Event
Asilya Suleymanova, Humbolt
Friday, March 17, 2017 13:30to14:30
Burnside Hall
Room 920, 805 rue Sherbrooke Ouest, Montreal, QC, H3A 0B9, CA
Spectral geometry of manifolds with conic singularities.
In 1966 Mark Kac asked his famous question 鈥淐an one hear the shape of a drum?鈥� that corresponds to the following mathematical problem. Suppose we are given a sequence of eigenvalues of some geometrical operator on a manifold. What information about geometry of the manifold can be derived from the sequence of eigenvalues? If a manifold is allowed to have singularities, then one can ask a similar question 鈥淐an one hear the presence of a singularity?鈥�. In the talk we investigate this problem for manifolds with conic singularities. Our main tool is small-time asymptotic expansion of the heat trace. We begin the talk with an overview of the heat kernel of Laplace operator on a manifold. In case of a compact smooth manifold the heat trace expansion gives some geometrical information such as dimension, volume and total scalar curvature of the manifold. In case of a manifold with conic singularities the heat trace expansion is more difficult than in the smooth case. Using Singular Asymptotics Lemma of Bruening and Seeley we obtain information about a singularity from the heat trace expansion.