I will give an overview of infinite-dimensional Riemannian geometry, especially those arising in physical applications. In particular I will discuss the sign of the curvature for the volume-preserving diffeomorphisms and the space of inextensible strings, and what it means. The talk will assume some knowledge of finite-dimensional Riemannian geometry but not much beyond that.
Curvatures of infinite-dimensional manifolds - Part II
Calabi-Yau varieties are of interest to physics, due to their appearance in string theory, and in mathematics, thanks to the tractable nature of many examples on which one can do enumerative algebraic geometry and algebraic number theory. In both cases, identifying Calabi-Yau varieties related by mirror symmetry is important. Calabi-Yau varieties with complex multiplication have special properties useful for rational conformal field theory and number theory. Very little is known about the transcendence of periods of higher forms, and the abundance of examples of Calabi-Yau varieties makes them a good place to look for results on transcendence of such periods. We present some new results on normalized period matrices for higher forms which generalize to certain families of Calabi-Yau varieties Th. Schneider's criterion for CM on elliptic curves, which is a result from transcendental number theory. We focus on the so-called Borcea-Voisin towers that are of interest in mirror symmetry. Some preliminary results with M.D. Tretkoff were presented in a previous lecture in Boulder. Genuinely new results on unnormalized periods of higher forms on Calabi-Yau varieties remain out of reach for the moment, and we hope such problems will make for good discussions. We aim to make our talk fairly accessible.
A criterion for complex multiplication on Calabi-Yau varieties
Curvatures of infinite-dimensional manifolds - Part II