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	Plenary Lecture MALLIAVIN CALCULUS OF BISMUT TYPE IN SEMI-GROUP THEORY Professor Remi Leandre Institut de Mathematiques. Universite de Bourgogne 21000. Dijon. FRANCE E-mail: Remi.leandre@u-bourgogne.fr Abstract: The talk is dovided in 5 parts: -)In the first part, we translate Malliavin Calculus of Bismut type in semi group theory, giving a sufficient condition in order a diffusion semi-group has a smooth density. -)n the second part, we translate Varadhan estimates for hypoelliptic diffusions in semi-group theory, estimates we got a long time ago by using the Brownian motion. -)In the third part, we translate in semi-group theory the Ben Arous and us theorem giving a sufficient condition in order an heat-kernel is strictly positive, where the Bismutian semi-distance plays an important role. -)In the fourth part, we translate one of the result we got a long time ago by probability by using a stochastic analog of the Rothschild-Stein inhomogeneous division method. -)In the fifth part, we translate in semi-group Bismut's ways of the Malliavin Calculus for Poisson process where the generator is an integro-differential operator. Brief Biography of the Speaker: 1957: Born in Belfort (France). 1978/1981: Student Ecole Normale Superieure Saint-Cloud. 1980: Agregation de Mathematiques. Rank: 5. 1982/1984: National service at the University of Monastir (Tunisia). 1984: These de Troisieme cycle. Directeur: J.M. Bismut. 1984/1988: assistant at the Universite de Besancon. 1987: Habilitation a diriger les recherches. Universite de Besancon. Directeur: J.M. Bismut. 1988: Bronze medal of C.N.R.S. 1988/1994: Charge de recherches Premiere classe C.N.R.S. Universite de Strasbourg. 1989: Rollo Davidson Price. 1994/2004: Directeur de recherches Seconde Classe. C.N.R.S. Universite de Nancy. 2003: Editor of a special issue ”Probability and geometry” of the Journal ”Infinite dimensional analysis, quantum probability and related topics”. 2004: Directeur de recherches Seconde Classe. Universite de Dijon. Organiser of various conferences: * ”Loops spaces” (Strasbourg 1994)with S. Paycha and T. Wuerzbacher. * ”Probability and geometry” (Nancy 1999). * ”Infinite dimensional analysis” (C.I.R.M. 2001). * I have organized a session ”Analysis on path spaces” at the Satellite Conference of Stochastic Analysis of I.C.M. at Beijing (2002). * ”Infinite dimensional analysis and path integrals” (C.I.R.M 2004) with C. De Witt-Morette. * ”Stochastic analysis and mathematical physics” (Dijon. 2006).

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