Professor Leonard Gross is one of the leading researchers in mathematical physics. He has made major contributions to analysis on Hilbert space, probability theory, logarithmic Sobolev inequalities, and quantum field theory. Of highest priority in the current proposal is the construction of quantized Yang-Mills fields. He hopes to combine his previous work on abelian gauge goups with his previous work on nonabelian path dependent field strengths and with techniques from stochastic differential equations to obtain compactness of Euclidean measures. Second, Professor Gross plans to seek a Euclidean extension of his earlier work on Clifford algebra hypercontractivity for Fermions. Such an extension is expected to be closely related to the Osterwalder-Schrader formulation of Euclidean quantum field theory, Third, Professor Gross plans to explore the existence of logarithmic Sobolev inequalities on noncompact Lie groups and on hyperbolic n-space via the method of central limit type theorems on groups.