I like to think about mathematics through the lens of geometric representation theory, which roughly means that I take symmetry seriously and that I always want to realize symmetries as symmetries of some hopefully very geometric object. This perspective has its roots in attempts, dating from the early days of quantum mechanics, to promote symmetries of classical mechanical systems to quantum-mechanical symmetries, and the subsequent realization that "a representation...should be seen as a quantum object" (Witten, QFT & the Jones Polynomial).
Today, as we begin to understand some aspects of quantum field theory and string theory, we are seeing the development of new mathematical methods for packaging and describing the data that quantum, stringy, and perhaps also "brany" geometries assign to a spacetime.
Some very exciting work in this direction is Kevin Costello's description of local observables in perturbative quantum field theories using factorization algebras. This approach extends Jacob Lurie's famous result on the classification of topological field theories, and it admits close connections to other structures mathematicians have used to describe quantizations and quantum field theories.
One consequence of the new stringy approaches to geometry is the discovery of relations among prima facie different geometrical spaces which become manifest only at the stringy or brany level. For instance, one simple classical duality, which may be presented as a curiousity in a first course in electromagnetism, is "electric-magnetic duality," the statment that Maxwell's equations of electromagnetism look the same under a certain switching of the electric and magnetic fields; this duality, in a stronger form, reoccurs throughout supersymmetric gauge theory and ultimately underlies the geometric Langlands correspondence. Other new kinds of quantum duality are at play in the rapidly growing field of cluster geometry and its veritable zoo of examples and associated structures (Hall algebras, Donaldson-Thomas invariants, and higher Teichmüller spaces, to name just a few).
My current research draws heavily from the class of geometric dualities with the collective name of "mirror symmetry:" physically, these result from the discovery that sometimes different superstring theories, after compactification on different manifolds, can produce the same quantum field theories. The corollary that the corresponding categories of boundary conditions (D-branes) should be equivalent suggests Maxim Kontsevich's Homological Mirror Symmetry program. Its central idea is that all the symplectic geometry of a symplectic manifold, encompassed by its category of "A-branes," ought to be equivalent to the algebraic geometry of a certain "mirror" algebraic variety, as encompassed by its category of "B-branes."
Traditionally, the category of A-branes for a symplectic manifold was understood as a Fukaya category, a category with objects Lagrangian submanifolds (with some extra data) and A∞ structure determined by counts of holomorphic disks. But from a conjecture of Kontsevich (2009), with proof in development by Sheel Ganatra, John Pardon, and Vivek Shende, and following earlier work by David Nadler & Eric Zaslow in the case of cotangent bundles, we now understand that all the symplectic geometry of a Weinstein manifold is contained in its skeleton, a (generally singular) Lagrangian which is the stable set for Liouville flow. In particular, the Fukaya category of a manifold can be obtained from a certain cosheaf of categories on its skeleton. Families of highly structured Weinstein manifolds are likely to admit classes of skeleta which are organized in interesting ways; a major theme of my research is the search for the underlying geometric and categorical structure of these skeleta.
Currently, I'm thinking about, inter alia, the following topics:
Skeleta of holomorphic symplectic manifolds
Mirror symmetry for various kinds of hyper-Kähler spaces
Deformations of derived critical loci
Dimer models and mirror symmetry
I am on tenure as an NSF Graduate Fellow and am not teaching this semester. In fact, I am out of the country this semester. I will return to Berkeley in January.
I'm away so I'm not running any seminars right now. Next semester, I will be organizing a reading group on derived algebraic geometry, following A study in derived algebraic geometry. Email me if you want in.
"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."