Algebraic Topology

matthew Ando和Randy McCarthy将研究稳定同伦理论与以椭圆上同调为中介的数学物理和数论的关系，以及稳定同伦理论与以k理论为中介的代数的关系。Ando将使用椭圆上同调来研究弦理论和椭圆曲线理论对紧流形拓扑的解释。他特别感兴趣的是将最近关于开放弦理论和膜的研究成果应用于椭圆上同调问题。McCarthy将发展双模的Witt结构理论和交换环谱的光滑和De Rham上同调理论。前者为Hesselholt和Madsen的计算提供了新的见解和实质性的推广，而后者提供了发展结晶陈氏特征的可能性，并将Bloch的工作扩展到交换环谱。在过去的几年里，拓扑学中意想不到的新结构出现了惊人的扩散，涉及到与数学和物理其他领域的新的相互作用。安藤和麦卡锡对阐明这些新的联系特别感兴趣。安藤的研究试图为拓扑、高能物理和由椭圆上同调表示的数论之间深刻而令人惊讶的关系给出明确的形式。代数拓扑学的传统范式是利用代数对拓扑学现象进行建模。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.