We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral sequences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the RiemannRoch theorem for curves and Bott’s theorem in representation theory (following [8]) using the algebraic-geomet...

We introduce and study some operational quantities associated to a space ideal A. These quantities are used to define generalized semi-Fredholm operators associated to A, and the corresponding perturbation classes which extend the strictly singular and strictly cosingular operators, and we study the generalized Fredholm theory obtained in this way. Finally we present some examples and show that...

Recently, the ideal free dispersal strategy has been proven to be evolutionarily stable in the spatially discrete as well as continuous setting. That is, at equilibrium a species adopting the strategy is immune against invasion by any species carrying a different dispersal strategy, other conditions being held equal. In this paper, we consider a two-species competition model where one of the sp...

Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by HtrD, is defined to be the injective dimension of the algebra D⊗k D ◦. We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The ...

Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by HtrD, is defined to be the injective dimension of the algebra D⊗k D ◦. We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The ...

We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base space of a fibre space of an irreducible symplectic manifold is the same to that of a projective space if the base space is smooth.

This paper is a sequel to the one in this same session which surveys recent results on the role of convexity in Hamilton-Jacobi theory. We describe here how value functions in optimal control can be represented as upper and lower envelopes involving so-called kernel functions. Particularly noteworthy is a lower envelope formula given in terms of the dualizing kernel, which is a value function i...