Homotopy Theory and Applications

michael J. Hopkins， Lars Hesselholt和Haynes R. miller Hesselholt教授与Ib Madsen和Thomas Geisser合作，在理解与环相关的切环痕迹空间方面取得了深刻的进展。这些空间是用来描述高维流形的微分同构群的，Hesselholt的工作是该计划的重要一步。Hesselholt工作的核心是他发现了p进算术代数几何的环切迹空间和de Rham-Witt复合体之间的联系。这导致将de Rham-Witt复形推广到一种混合特征情况，在这种情况下，它与p进消失循环有关。这种关系进一步表明，存在一个动机理论背后的睫状体切开术的痕迹空间。在这项资助的支持下，Hesselholt将继续研究p进算术代数几何与拓扑计算之间的惊人关系，这些计算与理解微分同构群有关。霍普金斯教授最近与Matthew Ando和Charles Rezkhas合作，发现了一种使用代数拓扑来处理模形之间同余的新方法，他计划在关注该领域的几个开放问题的同时继续研究。在另一项合作中，霍普金斯教授、丹·弗里德（Dan Freed）和康斯坦丁·泰勒曼（Constantin telemann）发现了Verlinde代数的拓扑表达式。Verlinde代数是数学物理学和环群表示理论中出现的代数结构。他们三人计划更深入地探索这种联系，他们的长期目标是发现拓扑量子场论的拓扑来源。霍普金斯还将继续他与Isadore Singer的“微分函数空间”理论的工作，该理论提供了代数拓扑的细化，特别适合满足数学物理的要求。Miller教授将继续他在Landweber精确和椭圆上同调理论方面的工作，他与Bill Dwyer在拓扑Hochschild同调和几何自由环空间方面的工作，以及他在结理论和现代Hopfalgebras理论中出现的代数结构的同调理论的研究。代数拓扑学领域兴起于19世纪后期，当时数学家们注意到在看起来非常不同的研究领域之间存在着深刻的相似性。到20世纪初，代数拓扑学的基本理论已经形成，它被用来解释、编纂和揭示几何的许多定性方面。在接下来的一百年里，代数拓扑学以惊人的速度发展，到目前为止，似乎当代数学的每个分支在阐述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学小组运行着一个庞大而多样的项目，强调代数拓扑学、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.