Date:
Thu, 16/04/202612:00-14:00
Location:
Ross 70
Title: Critical sets of harmonic functions
Abstract: Naber and Valtorta showed that if a harmonic function in the unit ball in R^d has frequency bounded by N, then the (d−2)-dimensional Hausdorff measure of its critical set in the half ball is bounded by exp(CN^2). It is conjectured that the optimal bound should instead be polynomial in N. This is known in dimension two, but the higher-dimensional case remains surprisingly subtle. In this talk, we will review the Naber–Valtorta approach and show that in dimension three the estimate can be improved to a bound that is much closer to polynomial. The talk is based on ongoing joint work with Ben Foster and Josef Greilhuber.
Abstract: Naber and Valtorta showed that if a harmonic function in the unit ball in R^d has frequency bounded by N, then the (d−2)-dimensional Hausdorff measure of its critical set in the half ball is bounded by exp(CN^2). It is conjectured that the optimal bound should instead be polynomial in N. This is known in dimension two, but the higher-dimensional case remains surprisingly subtle. In this talk, we will review the Naber–Valtorta approach and show that in dimension three the estimate can be improved to a bound that is much closer to polynomial. The talk is based on ongoing joint work with Ben Foster and Josef Greilhuber.