**Title:** Local Geometry of the Nodal Sets of Eigenfunctions

**Speaker:** Yaiza Canzani

**Speaker Info:** Harvard

**Brief Description:**

**Special Note**: **Note the unusual date/time**

**Abstract:**

Let (M,g) be a compact real analytic surface with no boundary, and let H denote a closed analytic curve in M. Write $\varphi_\lambda$ for the eigenfunctions of the Laplacian $\Delta_g$ with eigenvalue $\lambda^2$. When $M$ is the two-torus and H has non-vanishing curvature (Burgain-Rudnick) or when M is an arithmetic surface and $H$ is a geodesic circle (Jung), it has been shown that H is a \emph{good} curve in the sense that $\| \varphi_{\lambda} \|_{L^2(H)} \geq e^{-C \lambda}$ for some C>0 and all $\lambda >\lambda_0$. In these cases it was proved that$$ \# \{ \varphi_{\lambda}^{-1}(0) \cap H\}= O(\lambda). (*)$$

In this talk we show that the bound $(*)$ holds on the general surface $(M,g)$ provided $H$ is a good curve. This is joint work with John Toth.

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