Research interests
Within quantum information, my main interests are non-local games, and related topics such as quantum correlation sets, self-testing and device-independence, entanglement theory, and multi-prover interactive proofs. I'm also interested in new mathematical tools which might help us study these areas; my current focus on this side is approximate representation theory.
In Lie theory and algebraic combinatorics, some topics of interest are Kac-Moody groups and algebras, Schubert varieties, Coxeter groups, and hyperplane arrangements.
Below is a list of papers grouped by rough research topic. For a list of papers in chronological order, see my papers page.
Quantum information
Hyperlinear profile and entanglement requirements for non-local games:
For non-local games, we'd like to know how much entanglement is required to play optimally or near-optimally. In joint work with Thomas Vidick, we show that the amount of entanglement required by a linear system game is related to the hyperlinear profile of the solution group, which measures the growth rate of dimensions of approximate representations of the group. Using the connection with hyperlinear profile, we give an example of a non-local game which cannot be played optimally on any finite-dimensional Hilbert space, and with explicit lower bounds on the Hilbert space dimension required for near-optimal strategies. The connection between hyperlinear profile and entanglement raises the question of how fast the hyperlinear profile can grow. In a second paper, I give an example of a group where the growth rate is (sub)exponential.Tsirelson's problem and linear system games:
A linear system game is a type of non-local game built from a linear system over a finite field. Perfect strategies for these games correspond to representations of a certain finitely-presented group, called the solution group. It turns out that any finitely-presented group can be embedded in the solution group. This has some interesting applications in quantum information, including a resolution of the strong Tsirelson problem. In a follow-up paper, I refine this argument to show that there are non-local games which cannot be played optimally with any finite-dimensional Hilbert space.XOR non-local games:
Two papers on the structure of XOR non-local games. This class of non-local games is closely connected with semidefinite optimization and Clifford algebras. In the second paper I give an algebraic characterization of optimal strategies for XOR games, and use this to show that there are non-local games requiring a large amount of entanglement to play near-optimally.Lie theory and algebraic combinatorics
Nash blow-ups of Schubert varieties
Peterson translation has been used by Carrell and Kuttler to characterize smoothness of Schubert varieties. Peterson translation is defined using the Nash blow-up of a Schubert variety, and can be thought of as giving a combinatorial model for this blow-up. However, it's an open question as to whether this combinatorial model sees all the torus-fixed points of the Nash blow-up. Ed Richmond, Alex Woo, and I show that in the case of a cominuscule Schubert variety, the Nash blow-up is a Schubert variety in a two-step flag variety. Consequently the torus-fixed points can indeed be identified with Peterson translates in this csae.Rationally smooth Schubert varieties and Billey-Postnikov decompositions:
Ed Richmond and I have been studying rationally smooth Schubert varieties, with the goal of finding a complete classification. So far this has been completed in finite type. The main tool is a type of parabolic factorization in the Weyl group called a Billey-Postnikov decomposition, which seems to have a lot of applications (including to inversion hyperplane arrangements). The latest paper is an enumeration of smooth Schubert varieties in affine type A.Enumeration of parabolic double cosets
Parabolic double cosets in Coxeter groups are a frequently studied object, but we don't know how to enumerate them, even in the symmetric group. In this paper, we get some partial results by giving a formula for the number of parabolic double cosets with a fixed minimal element. The enumeration method is closely connected to the staircase diagrams used to enumerate smooth Schubert varieties.Inversion hyperplane arrangements:
These papers study the freeness of inversion hyperplane arrangements, which surprisingly turns out to be closely connected to rational smoothness of the corresponding Schubert variety. Using Peterson translation (which also comes from the geometry of Schubert varieties), I characterize free inversion arrangements via root-system pattern avoidance.The cotangent bundle of cominuscule Grassmannians
A result of Lakshmibai states that the cotangent bundle of the (ordinary) Grassmannian is an open subset of a Schubert variety in an affine two-step partial flag variety. In this short paper, Lakshmibai, Ravikumar, and I use Billey-Postnikov decompositions to extend this result to cominuscule Grassmannians.Lie algebra cohomology of affine Lie algebras:
My thesis papers. These papers apply a Lie algebra cohomology calculation to two problems: the definition of Kostka-Foulkes polynomials for affine Kac-Moody algebras in terms of a Brylinski filtration, and a version of the strong Macdonald theorems for twisted affine Kac-Moody algebras.Map enumeration:
An undergraduate research project. We count the number of two-vertex maps with respect to genus.Thesis
My Ph.D. thesis is titled Strong Macdonald Theory and the Brylinski Filtration for Affine Lie Algebras. For the (at the time) new results therein, it is probably better to read the associated papers "A Brylinski filtration for affine Kac-Moody algebras" and "Twisted strong Macdonald theorems and adjoint orbits" listed above. However, the thesis does contain some background material on semi-infinite cohomology, which might be useful in understanding these two papers.
Collaborators
If you are interested in my research, you might also want to check out the webpages of some of my collaborators: Richard Cleve, Ian Goulden, Ed Richmond, Falk Unger, Sarvagya Upadhyay.