Homotopy Theory and Its Applications

9803428米勒在早期的工作中，霍普金斯和米勒教授已经展示了如何将椭圆曲线的纯代数理论嵌入到同伦理论中。其结果是构造了一个新的对象--拓扑模形式的谱TMF，它是对经典模形式环的深刻丰富。霍普金斯教授希望证明，这个物体可以有几种用途。它应该得到一个广义的“Witten亏格”，将复杂的代数不变量赋给某些几何流形。另一方面，它应该解释某些theta函数所满足的深度算术同余(Hopkins教授早些时候猜想，R.Borcherds证明了)。霍普金斯的计划是将TMF的构造扩展到适当定义的“拓扑theta函数”的构造。米勒教授打算更早地与霍普金斯一起研究实K理论的更高类比；这类理论有一个无穷无尽的家族，基本上只研究了其中一个。他希望利用形式群的理论，对某些同伦群最近的精细计算给出概念性的和可推广的证明。他还将尝试将K理论和流形的谱理论之间的关系推广到椭圆类比。Hesselholt教授打算继续研究环的同伦理论上定义的“线性”不变量(拓扑循环同调)和深度算术“非线性”不变量(K-理论)之间的关系，并将他对TC的计算推广到更困难的情况。霍普金斯、米勒和赫塞尔霍尔特教授的工作走在了将同伦理论应用于数学的其他部分的前沿，特别是算术和保形场论。同伦理论提供了对组合学的巨大扩展，其中不仅跟踪等价类，而且跟踪各种对象彼此等价的方式。这些研究人员的工作是一个正式的背景，人们可以在这个背景下组织寻找在这些其他领域具有意义的物体，此外还具有巨大的内在复杂性和美感。***

9803428 Miller In earlier work, Professors Hopkins and Miller have shown how the purely algebraic theory of elliptic curves can be embedded into homotopy theory. The result is a construction of a new object, the spectrum TMF of topological modular forms, which is a deep enrichment of the classical ring of modular forms. Professor Hopkins hopes to show that this object can be used in several ways. It should receive a generalized "Witten genus," assigning sophisticated algebraic invariants to certain geometric manifolds. On the other hand, it should account for deep arithmetic congruences satisfied by certain theta-functions (conjectured earlier by Professor Hopkins and proven by R. Borcherds). Hopkins' program is to extend the construction of TMF to a construction of suitably defined "topological theta-functions." Professor Miller intends to pursue earlier work with Hopkins on higher analogues of real K-theory; there is an infinite family of these, of which only essentially one has been investigated. He hopes to use the theory of formal groups to give conceptual and generalizable proofs of recent elaborate computations of certain homotopy groups. He will also attempt to extend the relationship between K-theory and the spectral theory of a manifold to an elliptic analogue. Professor Hesselholt intends to continue his research into the relationship between a homotopy-theoretically defined "linear" invariant of rings (topological cyclic homology) and a deep arithmetic "nonlinear" invariant (K-theory), and to extend his computation of TC to more difficult cases. The work of Professors Hopkins, Miller, and Hesselholt stands at the forefront of the application of homotopy theory to other parts of mathematics, notably, arithmetic and conformal field theory. Homotopy theory offers a vast enlargement of combinatorics, in which one keeps track not only of equivalence classes but also of the ways in which various objects are equivalent to each other. The w ork of these investigators serves as a formal background against which one may organize the search for objects with meaning in these other areas, in addition to possessing great intrinsic complexity and beauty. ***