This semester (Winter 2026) I am teaching a topics course focused around mirror symmetry for toric varieties.
I am also co-organizer (with Nick Rozenblyum) of the Geometry & Physics Seminar hosted in the Fields Institute's Stewart Library, Mondays 2:00pm - 4:00pm.
Past seminars
I have served as organizer (or co-organizer) of the following seminars:
arXiv:1702.03255, with David Nadler, Mirror symmetry for honeycombs. Trans. Amer. Math. Soc. 373 (2020), 71β107.
Research description (ca. Summer 2025)
1. Historical background
My research 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 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. If X is hyperKähler, then the theory has N=4 supersymmetry, manifesting as Verbitsky's so(1,4) action on the cohomology of X.) 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.
Thanks to work of David Nadler, inspiring a masterful three-part series of works (1, 2, 3) by John Pardon, Sheel Ganatra, and Vivek Shende, realizing a vision of Maxim Kontsevich, it is now understood that when X is a Weinstein manifold -- a noncompact symplectic manifold generalizing a cotangent bundle -- the symplectic geometry of X is largely encoded in a usually singular Lagrangian subset L inside of X, now called the skeleton of the symplectic manifold X. (If X = T*M is a cotangent bundle, then L=M is the zero section.) In the Weinstein case, then, homological mirror symmetry is reduced to the problem of understanding how the Lagrangian L is related to the base of an integrable system on X. This is a rich geometric problem, with many open avenues of study.
3. 3d & 4d 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 highly 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. 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. (The beginnings of this research program can be found in Teleman's 2014 ICM address; the 2026 ICM address of Davids Ben-Zvi and Nadler explains developments since then.) 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.
In retrospect, our theorem can be understood as the first step towards a "fully extended" statement of relative Langlands duality. This ambitious program constructs 3d mirror dual pairs from the perspective of 4d mirror symmetry, better known as the geometric Langlands program. From the perspective we have been developing above, this is the duality relating a dual 4d A- and B-model, topological twists of 4-dimensional gauge theories for dual reductive groups G and G^. If you really believe in the cobordism hypothesis, this duality ought to be understood as an equivalence between a pair of 3-categories. On the B-side, the foundations for the relevant 3-category have been laid by Dima Arinkin and Germán Stefanich, but the A-side remains opaque. Further development of this theory promises to deepen our understanding of algebraic geometry, symplectic geometry, and supersymmetric quantum field theory, not to mention the many other fields (number theory chief among them) inspired by developments in the geometric Langlands program.
*: This is my "unofficial" research statement; my most recent "official" statement is available by request.
Other
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."
