Algebraic combinatorics

我的建议涉及对某些代数空间的研究，这些空间的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(仍然被非常活跃地研究的)例子集在于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的组合理解，即：年轻的画面。在当前该领域的基本问题中，更好地理解了Macdonald多项式；我们的组合方法已经建立了许多新的结果，并揭示了新的研究途径。我们的方法是研究某些由n个变量多项式组成的空间，这些空间被刻画为一族偏微分方程组的解集。特别地，这些多项式是三次调和或对角调和。考虑了几个重要的推广，它们的描述引起了涉及对称函数的新的有趣的公式。最近我的研究领域的一个补充涉及到根据实平面代数曲线对的结构对所有复多项式空间的一种全新的组合分类。

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.