Irrationality and transcendence of certain numbers. 8. Prime number theorems (including the Riemann hypothesis). 9. The proof of the most general reciprocity law in arbitrary number fields. 10. Decision on the solvability of a Diophantine equation. 11. Quadratic forms with any algebraic coefficients. 12. The extension of Kronecker's theorem on Abelian fields to arbitrary algebraic fields. 13. Impossibility of solving the general seventh degree equation by means of functions of only two variables. 14. Finiteness of systems of relative integral functions. 15. A rigorous foundation of Schubert's enumerative calculus. 16. Topology of real algebraic curves and surfaces. 17. Representation of definite forms by squares. 18. The building up of space from congruent polyhedra. 19. The analytic character of solutions of variation problems. 20. General boundary value problems. 21. Linear differential equations with a given monodromy group. 22. Uniformization of analytic relations by means of automorphic functions. 23. The further development of the methods of the calculus of variations. [24.] The simplicity of proofs (omitted).

29. Logicism is a philosophical, foundational, and foundationalist doctrine that can be advanced with respect to any branch of mathematics. Traditionally, logicism has concerned itself especially with arithmetic and real analysis (Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/logicism/)http://plato.stanford.edu/entries/logicism/ Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.(Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/intuitionism/).http://plato.stanford.edu/entries/intuitionism/ The guiding idea behind formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess. Finally, it is often the position to which philosophically naïve respondents will gesture towards, when pestered by questions as to the nature of mathematics..(Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/formalism-mathematics/)http://plato.stanford.edu/entries/formalism-mathematics/