Representation of Galois groups and descent in algebraic K-theory

DMS-0104162冈纳卡尔松这个项目包括几个不同的研究方向。Carlsson将研究他为场的代数K-理论谱构建的同伦理论模型，该模型建立在场的绝对伽罗华群的表示理论之上。他将试图证明该模型确实等价于场的K理论，并计算出这一结果对Quillen-Lichtenbaum猜想的影响，以及这一结果与Bloch-Kato猜想之间的关系。他还将研究代数拓扑学在高维数据分析中的潜在应用。他希望改进他和V.de Silva为同源计算开发的软件，目标是识别奇点等特征以及维度大于3的数据集的全局拓扑。他还计划研究计算问题中出现的同伦理论问题，这些问题与从大型单纯复形中抽样子复形有关。Kiem计划计算曲线的奇异模空间的交数。这是一个重要的问题，因为这种情况下的交数与著名的几何不变量有关，例如卡森不变量。过去两个世纪数学中的一个伟大主题是算术和几何之间的密切关系。对这一主题的研究是由Abel和Galois在19世纪初开始的，在本世纪它的进一步发展使我们获得了关于方程组的整数或有理解集合的非常精确的信息。这个项目的目标是探索这一主题的另一种表现形式，即所谓的代数K场论。代数K-理论是一种依附于算术对象的几何结构，称为域。域是算术对象，可以在其中进行加法、乘法和除法。代数K-理论将几何对象附加到这些域上，以这种方式可以很容易地提取已经很好理解的域的不变量。代数K-理论也包含许多鲜为人知的不变量，理解这些不变量是拓扑学家们自20世纪70年代初S定义更高代数K-理论以来一直努力的目标。关于代数K-理论，已经提出了两个重要的猜想：Quillen-Lichtenbaum猜想和Bloch-Kato猜想。他们将把一个场的代数K-理论与它的“绝对伽罗华群”的性质联系起来，绝对伽罗华群是一个对象，它包含了所讨论的域上的方程组的解集的所有可能的对称性。这个项目的目的是非常清楚地理解代数K-理论是如何从这一大群对称的信息中建立起来的，并利用这种理解来接近上面提到的两个中心猜想。

DMS-0104162Gunnar CarlssonThis project includes several different directions of research. Carlsson will study a homotopy theoretic model he has constructed for the algebraic K-theory spectrum of a field, which is built out of the representation theory of the absolute Galois group of the field. He will try to prove that the model is indeed equivalent to the K-theory of the field, as well as to work out the consequences of this result for the Quillen-Lichtenbaum conjectures and the relationship between this result and the Bloch-Kato conjecture. He will also investigate potential applications of algebraic topology in high dimensional data analysis. He expects to improve software which he and V. De Silva have developed for homology computation, with the goal of identifying features such as singular points as well as global topology for data sets of dimension greater than three. He also plans to study homotopy theoretic issues which arise in computational questions, relating to sampling subcomplexes from large simplicial complexes. Kiem plans to compute intersection numbers for singular moduli spaces of curves. This is an important problem since the intersection numbers in this case are related to well-known geometric invariants, such as the Casson invariant.One of the great themes in mathematics over the last two centuries is the strong relationship between arithmetic and geometry. The study of this theme was initiated by Abel and Galois in the early 19th century, and in this century its further development has resulted in our obtaining very precise information concerning sets of integer or rational solutions to systems of equations. The goal of this project is to explore another manifestation of this theme, in the form of the so-called algebraic K-theory of fields. Algebraic K-theory is a geometric construction attached to arithmetic objects, called fields. Fields are arithmetic objects in which one can add, multiply, and divide. Algebraic K-theory attaches geometric objects to these fields, and in such a way that already well understood invariants of fields can be extracted easily. Algebraic K-theory also contains many less well understood invariants as well, and the goal of understanding these invariants has been one toward which topologists have been striving since the early 1970's, when Quillen defined higher algebraic K-theory. Two important conjectures have been formulated concerning algebraic K-theory, the Quillen-Lichtenbaum and Bloch-Kato conjectures. They would relate the algebraic K-theory of a field to properties of its "absolute Galois groups", an object which incorporates all possible symmetries of sets of solutions of sets of equations over the field in question. This project aims to understand very clearly how algebraic K-theory is built out of information of this large group of symmetries, and to use this understanding to approach the two central conjectures mentioned above.