We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrodinger equation showing the solitonic nature of those.

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Magnetic vortex filament flows

Commutative Geometry Classical Hamiltonian Systems Algebraic Quantization Geometry of Algebraic Quantization Geometric Quantization Supergeometry Deformation Quantization Non-Commutative Geometry Geometry of Quantum Groups.

Geometric and Algebraic Topological Methods in Quantum Mechanics

. Let G be a Lie supergroup and H a closed subsupergroup.We study the unimodularity of the homogeneous supermanifold G/H, i.e. theexistence of G-invariant sections of its Berezinian line bundle. To that end, weexpress this line bundle as a G-equivariant associated bundle of the principalH-bundle G → G/H. We also study the ﬁbre integration of Berezinians onoriented ﬁbre bundles. As an application, we prove a formulaR of ‘Fubini’ type: G f = (−1) dimh 1 ·dimg/h R G/H R H f, for all f ∈ Γ c (G,O G ).Moreover, we derive analogues of integral formulae for the transforma-tion under local isomorphisms G/H → S/T, and under the products of Liesubsupergroups M · H ⊂ U. The classical counterparts of these formulae havenumerous applications in harmonic analysis.Mathematics Subject Classiﬁcation 2000: 58A50, 58C50, 53C30.Key Words and Phrases: supermanifold, Lie supergroup, homogeneous super-space, Berezin integral, invariant Berezinian form, unimodularity, Fubini for-mula, ﬁbre integration.

Invariant Berezin integration on homogeneous supermanifolds

This is the Preface to the special issue of 'International Journal of Geometric Methods in Modern Physics', v.3, N.1 (2006) dedicated to the 50th aniversary of gauge gravitation theory. It addresses the geometry underlying gauge gravitation theories, their higher-dimensional, supergauge and non-commutatuve extensions.

Gauge gravitation theory from the geometric viewpoint

Generalizing differential geometry of smooth vector bundles formulated in algebraic terms of the ring of smooth functions, its derivations and the Koszul connection, one can define differential operators, differential calculus and connections on modules over arbitrary commutative, graded commutative and noncommutative rings. For instance, this is the case of quantum theory, SUSY theory and noncommutative geometry, respectively. The relevant material on this subject is summarized.

Lectures on Differential Geometry of Modules and Rings

Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces

Supergauge theories in graded manifolds

Some geometrical aspects of the classical hyperbolic Landau-Hall problem are discussed. The Lie algebra of infinitesimal symmetries of this problem is explicitly given, turning out that it is isomorphic to so(2, 1) and that its associated Noether invariants are the hyperbolic angular momenta. The Hamiltonian formulation is also given, allowing us to obtain the manifold of orbits of constant energy for this problem using symplectic reduction techniques

Antonio López Almorox

Papers

Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces

Supergauge theories in graded manifolds

Geometrical aspects of the Landau-Hall problem on the hyperbolic plane