In my talks (17/10 and 14/11) I will sketch the construction of the
principal series of unitary representations of
and the coadjoint
orbits they correspond to. I will also try to explain the geometric
quantization of these orbits and show that the infinitesimal
representations thus obtained are indeed the ones corresponding to the
principal series. If there is any time left I might say something about
the Kirillov character formula for
I.M. Gelfand, R.A. Minlos, Z.Ya. Shapiro, Representations of the
Lorentz groups and their applications, Pergamon Press, 1963.
A. Kirillov, Elements of representation theory, Springer.
G.W. Mackey, Induced representations of locally compact groups I, Ann. Math.
When trying to construct a Riemannian geometry on the path space of
a Riemannian manifold several approaches could be thought about.
The local chart approach, considering the path space as an infinite
dimensional manifold and the basic tangent space the Cameron-Martin
Hilbert space, leads to the study of the so-called Wiener-Riemann
manifolds. Several difficulties appear in this study, namely the
difficulty of finding an atlas such that the change of charts is
compatible with the probabilistic structure (preserves the class of
Wiener measures together with the Cameron-Martin type tangent
spaces) and the non-availability of an effective computational
procedure in the local coordinate system. Indeed, in infinite
dimensions, the summation operators of differential geometry become
very often divergent series. But the path space is more than a
space endowed with a probability: time and the corresponding
Itô filtration provide a much richer structure. In
particular, the parallel transport over Brownian paths can be
naturally defined by a limiting procedure from ODEs to SDEs. The
stochastic parallel transport defines a canonical moving frame on
the path space: the point of view we have adopted is the one of
replacing systematically the machinery of local charts by the
method of moving frames (as in Cartan theory). In this way it is
possible to transfer geometrical quantities of the path space to
the classical Wiener space and use Itô calculus to
renormalize the apriori divergent expressions. An effective
computational procedure is then achieved, where Stochastic Analysis
and Geometry interact, not only on a technical level, but in a
deeper way: Stochastic Analysis makes it possible to define
geometrical quantities, Geometry implies new results in Stochastic
Analysis. An application to assymptotics of the vertical
derivatives of the heat kernel associated to the horizontal
Laplacian on the frame bundle is discussed.
A. B. Cruzeiro and P. Malliavin -"Renormalized differential
geometry on path space: structural equation, curvature", J.Funct.
Anal. 139 (1996), p. 119 -181.
A. B. Cruzeiro, P. Malliavin and S. Taniguchi - "Ground state
estimations in gauge theory", Bull Sci.Math. 125, 6-7 (2001), p.
I will first give a brief overview of the Bogomolny equations for vortices, their moduli space of solutions, and the method of geodesic approximation. In the case
vortices on a sphere of radius
this moduli space is
, but the geodesic method cannot be directly applied, because the
solutions of the Bogomolny equations are
not known explicitly. I will then try to show how to circumvent this problem in the limit where
is close to
J. M. Baptista and N. S. Manton, The dynamics of vortices on
near the Bradlow limit, hep-th/0208001.
T. M. Samols, Vortex Scattering, Commun. Math. Phys. 145, 149 (1992).