Courant's theorem states that the k-th eigenfunction of the Laplace operator on a closed Riemannian manifold has at most k nodal domains. Given a ball of radius r, we will discuss how many of nodal domains can intersect a ball (depending on r and k). Based on a joint work (in progress) with S.Chanillo and E.Malinnikova.
Let K be a convex body with volume one and barycentre at the origin.
How small is the volume of the intersection of K and -K? I shall
discuss such lower bounds and present applications to the Hadwidger
covering/illumination conjecture. Based on joint work with H. Huang,
B. Slomka and B. Vritsiou.
In this talk I will discuss the recent developments in the inverse spectral theory of bounded planner domains with strictly convex smooth boundaries. I will present a joint work with Steve Zelditch in which we prove ellipses of small eccentricity are Laplace spectrally unique among all smooth domains.
