|Date:||Tue, February 7, 2017|
|Place:||Research I Seminar Room|
Abstract: Two cones do not need to intersect. In the plane, the picture changes when double cones are considered: any two intersect. In three dimensions and higher, this is no longer the case. Now let E be a euclidean space of fixed finite dimension. Consider a finite collection L of double cones (tip at the origin) and a map c : E→L which we think of as follows: two points P and Q in E are connected if P lies in the cone Q+c(Q) or Q lies in the cone P+c(P). Note that Q+c(Q) is the double cone parallel to c(Q) with tip at Q.
This way, one has defined a graph G with vertex set E.
a) Is G connected?
b) If so, can on bound the number of jumps needed to go from P to Q?
c) Can one bound the length of jumps?
It turns out that investigating these questions allows one to show that certain semi-norms on Sobolev spaces of fractional order are equivalent, which in turn implies the existence of corresponding stochastic processes. I shall, however, focus on the geometric arguments.