Thom Polynomials for Group Actions and Singularities

与一个不变的子品种的表示，我们认为它的上同调类在等变上同调，并称之为托姆多项式的品种（"奇异性“）。因此，Thom多项式是一个G-等变特征类。追溯等变上同调的定义，我们得到的知识的Thom多项式的奇点减少全球奇点理论问题同伦理论（即计算的特征类的基本拓扑情况）。Thom多项式计算轨迹在参数空间中的上同调类，在该参数空间上对象（映射芽、微分形式、排列、丛、线性映射的二进制等）退化。 一个很有前途的方向是寻找相应的好的概念在非常上同调理论，计算这些非常Thom多项式和发现拓扑应用。另一个重要的方向是研究几何或物理相关表示的全局奇异性理论，如Dynkin表示、曲面丛、复数和真实的数上的多语言轨迹、超平面排列，以及它们与度、结式和判别式的关系。在许多这些情况下，人们期望不同的积极性性质的托姆多项式，这可以有助于新兴的桥梁几何和组合。第三个目标是探索Thom多项式在几何不变论中的间接应用。在这里，托姆多项式可以计算各种不稳定的轨迹表示，因此，他们给自然的，几何定义的关系，在上同调环的G.I.T.在各种几何和拓扑情况下，奇点的出现是由于全局的原因。也就是说，一个空间、一个流形、一个簇或一个映射的整体拓扑迫使某些奇点出现。（一个简单的例子是克莱因瓶的全局拓扑，当映射到三维空间时，它会强制双点。奇点的这种全局行为是由它们的托姆多项式决定的。如果我们把全局拓扑不变量代入Thom多项式，我们就得到拓扑强迫的奇点数。这个一般的观点包含几个数学领域作为特殊情况，其中包括代数几何的“退化轨迹公式”，微分拓扑的浸入和一般多点公式，以及代数组合学的Schur，Schubert和Schlematic多项式理论。然而，托姆多项式是出了名的难以计算。虽然已有一些强有力的方法，其中一些是作者自己提出的，但还没有找到一种通用的方法。我们建议研究Thom多项式理论的基础，包括对其他上同调理论的可能推广。另一个挑战是理解自然无穷级数的内部结构的Thom多项式。这个方向承诺建立不同领域之间的联系，在不同的计算方法中发挥作用，例如代数拓扑，对称函数，插值理论，局部化，Groebner基理论。另一个主要目标是找到计算Thom多项式在拓扑，几何和不变理论中的应用。

Associated with an invariant subvariety of a representation we consider its cohomology class in equivariant cohomology, and call it the Thom polynomial of the variety (``singularity''). Thus, the Thom polynomial is a G-equivariant characteristic class. Tracing back the definition of equivariant cohomology we obtain that the knowledge of the Thom polynomial of singularities reduces global singularity theoretic problems to homotopy theory (namely, to the computation of the characteristic classes of the underlying topological situation). Thom polynomials compute the cohomology class of the locus in a parameter space over which an object (map germ, differential form, arrangement, a bundle, a digaram of linear maps, etc) degenerates. A promising direction is the search for the corresponding good notion in extraordinary cohomology theories, the computation of these extraordinary Thom polynomials and finding topological applications. Another important direction is working out global singularitytheory of geometrically or physically relevant representations, such asDynkin quiver representations, surface bundles, multisingualrity loci over the complex and the real numbers, hyperplane arrangements, as well as their connections to degree, resultant and discriminant formulas. In many of these cases one expects different positivity properties of the Thom polynomials, which can contribute to the newly emerging bridge between geometry and combinatorics. The third goal is to explore the indirect usage of Thom polynomials in Geometric Invariant Theory. Here Thom polynomials can compute various non-stability loci of representations, hence they give natural, geometrically defined relations in the cohomology ring of the G.I.T. quotients.In various geometric and topological situations singularities occurfor global reasons. That is, the global topology of a space, a manifold, a variety or a map forces some singularities to occur. (A trivial example is the global topology of the Klein bottle which forces double points when mapped into 3-space.) This global behaviour of singularities is governed by their Thom polynomials. If we substitute global topological invariants into the Thom polynomial we obtain the number of singularities forced by topology. This general point of view contains several mathematical areas as special cases, among others ``degeneracy loci formulas'' of algebraic geometry,immersion and general multiple point formulas of differentaial topology, and the theory of Schur, Schubert and quiver polynomials of algebraic combinatorics. Thom polynomials, however, are notoriously hard to compute. Although some powerful methods are known, some by the author, no universal method has beed found. We propose to study the basics of Thom polynomial theory, including possible generalizations to other cohomology theories. Another challenge is the understanding of the interior structure of natural infinite series of Thom polynomials. This direction promises to establish connections between the different areas that play roles in different computational approaches, e.g. algebraic topology, symmetric functions, interpolation theory, localizations, Groebner basis theory.Another main goal is to find applications of computed Thom polynomialsin topology, geometry and invariant theory.