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

gunnar E. carlsson这个项目研究了高代数k理论中的下降问题。近年来，以V. Voevodsky和a . Suslin为代表的研究人员在确定收敛于场k群的谱序列的初始项方面取得了很大进展。本课题试图证明代数k -理论谱，而不仅仅是它的同伦群，可以纯粹地用所讨论域的绝对伽罗瓦群和代数闭域的代数k -理论谱来描述。关键成分是附属于绝对伽罗瓦群复表示的等变k理论谱，它可以被视为高度结构化的环谱，以及在增广理想下的补全构造的派生版本。该模型已经被开发出来，剩下的就是证明该模型与一般场的代数k理论的一致性。这些结果的有效性一方面给出了绝对伽罗瓦群的表示环的推导补全，另一方面给出了域的米尔诺k群之间的显著联系，米尔诺k群是直接由域的算术定义的群。这个项目试图将一个场（一个代数对象，它概括了我们通常对实数、复数和有理数的概念）中的算术行为与包含原始场的代数方程的所有可能解的更大场的一组对称的所谓复杂表示的行为联系起来。这些表示包含了大量的几何信息，绘制这种（几何和算术信息之间的）联系一直是数学中一个长期存在的主题。这些看似不同的思想之间的关系导致了最近算术和代数几何领域的许多惊人发展，包括德林格对所谓的韦尔猜想的证明，以及最近对几何朗兰兹纲领的研究。目前的项目代表了这一思想循环的另一个方面。

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.