O. G. Smolianov’s scientific contributions

In [AGV] it has been proved that the Yang Mills equations are equivalent to the Laplace equation with respect to a generalized Levy laplacian. In that paper an infinite hierarchy of exotic laplacians was introduced, of which the Levy laplacian is the simplest example. In the present paper a one-to-one correspondence between laplacians and traces on *-algebras of operators is established. The Volterra laplacian corresponds to the trace class operators; the Levy laplacian to the algebra of multiplication by functions of the position operator; the exotic laplacians, to *-algebras of unbounded operators with exotic traces. The analogue, for exotic laplacians, of the characterization of the Levy laplacian as ergodic average of the second derivatives, is established. An existence and uniqueness theorem for the heat equation associated to the Levy laplacian is proved by means of an explicit formula for the fundamental solution.