Algebraic combinatorics

我的建议涉及研究某些代数空间，其描述和性质是通过研究有趣的组合结构来建立的。一个典型的（仍然是非常积极研究）的例子包括在描述有限群的重要表示到不可约表示的显式分解。这在从理论物理到代数几何的许多领域都是至关重要的，它自然涉及对某些组合结构的深刻组合理解，即：Young Tableaux。在该领域目前的基本问题，在于更好地理解麦克唐纳多项式，我们的组合方法已经建立了numerousnew结果，并揭示了新的研究途径。我们的方法是通过研究某些空间的n个变量多项式的特点是解决一个家庭的偏微分方程组。特别地，这些多项式是调和的或对角调和的。几个重要的推广被认为是，他们的描述产生新的有趣的公式涉及对称函数。 最近除了我的研究领域涉及一个全新的组合分类空间的复杂多项式的结构对真实的平面代数曲线。

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.