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. (This apparently simple relation is actually a shadow of S-duality, a strong-weak duality whose discovery radically reshaped our understanding of string theory.) 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. Homological mirror symmetry
The above example of "quantum geometry," in which X is a Riemannian manifold and we study the (1-dimensional) worldline of a particle moving in X, originates from the fact that the theory of maps from a 1-manifold into a Riemannian manifold possesses more structure than is immediately obvious: mathematically, the Riemannian metric can be used to produce an adjoint to the de Rham differential; physicists call this "1d N=1 supersymmetry." (If X is actually a Kähler manifold, then this theory has "N=2 supersymmetry," which is known to mathematicians as the Lefschetz SL(2) action on differential forms.) It turns out that the part of the theory which is invariant under this extra symmetry is completely topological, which means it depends only on the topology of the 1-manifold mapping into X, and not on its length or other invariants.
The situation in higher dimensions is even richer.
If X is a Kähler manifold, then the theory of maps from 2-manifolds into X has "2d (2,2)-supersymmetry:" there are a pair of different supersymmetry algebras acting on the theory, and each one provides a way of producing a theory which only depends on the topology of the 2-manifolds with which we probe X. One of these theories, the topological A-model, is sensitive only to the symplectic structure of X, whereas the other, the topological B-model, is sensitive only to the complex structure. A priori, these two are not related, but it turns out that the 2d N=(2,2) supersymmetry algebra has its own symmetry: the homological mirror symmetry program predicts that the topological A-model with target X is equivalent to the topological B-model with target some "dual" Kähler manifold X'.
As a consequence of this remarkable duality, symplectic invariants of X -- like counts of holomorphic curves, which depend in sensitive and complicated ways on the global geometry of X --may be expressed in terms of complex-geometric invariants, like period integrals, on X'. What makes this duality possible is the presence of an integrable system on X: a fibration by (possibly singular) Lagrangian tori. Indeed, the name "integrable system" was always meant to suggest that such structure on a symplectic manifold X allows its symplectic geometry to be solved -- i.e., its Hamiltonian flows may be expressed using equations. In this perspective, usually known as SYZ mirror symmetry, the combinatorial interface mediating homological mirror symmetry is the base of this torus fibration, as a topological manifold equipped with integral affine structure away from a codimension-2 singular locus.
While many proofs of homological mirror symmetry in particular settings have been somewhat ad hoc, my research program takes seriously the above perspective, seeking to derive the most general statements of homological mirror symmetry by beginning only with the integral affine base suggested by the SYZ program. Beginning with such a manifold, one ought to be able to build up a mirror pair of a symplectic manifold X and a nonarchimedean algebraic variety X'; the integral affine base should be understood as a kind of nonarchimedean skeleton of X'. In joint work with Vivek Shende, we have accomplished this at the ``large-complex-structure/large-volume limit point'' where the symplectic manifold X becomes noncompact and the algebraic variety X' becomes singular. In this case, the base of the SYZ fibration, which we call a "fanifold" to suggest its construction by gluing together toric fans, is the "F1 points" of the singular variety X'.
In future work, we will explain how to move away from this limit point to express mirror symmetry for a pair of compact, smooth Kähler manifolds. In this regime, one must deal with singular fibers of the SYZ fibration, which historically have been a major stumbling block in efforts to understand torus fibrations. Luckily, work of Gross-Siebert suggests that one may reduce to the case of certain "generic" singularities, of which there are a finite number in each dimension; I have explained the proof of a local form of homological mirror symmetry, near such a singularity, here.
In addition, I am interested in how the singularities of the SYZ fibration relate to the combinatorics which appears in cluster theory. Following pioneering work of Fock-Goncharov, it is conjectured that mirror symmetry for cluster varieties may be expressed as a kind of Langlands duality. As a proof of concept, Ian Le and I described the part of cluster duality which is associated to the simplest of the Gross-Siebert SYZ singularities. More general cluster varieties will require studying contributions from more general singularities; in future work, Ian and I will describe the singularities which appear when studying the Richardson variety (the complement of a distinguished anticanonical divisor in the flag variety G/B) and use this presentation to prove homological mirror symmetry. It is expected that our work will suggest a way forward for general cluster varieties as well as opening up new perspectives on the symplectic geometry of the flag variety itself.
Alternatively, many cluster varieties have realizations as "Betti" realizations (e.g., character varieties) of hyperkähler moduli spaces which also have "Dolbeault" realizations (e.g., Hitchin moduli spaces). Traditionally, representation theorists have studied these spaces via their Dolbeault realizations, in order to avoid real symplectic geometry altogether and restrict to the study of holomorphic Lagrangian subvarieties (for instance, taking these holomorphic Lagrangians as singular support conditions for perverse sheaves or D-modules), but no robust bridge has been built to compare the two approaches. In joint work with Michael McBreen, I am working to prove comparison theorems which could be used to relate perverse-sheaf computations from representation theory to calculations in the Fukaya categories of real symplectic manifolds.
3. 3d mirror symmetry
As we have seen above, a hyperkähler manifold X provides a very interesting target for quantum field theories of maps from surfaces. However, just as Hodge structure on a Kähler manifold was merely a 1d shadow (or "decategorification") of the much richer invariants provided by 2d field theories, so too are the 2d field theories with target X shadows of
a theory of maps from 3-manifolds to X. As was the case with our previous examples, such a theory is very supersymmetric (3d N=4), and one can produce from it a pair of topological theories, the 3d A-model and the 3d B-model. These two theories, and the relation between them called 3d mirror symmetry, are at the very heart of modern geometric representation theory. Beginning with groundbreaking work of Braden-Proudfoot-Webster and Braden-Licata-Proudfoot-Wesbter, most of the last decade in the subject has been spent trying to probe parts of these theories, including the Coulomb branch construction of Braverman-Finkelberg-Nakajima and work of Okounkov-Aganagic on elliptic stable envelopes.
The full structure of the 3d A-model and B-model remains opaque. Just as the 2d A-model is governed by the J-holomorphic curve equation, the 3d A-model is governed by the Fueter equation, whose behavior is not as well understood (although the field is progressing rapidly). Nevertheless, taking a cue from methods in the theory of Fukaya categories, one can guess that there should be another approach to the 3d A-model using the tools of sheaf theory. (A precursor to this idea, couched in riddles, can be found in Teleman's visionary ICM address.) Using Kapranov-Schechtman's categorification of perverse sheaves by "perverse schobers," Justin Hilburn, Aaron Mazel-Gee and I proposed that the 3d A-model to a cotangent bundle should be modeled by a 2-category of perverse schobers. Using this approach we were able to prove a 3d mirror symmetry theorem for toric cotangent stacks, relating a 2-category of perverse schobers on a linear torus quotient V/T to a 2-category of coherent sheaves on the Gale dual torus quotient.
Our theorem has opened up a new vista in the subject, pointing the way forward toward categorifications of a great deal of recent work in geometric reprentation theory. As a first application, we will categorify the seminal work by Braden-Licata-Proudfoot-Webster on hypertoric categories 𝒪 and Koszul duality: we prove an equivalence between a pair of 2-categories 𝒪 associated to a Gale dual pair of toric hyperkäher manifolds. These 2-categories 𝒪 possess a wealth of structure invisible to their 1-categorical counterparts, and unearthing these structures will deepen our understanding of representation theory and holomorphic symplectic geometry.
We expect that these structural features will play a central role in future work on 2-categories 𝒪, especially as we prepare to move beyond the abelian case to general quiver varieties or 3d Coulomb branches.
Besides the usual justifications for categorification, and the new insights into classical categories 𝒪, there are seveal applications of our program. For one, moving up a categorical level can often be used to "remove a loop" from a situation, passing from a statement involving loop spaces to one involving finite-type geometry. So we propose, for instance, that Bezrukavnikov's equivalence between two realizations of the affine Hecke category is the categorical trace of an equivalence between dual realizations of the finite Hecke 2-category. In forthcoming work, we will establish a statement of this form, presenting loopy categories as traces of finite-type 2-categories, in the abelian ("geometric Tate's thesis") case. Another application of our work is to knot invariants, the original motivation of most work on 3d mirror symmetry. The 2-categorical B-model of the Hilbert scheme of C2 has long been used to produce invariants of braids and knots, and pieces of this story have been found in the mirror A-model. The future development of our program will unite many pieces of these scattered calculations and help us develop a more sophisticated understanding of the origin of knot invariants from 3d field theory.
*: This is my "unofficial" research statement; my most recent "official" statement is available by request.
Seminars and Teaching
In Fall 2022, I will be teaching a topics course (Math 270Z) on sheaf-theoretic methods in symplectic topology. The focus of the class will be on learning to think "microlocally," using modern methods in symplectic geometry and homotopical algebra. More information can be found on the course's Canvas page.
I have served as organizer (or co-organizer) of the following seminars:
As a graduate student, I felt secure in the knowledge that I always had the support of my union, UAW Local 2865, and
I support the Harvard Graduate Students Union in their fight for better working conditions and a democratic workplace.
I recognize the unique struggles faced by groups historically shut out of mathematics and the academy, and as an advocate for equity and inclusion, I support efforts to reform our culture to help universities live up to their promise of providing a safe and welcoming place for all people to pursue academic inquiry.
"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."