Speaker
Dr
Hovhannes Khudaverdian
(School of Mathematics The University of Manchester)
Description
In mathematical physics it is very useful to consider differential operators acting on densities of various weights on a manifold $M$. To study the geometry of such operators one can consider an operator pencil ${\Delta_t}$ where for an arbitrary real $t$ an operator $\Delta_t$ acts on densities of weight $t$ defined on a manifold $M$. Pencils of this kind can be interpreted as differential operators on a certain algebra of functions on extended manifold $\hat M$.
For second order operators the study of their geometry naturally fits into a Kaluza-Klein framework. For such an operator the related geometry is defined by principal symbol ("metric on M"), a connection on volume forms ("gauge field") and a function related with the scalar term ("Brans-Dicke scalar"). This becomes useful to study important and beautiful geometrical properties of second order differential operators.
The extended manifold $\hat M$ can be identified with Thomas bundle dating back in projective geometry to 1920. We see that study of the extended manifold $\hat M$ provides constructions on intersection of classical differential geometry and gravitational theory. Such investigations can be traced to H.Weil, Veblen, T.Y.Thomas, Pauli and Jordan.
Author
Dr
Hovhannes Khudaverdian
(School of Mathematics The University of Manchester)
