### Universal Limitations of Generalized Spherical Designs

Analysis and Geometry Seminar

11th January 2018, 1:30 pm – 2:30 pm

Howard House, 2nd Floor Seminar Room

How many points does one have to place on a sphere so that the average value of every polynomial up to degree k on the sphere coincides with the average on these points? These spherical designs have been introduced in the 70s and studied intensively ever since. We give a vast generalization to general compact manifolds and to weighted averages. The techniques are new and use partial differential equations; the results are new even on *S*^{2}. If time allows, I will discuss a generalized Sturm Oscillation theorem that is based on similar ideas as well as a connection to the notion of pair correlation.

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