Abstract: A spline is a function built by gluing polynomials together so that they agree where they meet. This gluing condition can be recorded combinatorially on a labeled graph, turning an analytic problem into an algebraic one. When the graph comes from the symmetries of a reflection group (such as the symmetric group) the resulting collection of splines is simultaneously a ring, a module over the polynomials, and a graded representation of the group. In certain cases, it is also the equivariant cohomology ring of an algebraic variety, and so deep geometric questions can translate into questions about splines. This talk introduces splines on symmetry groups from first principles and shares what is known about their structure more generally.
Mathematics Seminars and Colloquia