Finite Groups and Periodic Theories in Stable Homotopy

复上同调理论的研究是代数拓扑学中一个由来已久且备受推崇的领域。另一个领域是(有限)群论构造，以其最现代的形式成为等变稳定同伦理论。结果近年来，最引人注目的是西格尔的Burnside环猜想和Ravenel的幂零猜想的证明，已经戏剧性地开始实现稳定同伦理论的这两个分支。从历史上看，这些分支机构基本上是相互独立的。一个例外是Atiyah将K*(BG)(一种边缘主义类型的对象)与G的复数表示环联系起来的定理。现在有迹象表明，相互关系要广泛得多。这是库恩最近和目前研究的主题。很难说对这种复杂的代数模式的研究将走向何方，但数学史有力地表明，掌握这些模式将在其他地方产生影响，无论是使用模式本身，还是使用为处理它们而开发的技术。

The study of complex cobordism and related cohomology theories is a long standing and well respected area of algebraic topology. Another area has been that of (finite) group theoretic constructions, becoming, in its most modern form, equivariant stable homotopy theory. Results in recent years, most spectacularly the proofs of Segal's Burnside ring conjecture and Ravenel's nilpotence conjecture, have dramatically begun to bring to fruition these two branches of stable homotopy theory. Historically, these branches have been basically independent of each other. An exception is Atiyah's theorem relating K*(BG) (a bordism type of object) to the complex representation ring of G. There are now indications of much more widespread interrelations. This is the subject of Kuhn's recent and current research. It is hard to say where the study of such intricate algebraic patterns will lead, but the history of mathematics strongly suggests that mastering them will have repercussions elsewhere, either in use of the patterns themselves or of the techniques developed for treating them.