Algebraic combinatorics

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My proposal concerns the study of certain algebraic spaces whose description and properties are establishedthrough the study of interesting combinatorial structures. A typical (and still very actively studied) set ofexamples consist in describing explicit decompositions of important representations of finite groups intoirreducible representations. This is of crucial importance in many fields going from theoretical physics toalgebraic geometry, and it naturally involves a deep combinatorial understanding of certain combinatorialstructures, namely: Young Tableaux. Among the current fundamental problems in the area, lies a betterunderstanding of Macdonald polynomials; and our combinatorial approach has already established numerousnew results and revealed new avenues of research. Our approach goes through the study of certain spaces of n variable polynomials  characterized as solution sets of a family of partial differential equations. In particular these polynomials are  harmonic or diagonnaly harmonic. Several important generalizations are considered, and their description give  rise to new interesting formulas involving symmetric functions.  A recent addition to my areas of research concerns an entirely new combinatorial classification of the space of  complex polynomials in term of the structure of the pair of real plane algebraic curves.