Grant: Discovery Grants Program - Individual

A major task for mathematicians is to understand the types of spaces that we live in well enough to be able to list all their possible shapes and accurately describe each of these 3-dimensional shape's structure. To do this, mathematicians convert geometric spaces and geometric problems into the language of algebra, where calculations can be made. For instance, to each 3-dimensional space we can associate an algebraic object called a "group", and in many cases spaces with the same group are necessarily the same. Thus understanding the different possible 3-dimensional spaces is determined by the algebraic problem of understanding the different possible groups which can arise. One of the principal goals of my research proposal is to study the possibility of translating certain properties of 3-dimensional spaces into algebra. The geometric property in question is the ability to cut a space into disjoint surfaces which piece together locally like a deck of cards. Not all 3-dimensional spaces can be cut up like this and it is expected that only those whose groups can be ordered in algebraically coherent way do. This is what I intend to investigate. It also seems that the existence of such a splitting of the space is equivalent to a certain analytic condition reflecting higher dimensional geometry, and this is quite surprising as there is no compelling heuristic which explains why these three conditions should be connected. On the other hand, there is no known space for which they differ and many infinite families for which they are known to be the same. Another goal of this part of my research program will be to investigate whether the analytic condition is the same as the geometric one. A second part of my research proposal concerns the Dehn filling operation, a method for transforming one 3-dimensional space into another. Many of the basic problems of 3-dimensional spaces can be analysed in terms of Dehn filling, and one of my main goals is to contribute to our understanding of this operation and then to apply this work to study the different shapes of 3-dimensional spaces. Advances in our knowledge of the form of 3-dimensional spaces have led us to the point where we can fruitfully investigate relations between them. In a third part of my proposal I will study the qualitative behavior of infinite families of projections, or non-zero degree maps, of one 3-manifold onto another, with the goal of showing that under suitable conditions, such families arise as by-products of a fixed projection. This will lead to a better understanding of how such spaces relate. Finally another important relation between spaces is that of commensurability. Two spaces are called commensurable if they can be finitely unfolded to yield the same space. A final goal of my proposal is to understand how many different spaces of a given type can be commensurable and precisely describe those which are commensurable to some other space.