### 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 interests are in decision
problems in operator algebras, and 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

###### Combinatorics of linear system games

Together with Connor Paddock, Vincent Russo, and Turner Silverthorne, I've
been looking at graph incidence games, which are linear system games in
which every variable occurs in exactly two equations. A result of Alex Arkhipov
states that games of this type have quantum advantage only when the graph
is non-planar. It turns out that this isn't the end of the story, as we show
that any quotient closed property of solution groups of such games has a
forbidden minor characterization. It's pretty fun to try to work out the
forbidden minors for different properties; so far we've been able to find
the forbidden minors for finiteness and abelianess.

###### Membership problems for quantum correlations

Honghao Fu, Carl Miller, and I have been looking at the membership problem
for quantum correlations when the number of measurement settings and outcomes
is fixed. We show that this problem is undecidable for both the commuting-operator
correlations and the closure of the finite-dimensional correlations. This shows that
these correlation sets don't have a description as a rational semialgebraic set (or
any other description that allows you to decide membership).

###### Multi-prover proofs and perfect zero-knowledge

Nonlocal games are closely connected with multiprover proofs. Using the
connection between group theory and nonlocal games in some of the other
papers below, Matt Coudron and I give complexity lower bounds on the task
of computing the (approximately-)commuting-operator value of a nonlocal
game. These lower bounds are also lower bounds on the class of perfect
zero knowledge multiprover proofs with commuting-operator strategies.
Following this line, Alex Grilo, Henry Yuen, and I show that the complexity
class MIP* of multiprover proofs with entangled strategies is contained in
PZK-MIP*, the class of perfect zero knowledge multiprover proofs with
entangled strategies.

###### 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

###### The isomorphism problem between Schubert varieties

Given all the work that's been put into understanding Schubert varieties
over the years, it's natural to ask: when are two Schubert varieties
isomorphic? Ed Richmond and I answer this question for Schubert varieties
in the full flag varieties of Kac-Moody groups. It seems to be an interesting
open question to answer this for other classes of Schubert varieties.

###### 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 partial flag variety.
Consequently the torus-fixed points can indeed be identified with Peterson
translates in this case.

###### 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.