Grant: Discovery Grants Program - Individual

Category theory aims at detecting common aspects of various mathematical theories, and at establishing theorems that are therefore applicable to quite distinct areas of the mathematical and natural sciences. Its versatility may lead to the provision of mathematical models in areas where traditional set-theoretic foundations may have failed. The proposed project lies in the interface between algebra and topology and extends to topics in analysis, algebraic geometry, and theoretical computer science. We wish to - provide a comprehensive framework and theory for categories of interest in analysis, topology, and an area of theoretical computer science, called domain theory; - develop within that framework generalizations of the so-called Hausdorff and Gromov distances, with the goal of making these analytic tools available to a broader array of mathematical applications; - provide generalized notions of topological space and topos for potential applications in areas of algebraic geometry, domain theory and quantum physics where traditional theories may have proven to be inadequate; - strengthen recent advances in making the primary setting for homotopy theory, Quillen model categories, more constructive and accessible to computational methods, and extend these efforts to an important categorical device in stable homotopy theory: triangulated categories; and - develop a categorical theory of torsion which draws equally from factorization techniques and from the theory of fibrations, and apply it to Quillen model categories.