Homotopy Theoretic Investigations in Higher K-theory, High-dimensional Data Analysis, and High Dimensional Manifold Theory

Gunnar E. CarlssonThis project concerns the descent problem in higher algebraic K-theory. 在确定收敛于域的K-群的谱序列的首项方面，最近有许多进展，特别是V. Voevodsky和A.苏斯林 这个项目试图证明代数K-理论谱，而不仅仅是它的同伦群，可以纯粹用所讨论的域的绝对伽罗瓦群和代数闭域的代数K-理论谱来描述。 关键成分是等变K-理论谱附加到绝对伽罗瓦群的复表示，这可以被视为一个高度结构化的环谱，并在一个增广理想的完成建设的衍生版本。该模型已经被开发出来，剩下的就是证明该模型与代数K理论的一般领域的协议。 这些结果的有效性应给予惊人的联系，一方面，派生完成的代表环的绝对伽罗瓦组，并在其他米尔诺K群的领域，这是团体直接定义的算术领域。这个项目旨在连接的算术行为在一个领域（代数对象，概括了我们通常的概念，真实的，复杂的，有理数）和行为的所谓的复杂表示的一组对称的一个更大的领域，包含所有可能的解决方案，以代数方程的原始领域。 这些表示包含了大量的几何信息，并且绘制这种（几何和算术信息之间的）联系一直是数学中的一个长期主题。这种看似不同的想法之间的关系一直负责许多最近引人注目的发展算术和代数几何，包括Delinge的证明所谓的韦尔几何以及最近的工作几何朗兰兹计划。 本项目代表了这一思想圈的另一个方面。

DMS-0406992Gunnar E. CarlssonThis project concerns the descent problem in higher algebraic K-theory. There has been much recent progress in identifying the initial term of a spectral sequence converging to the K-groups of a field, notably by V. Voevodsky and A. Suslin. This project is an attempt to prove that the algebraic K-theory spectrum, not just its homotopy groups, can be described purely in terms of the absolute Galois group of the field in question, and the algebraic K-theory spectrum of an algebraically closed field. The key ingredients are the equivariant K-theory spectrum attached to complex representations of the absolute Galois group, which can be regarded as a highly structured ring spectrum, and a derived version of a completion construction at an augmentation ideal. The model has already been developed, and what remains is to prove the agreement of the model with algebraic K-theory of the field in general. The validity of these results should give striking connections between, on the one hand, the derived completion of the representation ring of the absolute Galois group, and on the other the Milnor K-groups of the field, which are groups defined directly from the arithmetic of the field. This project seeks to connect the behavior of the arithmetic in a field (an algebraic object which generalizes our usual notions of real, complex, and rational numbers) and the behavior of so-called complex representations of a group of symmetries of a larger field containing all possible solutions to algebraic equations of the original field. These representations contain a great deal of geometric information, and drawing this kind of connection (between geometric and arithmetic information) has been a longstanding theme in mathematics. The relationship between such seemingly distinct kinds of ideas has been responsible for many of the recent striking developments in arithmetic and algebraic geometry, including Delinge's proof of the so-called Weil conjectures as well as the recent work on the Geometric Langlands program. The present project represents another facet of this circle of ideas.