Suresh Eswarathasan: Lower bounds for the Weyl remainder on Euclidean domains

The remainder term $R(\lambda)$ for the spectral counting function $N(\lambda)$ likely encodes a great deal of dynamical information for the system at hand.  For $\Omega \subset \mathbb{R}^n$, a piecewise smooth bounded domain, we prove an omega bound that depends on the dimension of the fixed point set of the billiard map; the approach taken is through boundary trace expansions.  This is the first lower bound established in settings with boundary in dimension 2, at least to the knowledge of the authors.  As a corollary, $R(\lambda)$ for the Bunimovich stadium is $\Omega(\lambda^{1/2})$, hence confirming a conjecture of Sarnak.