The value of symmetry in mathematics is universally recognized as one of the great concepts used in the analysis of complex problems. In nature as well as in abstraction, the most symmetric choice for a solution often turns out to be the correct one or the most favorable among all candidates. Work to be done on this project continues an extended program in the development of a unified theory of symmmetrization deriving from a certain master inequality formulated and proved in 1974 (with antecedents going back to the fifth book of Pappus, at least) . Particular emphases will be placed on extremal problems from complex function theory, partial differential equations and quasiconformal mapping. Examples include finding the best constant in Landau's covering theorem giving the best inequality between the diameter of the image of a holomorphic function and its derivative at some point. In partial differential equations, work will be done in determining the shape of those domains of fixed diameter which minimize the heat kernel. The distortion of quasiconformal maps in any given direction is well documented. Very little is known on the limitations of how these maps can distort area. Work will focus on this question. It is axiomatic in mathematics that by asking the right questions, one is often led to the answer of a difficult question with comparative ease. Often right questions involve new insights into symmetry. The purpose of this research is to continue expanding the scope of general symmetry principles which have the potential for broad applicability.
