The notions of space, geometry and change can all be articulated by that of an inner product space. A geometric motivation for such a development could be that of understanding the dynamics of lines in an analytic way. The inner product space ℝ^n is a prime example of this motivation. Such spaces readily find their application not only in mathematics, but also in physics and engineering.
A natural extension, from the geometric point of view, would be to seek an understanding of the dynamics of any kind of shape, which would take on the role of “lines” within some prescribed inner product space. In this way, we seek to understand specific instances of subsets of the powerset of ℝ^n as inner product spaces. We wish to also consider this idea in a theoretical sense with the faculty of Riemannian manifolds, to further understand general instances of these spaces.
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.