Lojasiewicz Inequalities and the Zero Sets of Harmonic Functions

Abstract: Whereas $C^\infty$ functions can vanish (almost) arbitrarily often to arbitrarily high order (e,g, $f(x) = e^{-1/x}$ vanishes to infinite order at zero), the zero sets of analytic functions have a lot more structure. For example, you learn in intro to complex analysis that the zeroes of a Holomorphic function are isolated.

The Lojasiewicz inequalities (partially) quantify this extra structure possessed by analytic functions. Developed originally by algebraic geometers, Lojasiewicz inequalities have been used with great success to study geometric flows. In this talk, I will give a brief introduction to these inequalities and then discuss some joint work (and maybe some work in progress) with Matthew Badger (UConn) and Tatiana Toro (U Washington), in which we use Lojasiewicz inequalities to study the zero sets of harmonic functions and, more interestingly, sets which are infinitesimally approximated by the zero sets of harmonic functions.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.