Anthony Joseph
The Donald Frey Professor
My field of research is algebra and more particularly the representation
theory of Lie systems. Briefly, representation theory provides a grand
unification of the relations of classical functions which themselves
enter into almost every scientific arena. In the last two decades more
complex algebraic relations have been unraveled from conformal field
theory and statistical physics. Deep new methods have spun out solutions
to these extraordinarily intricate equations. Of particular interest is
the theory of "crystals" which derive from quantum groups and the global
bases which result. These developments will be of fundamental importance
to the mathematics and physics of the twenty-first century. Recently I
solved two twenty year old problems, one concerning the orbit method
relating geometry to representation theory and the second concerning
tensor product decompositions using notably global bases. My most recent
work concerns the geometry of regular co-adjoint orbits of parabolic
subalgebras of simple Lie algebras. From this one can expect to be able
to construct completely integrable dynamical systems analogous to the
Toda lattice.
Our group had previously been a part of a European TMR network "Algebraic Lie Representations". It is now part of the European RTN network, "Flags, Quivers and Invariant Theory in Lie Representation Theory".
Recent Publications
- [with V. Hinich] Orbital variety closures and the convolution product in Borel-Moore homology. Selecta Math., to appear.
- [with F.Fauquant-Millet] Semi-centre de l’algebre enveloppante d’une sous-algebre parabolique d’une algebre de Lie semi-simple. Ann. Ec. Norm. Sup., to appear.