Primes, knots and periodic orbits
speaker: Christopher Deninger (University of Münster )abstract: In the 1960s Manin and Mazur noted that from the viewpoint of étale topology there was an intriguing analogy between prime numbers embedded into the spectrum of the integers and knots in 3-space. Later Kapranov, Reznikov, Morishita and other authors discovered further intriguing analogies between number rings and the topology of 3-manifolds. For example, the Iwasawa zeta function corresponds to the Alexander polynomial of a knot. The search for a cohomology theory related to the Riemann zeta function led to the discovery of analogies between number rings and a class of 3-dimensional dynamical systems, where the primes would correspond to the periodic orbits. For example, Riemann's explicit formulas in analytic number theory correspond to a transversal index theorem in the dynamical context, proved by Álvarez-López and Kordyukov. The dynamical systems analogy refines the previous analogy because forgetting the parametrization, a periodic orbit gives a knot. Recently, we have constructed foliated dynamical systems for number rings and even for all arithmetic schemes that have some but not yet all the expected properties.
timetable:
Fri 11 Mar, 16:00 - 17:00, Aula Dini
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