Felice Ronga professeur ordinaire Section de mathématiques Avertissement. Nous avons le regret de vous informer que le professeur Felice Ronga est décédé en 2007.

Géométrie II

Géométrie algébrique

Géométrie II 2005-2006,   analyse II réelle 2004-2005.

Travaux pratiques en Maple :

• Introduction à Maple
• Etude de fonctions à une et deux variables
• L-Systèmes
• Trajectoires de champs de vecteurs

Recent publications

• [1] (with J.-P. Françoise) The decidability of real algebraic sets by the index formula, in "Real Algebraic Geometry", Springer Lecture Notes in Math. 1524 (1992), 235-239.
• [2] (with Th. Vust) Stewart Platforms without Computer ?, in Proceedings of the Conference "Real Analytic and Algebraic Geometry", Trento 1992, W. de Gruyter Verlag (1995), 197-212.
• [3] Un procédé d'élimination effective et quelques applications, Annales de l'Institut Fourier Tome 45 (1995)-Fascicule 2, 421-435.
• [4] (with J.-M. Gamboa) On open real polynomial maps, Journal of Pure and Applied Algebra 110 (1996), 297-304.
• [5] Points d'inflexion sur les courbes réelles : un travail de Felix Klein à la lumière de la théorie des singularités d'applications, Gazette des mathématiciens no 74, octobre 1997. Repris dans "Où en sont les mathématiques ?", éditeur Jean-Michel Kantor, coédition VUIBERT/SMF (2002).
• [6] The number conics tangent to 5 given conics : the real case, revista matemática de la Universidad Complutense de Madrid, Vol. 10 no.2 (1997), 391-421.
• [7] Klein's paper on real flexes vindicated, Singularities Symposium - Lojasiewicz 70, B. Jakubczyk, W. Pawlucki and J. Stasica editors, Banach Center Publications 44 (1998).
• [8] A real Riemann-Hurwitz theorem, Boletim da Soc. Bras. de Mat., Vol. 31, N0.2 (2000), 175-187.
• [9] in collaboration with Ivan PAN and Thierry VUST :
  Transformations birationnelles quadratiques de l'espace projectif complexe à trois dimensions, Annales de l'Institut Fourier Tome 51 (2001)-Fascicule 5, 1153--1187.
• [10] Schubert calculus according to Schubert, J^UMP* (2004 -- revised february 16, 2006)
• [11] The integral Thom polynomial for \Sigma^1,1, J^UMP* (2005)
• [12] in collaborazione con Thierry Vust:
  Diffeomorfismi birazionali del piano proiettivo reale, Commentarii Mathematici Helvetici 80 (2005), 517-540

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  * The Journal of Unpublishable Mathematical Papers

Current research efforts

• Real enumerative problems of algebraic geometry.
  
  • Trying to understand when they are maximal (i.e. the number of solutions that can be obtained over the reals equals the number of solutions over the complex), or complete (i.e. all the reasonable numbers of solutions, between the maximal and minimal number, can be realised).
    If N denotes the dimension of the projective space of plane curves of degree n, the number of curves of degree n through N-k points, tangent to k lines equals (2(n-1))^k over the complex, provided that k<2n-1; in [8] I show that the problem is maximal and complete for k=1, but don't know nothing for k=2 (for n=3 already).
    Note that the number of conics through 3 points, tangent to 2 lines is either 4 or zero, depending on the 3 points being or not in the same connected component of the complement of the 2 lines. Therefore this problem is maximal but not complete.
  • How many points of hyperosculation can there be on a real plane cubic? There are 24 over the complex.
• Prove or disprove that on a real smooth quartic projective surface there can be at most 48 lines.
• Effective geometry.
  Find an effective Sard theorem for polynomial maps, e.g. from Rⁿ to R².

Quelques images mathématiques

Images engendrées par la catastrophe de l'ombilic hyperbolique

Niveaux de fonctions