We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.

In this paper, by combining modular forms and characteristic forms, we obtain general anomaly cancellation formulas of any dimension. For 4 k+ 2 dimensional manifolds, our results include the gravitational anomaly cancellation formulas of Alvarez-Gaum and Witten in dimensions 2, 6 and 10 (\cite {AW}) as special cases. In dimension 4 k+ 2, we derive anomaly cancellation formulas for index gerbes. In dimension 4 k+ 2, we obtain certain results about eta invariants, which are interesting in spectral geometry.

No abstract uploaded!

We analyze spectral properties of a leaky wire model with a potential bias. It describes a two-dimensional quantum particle exposed to a potential consisting of two parts. One is an attractive -interaction supported by a non-straight, piecewise smooth curve L dividing the plane into two regions of which one, the interior, is convex. The other interaction component is a constant positive potential V0 in one of the regions. We show that in the critical case, V0 = 2, the discrete spectrum is non-void if and only if the bias is supported in the interior. We also analyze the non-critical situations, in particular, we show that in the subcritical case, V0 < 2, the system may have any finite number of bound states provided the angle between the asymptotes of L is small enough.

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).