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The problem of whether or not a smooth closed manifold admits a Riemannian metric whose scalar curvature is everywhere positive (PSC-metric) is an old one and a great deal is known about it. More recently, attention has shifted to the problem of understanding the topology of the space of such metrics and many interesting results have been forthcoming. In this talk we will discuss some analogues of this problem for manifolds with boundary and also manifolds with Bass-Sullivan singularities. We will look at some new results arising from joint work with Boris Botvinnik.
An important theme in Mathematics is the interplay of Mathematical structures from seemingly disparate areas. A H-space is such an example: a topological space with a continuous algebraic (multplication) structure: e.g. the unit circle in the complex plane under complex multiplication. A loop space is an even more refined example. Unveiling and studying such structure allows deeper topological and geometric understanding of the underlying space.