Luis Crespo

Universidad de Cantabria

 

Conference Symplectic and Algebraic Geometry of real and p-adic systems

 

Madrid, June 10–12, 2024

 

The notions of symplectic manifold and integrable system are usually formulated over the real field. However, some discoveries in mathematical physics (by B. Dragovich and others) lead to the question of whether these notions can be extended to p-adic fields. Á. Pelayo, V. Voevodsky and M. Warren laid the foundations for these definitions a decade ago. The aim of this minicourse is to present our new formulations and results for p-adic symplectic geometry. I will focus on the p-adic version of the Jaynes-Cummings system, an integrable system that has recently gained some attention in the real case. This is joint work with Álvaro Pelayo.

 

 

MONDAY June 10th, 12:10-13:30, Aula Miguel de Guzmán

Lecture 1: p-adic manifolds. This session will start with an introduction to the p-adic numbers and some analytic properties. Concretely, I will talk about differentiable and analytic p-adic functions and initial value problems in general, and about elementary functions in particular (exponential, sine and cosine), which, despite being defined in a similar way to the real ones, show very different properties. Next, I will introduce the concepts of symplectic manifold and integrable system in the p-adic field.

 

 

MONDAY June 10th, 15:00-16:20, Aula Miguel de Guzmán

Lecture 2: p-adic symplectic integrability. This session is devoted to present the p-adic oscillator and spin systems. In the real case, both systems have circles as fibers of the mo- mentum map. This is also true in the p-adic case, but the structure of a circle is much more involved and depends heavily on the value of p. For example, the action of the rotation group is not transitive in the circle: rotations only send points in the circle to points close enough to them.

 

 

WEDNESDAY June 12th, 10:00-11:20, Aula Miguel de Guzmán

Lecture 3: p-adic systems in mathematical physics. In the last session I will present the coupling of the two systems, oscillator and spin, known in the real case as Jaynes-Cummings system. I will talk about the image and fibers of the system and its critical points. Some aspects of the real system are still present in the p-adic system, some are not. For example, the fiber of (−1, 0) is the only one that may contain an isolated point (depending on p), but unlike the real case, there are more points in that fiber, and they are not isolated. Also, the fibers are not homeomorphic to “tori” (in the sense of products of circles).