From Topology to Combinatorics and Back

这个项目的主要主题是研究拓扑学、几何学和组合学之间的相互作用。四个领域将得到特别强调：紧致流形的三角剖分、Cohen-Macaulay复形、球面的有限线性商和拟阵。Euler-Poincare公式自提出以来已有100多年的历史，但紧致流形的三角剖分的组合性质在很大程度上仍然是未知的。事实上，对于任何维度5或更大的流形来说，所有可能的f向量(其记录每个维度中的面数)的完整表征是未知的。自从Hochster，Reisner和Stanley在1970年的S的基础工作以来，Cohen-Macaulay单纯复形的可能的f-向量就已经知道了，然而，重要子类的面数的分类，如球面和双重Cohen-Macaulay复形，在代数和拓扑组合学中仍然是一个基本的开放问题。尽管有限群的表示理论是一门非常发达的学科，但用于计算相应球商的许多拓扑不变量的工具，如Betti数，实际上是不存在的。拟阵是一种线性无关的组合抽象，可以通过各种拓扑空间来建模。它们的计数性质有许多应用，包括图的着色和流、网络可靠性和超平面排列的拓扑。数学中的两个递归和互补的思想是使用离散数据来逼近连续现象，并用光滑对象(如球体或多面体)来模拟后者。这个项目的指导原则是，这两种技术的结合是接近这两种情况的一种强有力的方法。例如，紧致流形通常被表示为多个变量的非奇异多项式方程的解，也可以通过使用更容易编码到计算机中的单纯复形来研究。反过来，这允许对原始空间的形状和几何进行更深入的研究。

The primary theme of this project is to study the interplay between topology, geometry and combinatorics. Four areas will receive special emphasis: triangulations of compact manifolds, Cohen-Macaulay complexes, finite linear quotients of spheres, and matroids. It has been over 100 years since the introduction of the Euler-Poincare formula, yet the combinatorial properties of triangulations of compact manifolds remain largely unknown. Indeed, a complete characterization of all possible f-vectors (which record the number of faces in each dimension) is not known for any manifold of dimension five or more. The possible f-vectors of Cohen-Macaulay simplicial complexes have been known since the foundational work of Hochster, Reisner and Stanley in the 1970's. However, the classification of face counts for important subclasses, such as spheres and doubly Cohen-Macaulay complexes, remains a fundamental open problem in algebraic and topological combinatorics. Even though the representation theory of finite groups is an extremely well developed subject, tools for computing many topological invariants, such as Betti numbers, for the corresponding spherical quotients are practically nonexistent. Matroids are a combinatorial abstraction of linear independence which can be modeled through a variety of topological spaces. Their enumerative properties have many applications including coloring and flows on graphs, network reliability, and the topology of hyperplane arrangementsTwo recurrent and complementary ideas in mathematics are the approximation of continuous phenomena using discrete data, and modeling the latter with smooth objects, such as spheres or polyhedra. The guiding principle of this project is that the combination of these two techniques is a powerful method of approaching both situations.For instance, compact manifolds, which are frequently presented as solutions to nonsingular polynomial equations in several variables, can also be examined by using simplicial complexes which are easier to encode into a computer. In turn, this allows a deeper study of the shape and geometry of the original space.