Interactions of Elliptic Cohomology with Other Subjects

AbstractAward：DMS-0504539首席研究员：诺拉甘特该项目研究椭圆余同调与其他数学和物理领域的相互作用，重点是等变现象和功率运算。从弦论中的轨道模型到2-范畴中的表示和特征理论，再到广义月光和稳定同伦理论。 Hopkins-Kuhn-Ravenel特征标理论在同伦理论的背景下得到了发展，并揭示了上述领域之间的许多形式上的相似性。 该项目的目标是更好地理解这些观察背后的机制。椭圆上同调中的幂运算已经与弦论中的乘积公式和广义Moonshine的扭曲Hecke算子联系在一起。这个项目进一步追求这些联系，旨在澄清这三个学科的文献之间的联系。用不那么专业的术语来说：椭圆上同调是数学和物理领域的丰富组合，从表面上看，这两个领域似乎没有太多关系。特别是在一组对称性的作用下，人们可以观察到看似无关的研究领域之间的任何类比。与来自不同数学领域的合作者，这个项目试图找到这些现象背后更深层次的机制的解释。

AbstractAward: DMS-0504539Principal Investigator: Nora GanterThe project investigates the interaction of elliptic cohomologywith other areas of mathematics and physics, with a focus onequivariant phenomena and power operations. Subjects treatedrange from orbifold models in string theory over representationand character theory in 2-categories to generalized Moonshine andstable homotopy theory. Hopkins-Kuhn-Ravenel character theoryarose in the context of homotopy theory and exposes many formalsimilarities between the above fields. The goal of the projectis to better understand the mechanisms underlying theseobservations. Power operations in elliptic cohomology werealready linked to product formulas in string theory and to thetwisted Hecke operators of generalized Moonshine. This projectwill further pursue these connections, aiming to clarify thelinks between the literature in these three subjects.In less technical terms: Elliptic cohomology is a rich mix ofareas of mathematics and physics, which on the surface do notappear to have much to do with each other. Especially in thepresence of an action by a group of symmetries, one can observemany analogies between seemingly unrelated research areas. Withcollaborators from various fields of mathematics, this projectseeks to find an explanation for the deeper mechanisms underlyingthese phenomena.