Math Colloquium - Restrictions of eigenfunctions to general subsets of surfaces

Suresh Eswarathasan (Dalhousie University) presents: Restrictions of eigenfunctions to general subsets of surfaces

Abstract: Let M be a compact surface, without boundary, in Euclidean 3-space.  We can consider some special functions, namely Laplace-Beltrami eigenfunctions, that have some fundamental importance.  Functions of this type arise in physics as modes of periodic vibration of drums/membranes or as the stationary states of a free quantum particle on a surface.

In the first part of the lecture, I will give a survey of results that demonstrate how the geometry of M affects the behaviour of these special functions, particularly their “size” which can be quantified by estimating integrals over certain geometric subsets of M.  In the second part of my lecture, I will discuss joint work (past and in-progress) with Malabika Pramanik (U. British Columbia) studying these integrals over general Borel subsets of $M$ and how our results generalize those presented in the first part of the lecture.

Explicit examples and formulas will be used throughout a majority of the talk.




Colloquium Room 319, Math Department, Chase Building, Dalhousie University