We realize the moduli spaces of cubic fourfolds with specified group actions as arithmetic quotients of complex hyperbolic balls or type IV symmetric domains, and study their compactifications. We prove the geometric (GIT) compactifications are naturally isomorphic to the Hodge theoretic (Looijenga, in many cases Baily–Borel) compactifications. The key ingredients of the proof are the global Torelli theorem by Voisin, the characterization of the image of the period map given by Looijenga and Laza independently, and the functoriality of Looijenga compactifications proved in the Appendix.

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The paper deals with the possibility to solve the heat equation backwards in time. More specifically, we treat the following problem. Given the temperature at a finite number of points of a homogeneous bar, how old can the heat distribution be? In the case that the temperature is given at equidistant points x_{1}, the problem is completely solved. In the case of nonequidistant x_{i}we find an upper bound for the age. Such a bound is also obtained when the information about the heat distribution is given by the value of a finite number of linear functionals.

We investigate spectral properties of the Laplacian in <i>L</i>² (<i>Q</i>), where <i>Q</i> is a tubular region in ³ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface with boundary, which consists of symplectic vector bundle on and a Lagrangian subbundle on\partial as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.