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Abstract: This is the first of a series of talks outlining Naber-Valtorta's paper "Volume estimates on the critical sets of solutions to elliptic PDEs". We will review orthogonal polynomial expansions of solutions to the Laplace and heat equations, and how these expansions help us study the elliptic and parabolic frequency functions. We will also discuss sharp almost-rigidity and almost-splitting theorems for the frequency functions and use these to study the set of frequency-pinched points.

https://sites.math.rutgers.edu/~yyli/Learningseminar.html