Inner Product Spaces on Subsets of the Powerset of Riemannian Manifolds

Rn has given us an analytic notion of space, geometry and change. Each readily finds its application, not only in mathematics, but also physics and engineering. A geometric motivation for such a development, in considering functions on Rn, could be that of understanding the dynamics of line segments in an analytic way. That is, the specification Rn as a real inner product space, giving rise to notions such as of derivative. A natural extension, from the geometric point of view, would be to seek an understanding of the behaviours of any kind of shape; given again, an intuitive set of rules. The most canonical approach to this problem would be to consider the powerset of Rn and more wieldy subsets and quotients of this set.

We seek to understand specific instances of subsets of the powerset of Rn as analytic spaces; applying them to define subsequent analytic notions analogous to those found in Rn. We wish to also consider this idea in a theoretical sense, to further understand what can and can’t be done with these spaces.

David Perrella

University of Western Australia

David Perrella is a recent graduate in mathematics and physics at UWA and will be continuing his studies in pure mathematics next year. His academic interests include topics such as differential geometry and calculus on manifolds. His project this summer for AMSI is of a similar nature. He is very driven by his personal quest to extract the deep facts lurking in fields of this kind. He also very much enjoys spaghetti on any day (or night) of the week.

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