Algebraic Topology

DMS-0306429 Matthew Ando和Randy McCarthy Ando和McCarthy将研究稳定同伦理论与椭圆上同调中介的数学物理和数论之间的关系，以及稳定同伦理论与K-理论中介的代数之间的关系。Ando将使用椭圆上同调来研究弦理论和椭圆曲线理论对紧致流形的拓扑有什么影响。他特别感兴趣的是把最近关于开弦理论和d-膜的工作与椭圆上同调中的问题联系起来。McCarthy将发展双模的Witt结构理论和交换环谱的光滑性和De Rham上同调理论。前者提供了新的见解和对Hesselholt和Madsen的计算的实质性概括，而后者提供了发展晶体陈特征的可能性，从而将Bloch的工作扩展到交换环谱。在过去的几年中，人们看到了拓扑学中预期的新结构的惊人扩散，涉及到与数学和物理的其他领域的新相互作用。Ando和McCarthy对阐明这些新的联系特别感兴趣。Ando的研究试图给出由椭圆上同调表示的拓扑学、高能物理和数论之间深刻而令人惊讶的关系的显式形式。传统的代数拓扑学范式是利用代数将现象模型化为拓扑学。麦卡锡的研究颠覆了这一范式，利用拓扑空间的深层结构来解决代数中的难题。

DMS-0306429Matthew Ando and Randy McCarthyAndo and McCarthy will study the relationship between stable homotopytheory and mathematical physics and number theory mediated by ellipticcohomology, and the relationship between stable homotopy theory andalgebra mediated by K-theory. Ando will use elliptic cohomology toinvestigate what string theory and the theory of elliptic curves haveto say about the topology of compact manifolds. He is particularlyinterested in bringing recent work on open string theories andD-branes to bear on problems in elliptic cohomology. McCarthy willdevelop a theory of Witt structures for bimodules and a theory ofsmoothness and De Rham cohomology for commutative ring spectra. Theformer offers new insight and substantial generalizations ofcalculations of Hesselholt and Madsen, while the latter offers thepossibility of developing a crystalline Chern character and soextending to commutative ring spectra the work of Bloch.The last few years have seen an astonishing proliferation ofunexpected new structure in topology, involving new interactions withother areas of mathematics and physics. Ando and McCarthy areparticularly interested in articulating these new connections.Ando's research seeks to give explicit form to the deep and surprisingrelationship between topology, high energy physics, and number theorysignaled by elliptic cohomology. The traditional paradigm ofalgebraic topology is the use of algebra to model phenomena intopology. McCarthy's research turns this paradigm upside down, usingthe deep structure of topological spaces to tackle hard problems inalgebra.