Twists of elliptic cohomology and K-theory

关于椭圆上同的许多微妙问题都与椭圆上同的曲折有关。例如，在扭曲等变椭圆上同调（Grojnowski-Ando）和扭曲等变k理论（Freed-Hopkins-Teleman）的代数拓扑中出现了环群的表示。Ando提出了在扭椭圆上同调和扭k理论之间构造一个比较映射，解释了环群的表示在这两个理论上出现的原因。另一个例子是，近年来拓扑场论的研究取得了惊人的进展，了解这些理论在家庭中的变化是至关重要的。Ando将研究椭圆上同调的最重要的扭曲是如何与构造二维拓扑场论的某些族的问题联系起来的。希望这将有助于了解椭圆上同性本身在家族中是如何变化的。椭圆上同调是拓扑、数论和物理学之间深刻而神秘的相互作用的轨迹。1998年，物理学家爱德华·威滕（Edward Witten）在美国数学学会（American Mathematical Society）的吉布斯讲座（Gibbs Lecture）中，将这种相互作用称为“21世纪数学的一小块碰巧落在了20世纪”，并提出，理解这种相互作用将是21世纪一个有趣而重要的问题。最近的研究表明，椭圆上同调的基本结构之一是它的“扭曲”：每当该学科取得重要进展时，它们就会一次又一次地出现。安藤的项目将直接关注这些扭曲，探索它们在代数、拓扑和物理中的表现。

Many subtle issues about elliptic cohomology concern its twists. For example, the representations of loop groups arise in algebraic topology in twisted equivariant elliptic cohomology (Grojnowski-Ando) and in twisted equivariant K-theory (Freed-Hopkins-Teleman). Ando proposes to construct a comparison map between twisted elliptic cohomology and twisted K-theory which explains why the representations of loop groups appear on both theories. For another example, in recent years there has been spectacular progress in the study of topological field theories, and it is of fundamental importance to understand how these vary in families. Ando will investigate how the most important twists of elliptic cohomology are related to the problem of constructing certain families of two-dimensional topological field theories. It is hoped that this will give insight into how elliptic cohomology itself varies in families. Elliptic cohomology is the locus of a deep and mysterious interaction between topology, number theory, and physics. In his 1998 Gibbs Lecture to the American Mathematical Society, the physicist Edward Witten called this interaction "a little piece of 21st century mathematics which happened to land in the 20th century", and proposed that understanding it would be an interesting and important problem for the 21st century. Recent work has shown that one of the fundamental pieces of structure of elliptic cohomology are its "twists": they crop up again and again each time there is important progress in the subject. Ando's project will focus directly on these twists, exploring their manifestations in algebra, topology, and physics.