The research funded by the award will be directed towards several closely related questions and activities. A first part of the project is to build the theory of algebraic stacks from scratch, with the goal of writing foundations for algebraic stacks with a minimal amount of assumptions on the base scheme, no separation axioms for algebraic spaces, and no separation axioms for algebraic stacks. This will be continually documented online in the stacks project, see http://math.columbia.edu/algebraic_geometry/stacks-git. Many algebraic stacks are quotient stacks, but this is not always the case. A second component of the project involves the question of whether every algebraic stack is always etale locally a quotient stack. This is related to the question of whether the cohomological Brauer group and the classical Brauer group coincide for separated smooth algebraic spaces over a field. This leads into other questions, especially the relation between period and index for Brauer classes. A third part of the project is to see whether there exists a natural degree map on the group of zero cycles on a GIT-stack. And finally, the project includes a fourth part aimed at studying moduli of stacky curves, which is a natural explicit example of a higher algebraic stack. Throughout the PI will moderate the stacks project mentioned above.
A question that has long fascinated physicists and philosophers is: What is space? For a mathematician a 3-dimensional manifold seems a good first approximation to space. After learning about special relativity a four dimensional space with a Lorentz metric seems closer. After learning about the standard model it seems that space comes endowed with certain vector bundles. And so on. In algebraic geometry students are taught early on that there are many different spaces, and that in fact the collection of all spaces forms itself some kind of space. A famous example is the moduli space of curves of genus g (Riemann surfaces) which is used by physicists in string theory. In a fundamental paper on moduli of curves in algebraic geometry, Deligne and Mumford pointed out that the space of curves has additional structure in that its points come endowed with certain finite groups (namely the automorphism groups of the corresponding curves). They coined the phrase "algebraic stack" to denote this type of space. It turns out that the language of algebraic stacks is an extremely useful tool in studying very classical objects such as vector bundles on curves and surfaces, moduli of elliptic curves and abelian varieties, etc, etc. The project will partly develop the foundations of algebraic stacks in a very general setting, and partly find new properties of these spaces, such as whether the finite groups attached to the points of an algebraic stack all in some natural way are contained in a single bigger group.
