Research | Kawiarnia Szkocka

Topics:

Lie systems and superposition rules

Lie systems are systems of first-order differential equations whose general solutions can be written as a function, the superposition rule, of a finite set of particular solutions and a set of constants. These systems have many geometric properties and applications in physics.

Scientific interests

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Supergeometry

From a physical point of view, manifolds with odd and even coordinates appear in several branches of physics, e.g. sypersymmetry. The mathematical approach to these objects is known as supergeometry. There is a huge interest on studying these manifolds. In particular, I am focused on the study of geometric properties of differential equations on supermanifolds and their applications in supermechanics, string theory and other topics.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Symplectic geometry and Classical Mechanics

The standard approach to Classical Mechanics leads to a manifold endowed with a non-degenerate two-form, a symplectic manifold. This is the natural setting for Classical Mechanics, where the Hamilton equations arise as the equation for the integral curves of a Hamiltonian vector field with respect a symplectic form. I'm interested in developing symplectic techniques to study mechanical systems dependents on time and applications of my results in Physics.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Poisson geometry

Symplectic manifolds are a particular case of Poisson manifolds, which are manifolds foliated in submanifolds which are symplectic. Poisson geometry can be employed to describe many relevant systems appearing in Physics which cannot be described through symplectic geometry. I'm interesting in the study of new Poisson geometric techniques to study such systems

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Lie bialgebra, Poisson--Lie groups and coalgebras

Classification and use in integrable systems. Hopf algebras and their geometric counterparts.

PhD students (past and present)

C. Sardon (2015 Cum Laude and Special Prize for doctoral theses)

D. Wysocki (2023 proposed Cum Laude for referees - rejected by Scientific Council).

Collaborators

J.F. Carinena

C. Sardón

F.J. Herranz

S. Vilarino

A. Blasco

J.C. Gallardo

J. Jover

A. Ballesteros

J.F. Grabowski