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- summer school
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The school aims to provide an introduction to a very dynamic area of research that lies at the intersection of number theory, p-adic analysis, algebraic geometry and singularity theory. In this area, the local zeta functions (p-adic, archimedean, motivic, etc) and certain counterparts known as Poincaré series (usual and motivic) associated with filtrations play a central role.
The Local zeta functions were introduced in the 50s by I. M. Gel'fand and G. E. Shilov. The main motivation was that the existence of meromorphic extensions for the Archimedean local zeta functions implies the existence of fundamental solutions for partial differential equations with constant coefficients.
In the 70s, J.-I. Igusa developed a uniform theory for local zeta functions and oscillatory integrals, with a polynomial phase, on fields of characteristic zero. In the p-adic case the local zeta functions are related to the number of solutions of polynomial congruences and to exponential sums mod pm. There are many (very difficult) conjectures that connect the poles of the local zeta functions with the topology of complex singularities.
Recently J. Denef and F. Loeser introduced zeta motivic functions, which constitute a vast extension of the p-adic zeta functions studied by Igusa. Another important object is the Poincaré motivic series recently introduced by A. Campillo, F. Delgado and S. Gusein-Zade, which also have to do with the topology of complex singularities.
It is also important to mention that the local zeta functions are related to Feynman and string amplitudes. There is also a growing interest in the understanding of the mathematical and physical problems that appear in both, classical and p-adic quantum theories. The main purpose of the school is to present these connections and some of the challenges they have opened.
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