Twists of elliptic cohomology and K-theory

椭圆上同调的许多微妙问题都与它的曲折有关。例如，圈群的表示出现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。 安藤提出在扭曲椭圆上同调和扭曲K-理论之间构造一个比较映射，解释了为什么两个理论都有循环群的表示。 另一个例子是，近年来拓扑场论的研究取得了惊人的进展，了解这些场论在不同的族中是如何变化的是至关重要的。 安藤将研究如何椭圆上同调的最重要的扭曲有关的问题，建立某些家庭的二维拓扑场论。 人们希望这将使洞察椭圆上同调本身如何在家庭中变化。椭圆上同调是拓扑学、数论和物理学之间深刻而神秘的相互作用的轨迹。 在1998年美国数学学会的吉布斯演讲中，物理学家爱德华·威滕（Edward Witten）将这种相互作用称为“20世纪碰巧出现的21世纪数学的一小部分”，并提出理解它将是21世纪有趣而重要的问题。 最近的工作表明，椭圆上同调结构的基本部分之一是它的“扭曲”：每当这个问题有重要进展时，它们就会一次又一次地出现。 安藤的项目将直接关注这些曲折，探索它们在代数，拓扑学和物理学中的表现。

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.