Monthly Archives: January 2018

András Vasy (Stanford)

The Analysis and PDE seminar will take place Monday, February 5, in 740 Evans from 4:10 to 5 pm.

Title: Fredholm theory and the resolvent of the Laplacian near zero energy on asymptotically conic spaces

Abstract: We consider geometric generalizations of Euclidean low energy resolvent estimates, such as estimates for the resolvent of the Euclidean Laplacian plus a decaying potential, in a Fredholm framework. More precisely, the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$ on asymptotically conic spaces $(X,g)$ of dimension at least $3$, and the main result is uniform estimates for $P(\sigma)^{-1}$ as $\sigma\to 0$ on microlocal variable order spaces under an assumption on the nullspace of $P(0)$ on the appropriate function space (which in the Euclidean case translates to $0$ not being an $L^2$-eigenvalue or having a half-bound state). These spaces capture the limiting absorption principle for $\sigma\neq 0$ in a lossless, in terms of decay, manner.

Semyon Dyatlov (UC Berkeley)

The Analysis and PDE seminar will take place Monday, January 29, in 740 Evans from 4:10 to 5 pm.

Title: Fourier dimension for limit sets

Abstract: For a finite measure $\mu$ on the real line, its Fourier dimension is defined using the rate of polynomial decay of the Fourier transform $\hat \mu$. The Fourier dimension of $\mu$ may be much smaller than the Hausdorff dimension of the support of $\mu$: a classical example is the Cantor measure on the mid-third Cantor set which has Fourier dimension equal to 0.

I will present a joint result with J. Bourgain showing that the Patterson-Sullivan measure on the limit set of a convex co-compact group of fractional linear transformations has positive Fourier dimension. The proof uses advanced tools from additive combinatorics (the discretized sum-product theorem) and exploits the fact that fractional linear transformations are (generally) not linear. An application is a new spectral gap result for convex co-compact hyperbolic surfaces.

Jeffrey Galkowski (Stanford)

The Analysis and PDE seminar will take place Monday, January 15, in 740 Evans from 4:10 to 5pm.

Title: Concentration of eigenfunctions: Averages and Sup-norms

Abstract: In this talk, we relate microlocal concentration of eigenfunctions to sup-norms and sub-manifold averages. In particular, we characterize the microlocal concentration of eigenfunctions with maximal sup-norm and average growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. This talk is based on joint works with Yaiza Canzani and John Toth.