Descripción del proyecto
Un seguimiento de la evolución temporal de los nudos en el electromagnetismo y la mecánica cuántica
La teoría de nudos, el estudio de curvas cerradas en tres dimensiones y sus posibles deformaciones sin que una de las partes atraviese a la otra, es cada vez más importante en los ámbitos de la biología, la química y la física. El proyecto KNOTDYNAPP, financiado por las Acciones Marie Skłodowska-Curie, tiene como objetivo estudiar la evolución temporal de los nudos en diferentes sistemas dinámicos. Se prestará especial atención en demostrar la existencia de soluciones de ecuaciones diferenciales que contienen nudos persistentes a lo largo del tiempo en el electromagnetismo y la mecánica cuántica. Para abordar determinados problemas matemáticos en dichos ámbitos, los investigadores tomarán prestadas técnicas de la geometría diferencial y de la topología de baja dimensión.
Objetivo
Mathematical knot theory plays an increasingly important role in biology, chemistry and physics. In this project we aim to study the time evolution of knots in different dynamical systems. We are particularly interested in differential equations that are motivated by electromagnetism and quantum mechanics. For such differential equations we aim to prove the existence of solutions that contain knots, which evolve as desired, and explicitly construct such solutions.
In the case of electromagnetic fields this refers to vector fields, representing the electric and magnetic part of such a field, that satisfy Maxwell's equations and have closed flow lines in the shape of a given knot for all time. In particular, we want the knot type of this closed flow line to be stable, i.e. it is not allowed to change over time.
In the case of quantum wavefunctions we are concerned with complex-valued functions that satisfy linear or non-linear Schrödinger equations and whose nodal set is knotted at a moment in time. We plan to develop a construction of such functions for which the time evolution of such a quantum vortex knot is determined by a prescribed surface, embedded in 4-dimensional space representing space and time.
We also study relations between topological properties of knots and the corresponding functions. For example, we investigate the connection between the fibration property of a knot K and the non-vanishing of a magnetic field induced by an electric current through a knotted wire in a shape that is isotopic to K.
These mathematical problems are approached with techniques from differential geometry, low-dimensional topology and the theory of differential equations. The proposal also discusses the two way transfer of knowledge between the host institute and the candidate.
Ámbito científico
- natural sciencesphysical scienceselectromagnetism and electronicselectromagnetism
- natural sciencesmathematicspure mathematicstopologyknot theory
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equations
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesmathematicspure mathematicstopologyalgebraic topology
Programa(s)
Régimen de financiación
MSCA-IF-EF-ST - Standard EFCoordinador
28006 Madrid
España