Mathematical Sciences: Algebraic K-Theory, Topological Cyclic Homology and Crystalline Cohomology

这个项目的一个目标是继续研究与代数k理论密切相关但更易于计算的理论，特别是拓扑Hochschild同调和由此产生的各种理论。这些研究的一个主要焦点是W(p)(A)，它是环A的拓扑Hochschild同调的不动点谱的同伦逆极限，它是圆的子群具有可被素数p整除的阶，其中的结构映射是由它们的不动点子空间的等变映射的限制引起的。当A可交换时，W(p)(A)的第0个同伦群由A的第p个Witt向量组成，并且存在一个从K（End(A)）到W(p)(A)的自然映射，该映射在第0个同伦群上是期望映射（来自Almkvist的工作）。如果我们使用W(p)(A)作为p-th Witt向量的推广，我们可以考虑W(p)(A)的同伦不动点谱（存在一个圆作用）作为p-th DeRham-Witt复形的可能推广。通过W(p)(A)的同伦轨道谱，从K(A)到K(End(A))（通过同构自同态）到W(p)(A)因子的复合映射，研究者希望这是对Bloch晶体陈氏特征的一个很好的修正。这个项目处理了一些在过去几十年里为将几何信息简化为一个计算主题而取得巨大成功的代数装置。所涉及的几何信息的性质是难点的关键。虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但这与几何对象的拓扑特性大不相同。这些属性包括连通性（整体）、打结性、无孔等等。所有对这些性质的系统研究，例如，如何判断她的两个几何物体在其中一个性质上是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。研究者的工作旨在更好地理解目前可用的一些强大的代数工具之间的关系，显然希望有了更好的理解，就有能力完善和提高它们。如此基本的东西的潜在价值是很难量化的，因为拓扑性质和问题出现在如此广泛不同的数学设置中，如控制卫星运动的微分方程和经济模型的平衡条件。＊＊＊

9415615 McCarthy One goal of this project is to continue the study of theories closely related to algebraic K-theory but which are more accessible to computation, in particular, topological Hochschild homology and various theories which are generated from it. A major focal point for these investigations is W(p)(A), which is the homotopy inverse limit of the fixed point spectra of topological Hochschild homology of a ring A with respect to the subgroups of the circle having order divisible by the prime p, where the structure maps are those arising from the restriction of equivariant maps to their fixed point subspaces. When A is commutative, the 0-th homotopy group of W(p)(A) consists of the p-th Witt vectors of A, and there is a natural map from K(End(A)) to W(p)(A) which on the 0-th homotopy group is the expected map (from work of Almkvist). If we use W(p)(A) as a generalization of the p-th Witt vectors, one is led to consider the homotopy fixed point spectrum of W(p)(A) (there is a circle action) as a possible generalization of the p-th DeRham-Witt complex. The composite map from K(A) to K(End(A)) (by the identity endomorphism) to W(p)(A) factors through the homotopy orbit spectrum of W(p)(A), and it is the investigator's hope that this is a good modification of Bloch's crystalline chern character. This project treats some of the algebraic apparatus that has been developed with huge success over the past several decades for reducing geometric information to a subject for calculation. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whet her two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. The investigator's work aims to understand better the relations among some of the powerful algebraic tools now available for doing this, the obvious hope being that with better understanding comes the ability to perfect and sharpen them. The potential value of something so basic is hard to quantify, since topological properties and questions arise in such widely varied mathematical settings as the differential equations that govern satellite motions and the conditions for equilibrium of models of the ecomomy. ***