(Someday I aspire to write a paper that does not have "Mirror symmetry" in the title.)
This statement, written as I enter my final year of graduate school, replaces an older attempt, written early in my graduate studies and only lightly edited since then. When one writes a research statement early on in graduate school, its content is aspirational. Such a statement is mostly a declaration of principles: here are some things which I believe to be good mathematics, and here is how I hope to contribute in a similar way.
The completion of one's graduate studies is therefore a time of reckoning: what have I done so far, what do I intend to do in the future, and most importantly, what does any of this have to do with good mathematics, as I understood it three years ago? In the following I hope, after revisiting my perspective on the sort of mathematics I enjoy, to engage in just this justification.
1. Historical background
My research, which is inspired by potential applications to geometric representation theory, lives nearest to a field which I would like to call "quantum geometry." Although the name sounds quite modern, I mean by this name to reference a set of shared goals which trace back to the 1920s and the initial mathematical development of quantum mechanics.
Historically, the phrase "geometric representation theory" has been used to describe a reënvisioning of the methods and goals of representation theory, beginning around 1981 with the Beilinson-Bernstein theorem. Since then, representation theory has been understood as a study of the quantum geometry of symmetric spaces associated to a group G. When G = SU(2), one incarnation of this geometry is in the space of G-harmonic polynomials on on the flag variety P1: these are just the spherical harmonics, whose appearance in wavefunctions for electrons in the hydrogen atom motivated the explosion of 20th-century research into representation theory.
What is the "quantum geometry" of a general Riemannian manifold X? We now understand that a natural way to probe the geometry of X is by studying the quantum-mechanical system of a particle moving in spacetime X. This is a 1-dimensional quantum field theory with supersymmetry; its Hilbert space of states is the space of differential forms on X, and its Hamiltonian is the Laplacian on forms. In other words, what one might mean by the quantum geometry of X is just Hodge theory: the study of the cohomology of X by means of harmonic forms. Moreover, as Witten explained so beautifully in "Supersymmetry and Morse theory," given a Morse function f: X -> R, one can perturb this Hamiltonian to one whose ground states are localized near critical points of f: to first order in perturbation theory, H has one zero mode for each critical point, and the degeneracies in perturbation theory are removed by a calculation of instantons, which are precisely Morse flow lines among critical points!
The above example illustrates one of the most important lessons of 20th-century geometry: the geometry of a space X is often encapsulated in moduli spaces of solutions to certain differential equations associated to X. In my own research I am interested in spaces X which are not just Riemannian but Kähler or hyper-Kähler, which we often take as the targets not of 1-dimensional but rather 2-dimensional or 3-dimensional sigma models, respectively. One reason that the study of such theories of so interesting, besides our natural interest in the geometry of X, is that these theories admit a remarkable feature coming from their origins in string theory: they are related among each other by dualities.
One simple manifestation of a stringy duality is the appearance in basic electromagnetism of "electric-magnetic duality": this is the realization that Maxwell's equations in vacuum look the same if one switches electric and magnetic terms. (Despite the simplicity of its appearance, this is actually an instance of S-duality, one of the deepest and most important dualities we know.) Dualities reveal to us that a pair of quantum field theories, which a priori bear no relation to each other, are actually the same; when both theories are sigma-models, with respective target spaces X and X', this means that "the quantum geometries of X and X' are the same," even though X and X' may be radically different as geometric or topological spaces!
2. My research
The duality on which my past research has focused is mirror symmetry, which relates the symplectic geometry of a space X to the complex geometry of a space X', where each sort of geometry is encapsulated in the category of boundary conditions of a certain topological quantum field theory. On the symplectic side, this category is the Fukaya category, encoding counts of holomorphic curves into X. The invariants obtained from the study of the holomorphic curve equation, whose first appearance in symplectic geometry is due to Misha Gromov, are very powerful yet difficult to compute; the good news is that recent advances in the theory of constructible sheaves have made computations much more tractable. Following work of several people, including the seminal work by David Nadler and Eric Zaslow in the setting of cotangent bundles and a conjecture of Maxim Kontsevich in the case of a general Weinstein manifold, as well as several other people since, we now understand how to relate symplectic invariants of Lagrangians in a Weinstein manifold to a category where computations are much simpler.
The key insight is that a Weinstein manifold has a Lagrangian skeleton, and from the perspective of that skeleton, locally, other Lagrangians "look like" constructible sheaves and their invariants can be calculated locally in categories of constructible sheaves and then glued together.
As a consequence of the above developments, any skeleton for a Weinstein manifold gives a presentation of its Fukaya category. So we can reframe many questions of mirror symmetry in the following form: given a Kähler manifold X' whose mirror X is Weinstein, how does the complex geometry of X' relate to the skeleton of X? One of my papers gives an answer to this question in the case where X is a hypersurface in (C*)^n, and its mirror X' is the boundary of a toric variety, thus proving Kontsevich's homological mirror symmetry conjecture in a quite general setting. One drawback of this approach is that it only works for Weinstein manifolds; however, by combining the techniques described above with recent work of Nick Sheridan on deforming Fukaya categories, we can often use results from the Weinstein case to deduce information about mirror symmetry for closed symplectic manifolds.
3. Future directions and work in progress
I am working on several sequels to my past work on affine hypersurfaces, most of them focused on broadening our understanding of skeleta of affine hypersurfaces (and other Kähler manifolds at large volume limit) and the relations among them. The structure of these skeleta is already very interesting in low dimensions: the relations among Fukaya categories of a 1-dimensional skeleton undergoing a transition are related to the octahedral axiom, and relations among Fukaya categories of 2-dimensional skeleta are related to cluster transformations. I am pursuing a deeper understanding of these skeleta, and especially, in the case of affine hypersurfaces, in the relation between skeleta and tropical or "cotropical" data, of which intriguing hints have already appeared in several cases. There is also some weak but tantalizing evidence of the relation between skeleta and stability conditions for Fukaya categories, which could be very helpful in understanding the (as yet mostly mysterious) space of stability conditions.
I am also interested in understanding (2d, as above) mirror symmetry for hyper-Kähler varieties. The simplest examples of this phenomenon are the toric hyper-Kähler varieties; in this case, a partial correspondence of branes on mirror varieties has already appeared, but many questions remain, especially about the geometry underlying mirror symmetry. Another very important class of examples of hyper-Kähler varieties (in particular, symplectic resolutions) is given by the Springer resolution T*(G/B). Roman Bezrukavnikov has a conjectural description of a mirror to the Springer resolution, and I am working to understand the skeleton of this mirror variety, which is built from a certain affine Springer fiber, and match it with the complex geometry of the Springer resolution.
*: This is my "unoffcial" research statement; my "official" statement is available by request.
Seminars and Teaching
This semester I am supported by the RTG grant on Number Theory and Arithmetic Geometry, so I will not be teaching.
I have served as organizer (or co-organizer) of the following seminars:
"Both students and intellectuals should study hard. In addition to the study of their specialized subjects, they must make progress both ideologically and politically, which means that they should study Marxism, current events and politics. Not to have a correct political point of view is like having no soul."