Dept Math & Stat UiT, Forskningsparken B459
22 Aug 10:15-12:00; 30 Aug 2024 13:15-15:00
Omid Makhmali (UiT)
"Riemannian surfaces and the rolling problem" (Working seminar)
In this pedagogical talk we will start by recalling the local description of Riemannian metrics in dimension 2. It would be helpful if the students had already passed a course on differential geometry of curves and surfaces, e.g. M. do Carmo's classical book would be a good reference. We will then formulate the rolling problem of a pair of surfaces and list a few research perspectives that arise from this formulation. Our main focus will be on a geometric structure that is inherent to the rolling problem i.e. a rank 2 distribution in dimension 5. In parallel to the theory of Riemannian surfaces, we will present the local theory of such distributions, without giving all the details. We will show how one can compute the fundamental invariant of such distributions arising from rolling surfaces. If time permits, we will finish by formulating two open problems and suggest some strategies that may result in solving them.
I will send some references next week. Meanwhile, if you want to have a feeling about non-holonomic distributions and related problems I highly recommend watching this talk by Robert Bryant.
Moreover, if you want to have an idea about some of the latest results on rolling surfaces and their far-reaching potential applications, you might find the following Nature article and it's accompanying video interesting.