Seminar:
Applied Mathematics
Event time:
Wednesday, March 30, 2022 - 12:00pm
Location:
https://yale.zoom.us/j/97458245891
Speaker:
Floran Feppon
Speaker affiliation:
ETH Zurich
Event description:
We propose a mathematical analysis of acoustic wave scattering due to a time modulated highly contrasted inclusion. The density of the inclusion is periodically modulated at a frequency which is much larger than the incident frequency, whence the denomination of “ultra fast” modulation. In most scenarios, we find that the effect of the fast time modulation is averaged over time and everything occurs as if the medium had a unmodulated effective density. However, when the modulation is finely tuned, a strong coupling between the inclusion and the incident wave arises: we show that the scattered wave carries high frequency components (oscillating at the frequency of the modulation), and upon explicit conditions on the modulation, we find the existence of exponentially growing outgoing modes, which suggests that such device could serve as an amplifier. Our analysis relies on a novel approach for the understanding of subwavelength resonance, which is based on the rewriting of the scattering problem in terms of the Dirichlet-to-Neumann operator.We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in Lp∩L∞, for any p∈(1,n), where n is the dimension of the manifold. In particular, our result applies to all known examples of 4-dimensional gravitational instantons. The result is obtained by a fixed point argument, based on novel estimates for the heat kernel of the Lichnerowicz Laplacian. It allows us to give a precise description of the convergence behaviour of the Ricci flow. Our decay rates are strong enough to prove positive scalar curvature rigidity in Lp, for each p∈[1,nn−2), generalizing a result by Appleton. This is joint work with Oliver Lindblad Petersen.