β Scribed by Hurtubrise J.C.
No coin nor oath required. For personal study only.
Spaces of particles have long been studied in homotopy theory, partly for their intrinsic interest but also for their role in describing the structure of loop spaces. Recently the structure of these spaces has been put to good use in understanding several moduli spaces of solutions to variational problems, such as the moduli of holomorphic maps of surfaces into certain complex manifolds, the moduli of instantons, and the Chow varieties. In these notes, we give a detailed description of the particle structures involved in the first two cases, and then explain how well-established results on the topology of particle spaces can be exploited to prove stability theorems for the topology of the moduli spaces, theorems which state that the moduli space approximates in a suitable homotopic sense the topology of the function spaces in which they sit, provided one stabilises with respect to a charge or degree.
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Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades this subject has been greatly enhanced by its relationships with other areas of mathematics, including differential geometry, topology and even theoretical physics, specifically g