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Tuesday, May 24, 2022
4:00 PM - 5:00 PM
Linde Hall 310

Mathematics Colloquium/Taussky-Todd Lecture

Applied l-adic cohomology
Philippe Michel, EPFL/TAN, Ecole Polytechnique Fédérale de Lausanne,

l-adic cohomology has its origins in the study of congruences in the ring of integers and specifically in the problem of counting solutions of system of polynomial equations modulo a prime number q. This is a complex theory, conjectured by A. Weil, constructed by A. Grothendieck and developed by P. Deligne, N. Katz, G. Laumon and others. The basic objects are l-adic sheaves over an algebraic variety over Fq; to these are associated "trace functions" on the set of Fq-points. For the affine line, the functions can also be considered as q-periodic functions over the integers; they can then "interact" with the basic functions from analytic number theory, like the characteristic function of the primes. In this talk I will highlight some classical problems from analytic number theory in which sophisticated trace function pop-up naturally and will explain how basic and not so basic methods from analytic number theory and l-adic cohomology allow to measure these interactions.

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