I will discuss the short-time asymptotics of the heat trace on two-dimensional bounded domains with corners, either in Euclidean space or in a closed surface. We consider Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems. Of particular interest is the case where the boundaries have nontrivial curvature all the way down to the corner. The interaction of curvature and corners produces some interesting phenomena that are reflected in the heat trace asymptotics and, therefore, in the Laplace spectrum. The results in this talk are joint with various combinations of Sam Looi (Caltech), Medet Nursultanov (Helsinki), and Julie Rowlett (Gothenburg).
