Homotopy Theory and Applications

DMS-0306519 Michael J.霍普金斯、Lars Hesselhot和Haynes R.米勒教授Hesselholt，与Ib马德森和托马斯Geisser合作，在理解与环相关的分圆迹空间方面取得了很大的进展。 这些空间被发明来描述高维流形的代数同态群，Hesselholt的工作是该程序中的重要一步。中央赫塞霍尔特的工作是一个连接，他发现之间的分圆迹spacesand德拉姆-维特复杂的p-进算术代数几何。这导致了德拉姆-维特复形的推广到一个混合特征的情况下，它是相关的sheafof p-adic消失循环。这种关系进一步暗示了分圆迹空间背后动机理论的存在。在该基金的支持下，Hesselholt将继续研究p-adic算术代数几何与拓扑学计算之间的惊人关系，以理解代数同构群。 教授霍普金斯，与马修安藤和查尔斯Rezkhas最近发现了一种新的方法之间的同余模使用代数拓扑，他计划追求的眼睛在该地区的几个开放的问题。 在另一项合作努力中，霍普金斯教授、丹·弗里德和康斯坦丁·泰勒曼发现了维林德代数的拓扑表达式--数学物理和循环群表示理论中出现的拓扑结构。 他们三人计划更深入地探索这种联系，长期的野心是发现拓扑量子场论的拓扑来源。 霍普金斯还将继续他的工作与辛格对他们的理论“微分函数空间”，这提供了精细化的代数拓扑结构，特别适合于满足thedrequirements的数学物理。 教授米勒将继续他的工作Landweber精确和椭圆上同调理论，他的工作与条例草案德怀尔拓扑Hochschild同源性和几何ofree循环空间，以及他的同伦理论的研究algebraicstructures所产生的结理论和现代理论的Hopfalgebras。该领域的代数拓扑结构出现在19世纪后期的世纪作为数学家注意到了深刻的相似之处非常不同的前瞻性研究领域。 到了世纪初，代数拓扑学的基本理论已经形成，并被用来解释、整理和揭示几何学的许多定性方面。在接下来的一百年里，代数拓扑学以惊人的速度向前发展，到现在为止，似乎当代数学的每一个分支在试图阐明其定性方面时都借鉴了某种形式的代数拓扑学。 麻省理工学院的代数拓扑小组运行着一个大型而多样化的项目，强调代数拓扑、代数几何、几何和数学物理之间的许多联系。

DMS-0306519Michael J. Hopkins, Lars Hesselhot, and Haynes R. MillerProfessor Hesselholt, working with Ib Madsen and Thomas Geisser, hasmade deep advances in understanding the cyclotomic trace spacesassociated to a ring. These spaces were invented to describing thediffeomorphism groups of high dimensional manifolds, and Hesselholt'swork is a significant step in that program. Central to Hesselholt'swork is a connection he descovered between the cyclotomic trace spacesand the de Rham-Witt complex of p-adic arithmetic algebraicgeometry. This has led to a generalization of the de Rham-Witt complexto a mixed characteristic situation, where it is related to the sheafof p-adic vanishing cycles. This relationship further suggests theexistence of a motivic theory behind the cyclotomic tracespaces. Under the support of this grant, Hesselholt will pursue thisstriking relationship between p-adic arithmetic algebraic geometry andthe computations in topology relevant to understanding diffeomorphismgroups. Professor Hopkins, working with Matthew Ando and Charles Rezkhas recently uncovered a new approach to congruences between modularforms using algebraic topology, which he plans to pursue with an eyeon several open problems in the area. In another collaborativeeffort, Professors Hopkins, Dan Freed and Constantin Telemandiscovered a topological expression for the Verlinde algebra--analgebraic structure arising in mathematical physics and in the theoryof representations of loop groups. The three of them plan to explorethis connection more deeply, with the long term ambition ofdiscovering the topological sources of topological quantum fieldtheories. Hopkins will also continue his work with Isadore Singer ontheir theory of ``differential function spaces,'' which offers arefinement of algebraic topology especially suited for meeting thedemands of mathematical physics. Professor Miller will continue hiswork on Landweber exact and elliptic cohomology theories, his workwith Bill Dwyer on topological Hochschild homology and the geometry offree loop spaces, and his study of the homotopy theory of algebraicstructures arising in knot theory and the modern theory of Hopfalgebras.The field of algebraic topology arose in the late 19th century asmathematicians noticed deep similarities between very differentlooking areas of research. By the beginning of the 20th century, thebasic theory of algebraic topology was in place, and it was being usedto explain, codify and reveal many qualitative aspects of geometry.During the next hundred years algebraic topology advanced at anamazing pace, and by now it seems as if every branch of contemporarymathematics draws on some form of algebraic topology when undertakingto articulate its qualitative aspects. The algebraic topology groupat MIT runs a large and diverse program emphasizing the manyconnections between algebraic topology, algebraic geometry, geometry,and mathematical physics.