Research

Algebraic Geometry

Broadly speaking, my field of interest is algebraic geometry. As the name suggests, algebraic geometry is a melding of the worlds of algebra and geometry, and concerns the study of geometric objects described by algebraic equations. This synthesis makes for a particularly potent theory, since it allows the use of both geometric intuition and algebraic computation. You’ve probably seen such a synthesis in action if you’ve ever used the derivative to locate the local extrema of a function. In that case, you used information about the curve’s defining equation (something algebraic) to help locate the curve’s local extrema (something geometric).

General algebraic geometry often works in the very same way. First, when studying the geometry of a certain object or situation, a geometric question naturally arises. To answer this question, you first translate this geometric question into an algebraic one. Using your well-developed algebraic skills and tools, you set about answering this algebraic question. After achieving your answer, you translate the result back into the original geometric language. Of course, this oversimplifies the picture, as both the translation steps and algebraic work could be (and usually are) quite difficult, but it does give a rough idea of how algebraic geometry can work in practice.

The basic objects of study in algebraic geometry are schemes, which can be thought of as geometric objects (like curves and surfaces) together with a bit of extra data, with the additional information providing the bridge between the worlds of geometry and algebra. (For the more interested reader, this additional information comes in the form of a sheaf describing the locally-defined functions.)

Group Covers of Schemes

Some of the first objects I studied in algebraic geometry were certain special classes of schemes (or, more correctly, maps of schemes) called group covers. A group cover consists of the following data: 1) a pair of schemes ; 2) a group (a set together with some additional, algebraic structure); 3) an action of the group on the scheme ; that is, an association to each element of the group an automorphism of the scheme , in a compatible way; and 4) a map from to identifying the scheme with the quotient scheme (the scheme obtained by identifying points of mapped to each other by elements of ).

For a mental picture, imagine the scheme (the covering scheme) as a geometric object (like a surface) possessing a group of nice symmetries. For example, the scheme might consist of the three parallel planes to the right, which has the nice property that the three planes can be interchanged while still preserving the overall picture. The group in this case would be the permutation group with six elements, corresponding to the six possible ways of rearranging the three planes. Identifying the points mapped to each other by the group action would mean identifying all three planes, and so (the quotient scheme) would be a single plane. Thus, we could speak of the scheme (the three parallel planes) as a -cover of the scheme (the single plane).

During the 1990s, group covers were carefully detailed for a large collection of groups (those called abelian). Part of the usefulness of the resulting theory was that it was constructive: it allowed you to start with a simple scheme and a group , and then, following an explicit recipe, build a more complicated (and hopefully more interesting!) covering scheme . It seemed natural to hope, then, that an understanding of group covers for more complicated (i.e., nonabelian) groups could assist in the construction of even more exotic schemes; this was the goal of my thesis, which I subsequently expanded upon here.

Quotients and Moduli Spaces

With group covers, the essential problem is constructing, from a group and a scheme , a new covering scheme on which acts such that . The converse problem is also very interesting; namely, given a scheme on which a group acts, construct a quotient scheme . At first glance, this might seem no different than our original problem, but in fact there are important distinctions between the two. In the case of group covers, the hypothetical quotient scheme is part of the given data (it is the scheme ). The question is not whether that scheme exists, but rather whether it can be realized as a quotient scheme, i.e., whether the covering scheme exists. For the converse problem, however, the existence of a quotient scheme is not guaranteed. Attempts to overcome this obstacle led to the development of geometric invariant theory (GIT), algebraic spaces and stacks. Each of these topics attempts to address the construction and existence of quotient objects, either working within the land of schemes (in the case of GIT) or by broadening the notion of scheme to include more exotic–and yet somehow still geometric–objects (in the case of algebraic spaces and stacks).

The problem of understanding quotients arises quite naturally in the context of moduli spaces. A moduli space is a geometric object, each of whose points represents an object of some fixed type. For example, suppose you were really interested in curves. It would be completely natural of you to collect all your curves into one big pile, or set. Abstracting a little, you might view that set as a collection of points, in which each point corresponded to one of your curves. After a bit of thought, you might also like to organize your set so that “nearby” points corresponded to curves that were “nearly the same.” In fact, you might decide it would be especially nice if your set had the structure of a space, so that as you traversed a path in your space, the curves corresponding to the points being traversed “moved” (or “deformed”) in a nice, continuous way. What you’ve just described is essentially the moduli space of curves, and the construction of such spaces is one of the main uses of GIT, algebraic spaces and stacks.

Algebraic stacks, in particular, have been incredibly successful at guaranteeing the existence of many moduli spaces, but their success has come at the cost of leaving the “concrete” world of schemes and algebraic spaces. To deal with this, algebraic geometers have come up with various notions of what it means for a scheme (or algebraic space) to “approximate” a stack. One such notion is that of a good moduli space, an idea developed by Jarod Alper in his Ph.D. thesis. I have currently joined Jarod in a project investigating the existence and properties of such spaces.

Tropical Geometry

Quite apart from the above discussion is the burgeoning new subfield of algebraic geometry known as tropical geometry. Loosely speaking, tropical geometry is the piecewise-linear geometry that results from applying a logarithmic (or valuative) map to an algebro-geometric object and extracting the “skeleton” of the resulting structure. For example, the logarithm of a general complex conic is pictured below-left, and the corresponding tropical curve is pictured below-right (images taken from Andreas Gathmann’s Tropical algebraic geometry):

In general, tropicalizations of curves are metric graphs, and tropicalizations of higher-dimensional algebro-geometric objects are more general piecewise-linear objects. The linearity of the tropical objects is one of the great advantages of tropical geometry, as it relates complicated algebro-geometric properties to problems in combinatorics and linear algebra which are, on the whole, much easier to study. I currently have several ongoing projects with Aaron Bertram looking to exploit precisely that linearity, including problems dealing with the tropical Grassmannian, tropical degeneracy loci and the tropical Petri equations.