Robert W. Easton

Background

I was born and raised in Lapeer, Michigan, a moderately-sized city in the thumb area of Michigan. If you’re not sure what the “thumb area” is, simply look at the palm of your right hand. That’s what (the lower peninsula of) Michigan looks like. Pretty much anyone from Michigan will explain his location in this way. For reference, Lapeer is about twenty miles east of Flint, sixty miles north of Detroit, and seventy-five miles north of Ann Arbor.

Author at age two

As a small child, it’s probably fair to say I didn’t have a head for math, so much as a head for a much larger body. As the picture to the right illustrates, at the age of two my head made up nearly half the mass of my entire body. Despite my giant head, I didn’t qualify as an exceptionally talented and gifted elementary school student. I mean this quite literally. My elementary school, Mayfield Elementary, had a program called TAGS (Talented And Gifted Students), and I wasn’t invited until my final year (sixth grade). That also happened to be the year they greatly cut funding for the program: they went from dissecting eyeballs the year before I was invited to simply drawing in blank notebooks the year I participated. Come to think of it, maybe I wasn’t really in TAGS after all.

Author at age eight (dramatization)

I attended Lapeer West for high school, and that is where I really began to take an interest in math and the sciences. At the time, I was fairly certain I wanted to be a mad scientist, which I took to mean “make weapons for the government”. This seemed to fit with my activities in TAGS, which mainly involved me drawing elaborate contraptions for brewing “molecular acid.” After graduation, I attended the University of Michigan. Initially, I planned to major in math and physics— the cornerstone subjects of any mad scientist’s studies —but by the end of my first year I discovered an innate dislike for the world of units (meters, joules, etc.) and so switched my interests to purely math. And by purely math I mean Pure Math, as opposed to Applied Math, which wandered too closely for my tastes to the world of units. I took as many pure math classes as I possibly could over the next three years, and slowly began to think of myself as a budding little number theorist (due in part, no doubt, to the excellent teachings of a certain Professor Trevor Wooley). As graduation neared, I planned to continue my life as a permanent student by attending graduate school in mathematics at Stanford University.

Andrew Schultz, Ravi Vakil, and author (Stanford graduation)

At Stanford, my interests gradually turned from number theory to algebraic geometry, beginning with a reading course on the subject led by Professor Ravi Vakil. Before long, the beauty of algebraic geometry, combined with Ravi’s energy and enthusiasm for the subject, fully persuaded me to choose it as my specialized field of study. These same qualities convinced my friend and classmate, Andrew Schultz, to specialize in the field as well, and in the summer after our second year at Stanford we became Ravi’s first officially-advised students. In May of 2007, after three more years of intense study, we graduated from Stanford. Huzzah!

After Stanford, I obtained a VIGRE fellowship for a three-year post-doctoral position at the University of Utah. During my first year at Utah, I taught a pair of undergraduate calculus courses while I continued to work on and extend the results in my Ph.D. thesis. I also participated in an exciting weekly research group focused on Gromov-Witten theory and Bridgeland stability, working with Aaron Bertram, Y.-P. Lee, Arend Bayer, Joro Todorov, and Yunfeng Jiang.

I spent the past year on leave from the University of Utah, the fall semester as a visiting assistant professor at the University of Washington (at the generous invitation of Professor Sándor Kovács) and the spring semester in Berkeley at MSRI (for the giant algebraic geometry program). I have since returned to Utah to complete the final two years of my post-doc position. And after that? In the wise words of Yoda, “Difficult to see. Always in motion is the future.”

Research Interests

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 structures. This synthesis makes for a particularly potent theory, since it allows one to rely both on geometric intuition and algebraic precision. You’ve probably seen such a synthesis in action if you’ve ever used the derivative to locate the critical points of a curve. 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 answer 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) plus more, 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.) For my thesis, I focused on a special class of schemes called group covers. Group covers 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 complicated geometric object (like a surface) possessing a group of nice symmetries. For example, the scheme might simply consist of two parallel planes, which has the nice property that the two planes can be interchanged (“flipped”) while still preserving the overall picture. The group in this case might be the group consisting of two elements: one element corresponding to the trivial symmetry (“do nothing”), and the other element to the interchange of the two planes (“flip”). Identifying those points mapped to each other by the group action would mean identifying the two parallel planes, and so (the quotient scheme) would be a single plane. Thus, we could speak of the scheme (the two parallel planes) as a double-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 one 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.

A more in-depth explanation of my research, including topics not mentioned here, is available here.