My research in a nutshell
I enjoy many forms of algebra. In modern mathematics, this means the study of rules for combining mathematical expressions, such as multiplying polynomials or composing symmetries. One of my favorite aspects of algebra is algebraic geometry, the relation between the algebraic properties of a system of equations and the geometry of its set of solutions.
Recently, my research has focused on cluster algebras, which generalize the algebras of functions on many notable spaces, such as spaces of matrices, reductive Lie groups, Grassmannians and Teichmüller space. Roughly speaking, these algebras have many special coordinate systems - called clusters - together with a recursive rule - called mutation - which allows any cluster to be reconstructed from any other cluster. This simple setup has far-reaching consequences, which includes good geometric properties in many cases, but not all. Part of my research has been the introduction and investigation of locally acyclic cluster algebras, a definition intended to characterize those cluster algebras which are geometrically well-behaved.
If you would like to read more about my research, you can read my research statement.
My papers
- Superunitary regions of cluster algebras (arXiv)
Joint with Emily Gunawan
This paper considers the region on which every cluster variable in a cluster algebra is at least 1. For finite-type cluster algebras, we show these inequalities carve out a set which is homeomorphic to the generalized associahedron, a polyhedron whose faces encode the relations between clusters.
This has a notable application. Friezes of Dynkin type are configurations of numbers with rows corresponding to the vertices of a Dynkin diagram, and satisfying a recursive mesh relation. We show that positive integral friezes of Dynkin type are parametrized by a discrete subset of the superunitary region, and are therefore finite (settling an open conjecture). - Juggler's friezes (arXiv)
Joint with Roi Docampo
SL(k)-frieze patterns are infinite strips of numbers in which every diamond of size k has determinant 1. These configurations have several remarkable properties, including periodicity, a connection to linear recurrences, and a parametrization by part of a Grassmannian.
In this paper, we define a generalization of SL(k)-frieze patterns consisting of strips of numbers with a ragged lower edge, called juggler's friezes (for the juggler's functions that give them their shape). We extend the previously mentioned results to juggler's friezes, including a parametrization by positroid varieties. - Linear recurrences indexed by Z (arXiv)
We consider infinite systems of linear equations in variables indexed by the integers, in which each variable is expressed as a linear combination of the previous variables. We construct a unique reduced representative of any such system, and use this to construct a solution matrix whose columns form a fundamental system of solutions. - Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures (arXiv)
Joint with Mustafa Hajij, Ghada Zamzmi, and Matthew Dawson
We consider the application of deep networks to the construction of numeric representations of several algebraic objects, such as R-matrices and the Temperley-Lieb algebra. Since each R-matrix gives an invariant for knots, we briefly consider what kinds of knot invariants are produced by the algorithm. - Non-commutative resolutions of toric varieties (arXiv)
Joint with Eleonore Faber and Karen E. Smith
We construct a canonical non-commutative resolution of an arbitrary toric variety, by considering the endomorphism ring of the direct sum of conic modules (a finite set of combinatorially-defined modules). Our main result is that this ring has finite global dimension, implying the finite global dimension of several related rings (such as the ring of algebraic differential operators in positive characteristic). - The twist for positroid varieties (arXiv)
Joint with David Speyer.
In a preprint first circulated in 2003, Postnikov introduced a generalization of Berenstein, Fomin, and Zelevinsky's theory of wiring diagrams. Two constructions were associated to a bipartite planar graph (though not in this language):- A boundary measurement map to a positroid variety, whose coordinates were partition functions counting matchings in the graph.
- A cluster of Pl"ucker coordinates on a positroid variety, expected to be part of a cluster structure.
- Lower bound cluster algebras: presentations, Cohen-Macaulayness, and normality (arXiv)
Joint with Jenna Rajchgot, and Bradley Zykoski.
Lower bound cluster algebras are an approximation of cluster algebras which only consider cluster variables one mutation away from an initial seed. In this paper, we give a general presentation of any lower bound algebra. We use a Gröbner degeneration of these relations to prove that lower bound algebras are normal and Cohen-Macaulay. This was an REU project at the University of Michigan. - The greedy basis equals the theta basis (arXiv)
Joint with Man Wai Cheung, Mark Gross, Gregg Musiker, Dylan Rupel, Salvatore Stella, and Harold Williams.
Theta functions are generating functions counting certain tropical curves, which collectively form a canonical basis for many cluster algebras. Rank 2 cluster algebras also have a greedy basis, defined by a recursion among the coefficients. This paper proves the two basis coincide for any rank 2 cluster algebra, by constraining the monomial support of the theta functions. This work was produced as part of the AMS's Mathematical Research Communities Program. - The existence of a maximal green sequence is not invariant under quiver mutation (arXiv)
A maximal green sequence for a quiver is a sequence of quiver mutations which sends g-vectors to their negatives. By translating these sequences into paths in an associated scattering diagram, we prove that a maximal green sequence for a quiver induces a maximal green sequence in any subquiver. This is used to provide a quiver with no a maximal green sequences, but which is mutation-equivalent to a quiver with a maximal green sequence. - Cluster algebras of Grassmannians are locally acyclic (arXiv)
Joint with David Speyer.
By the work of Scott, the homogeneous coordinate ring of a Grassmannian is a cluster algebra. By the work of Postnikov, this should extend to any open positroid variety. This paper demonstrates that each of these cluster algebras is locally acyclic, a geometric property which implies a number of important properties. - A=U for locally acyclic cluster algebras (arXiv)
This short note demonstrates that locally acyclic cluster algebras coincide with their upper cluster algebra. This was proven earlier in Locally acyclic cluster algebras; however, the proof which appeared there cited an earlier work which required the cluster algebra was totally coprime. This paper reproduces that proof without this unnecessary requirement. - Singularities of locally acyclic cluster algebras (arXiv)
Joint with Angelica Benito, Jenna Rajchgot, and Karen E. Smith.
This note considers cluster algebras in positive characteristic. We show that the Frobenius endomorphism of any upper cluster algebra has a canonical Frobenius splitting. We use this splitting to show that locally acyclic cluster algebras are F-regular. As a consequence, locally acyclic cluster algebras have (at worst) canonical singularities, even in characteristic zero. - Computing upper cluster algebras (arXiv)
Joint with Jacob Matherne.
This paper introduces an algorithm for generating elements of upper cluster algebras, which can be used to produce presentations in many cases. We produce explicit presentations for many examples. - Skein algebras and cluster algebras of marked surfaces (arXiv)
This paper extends the Kauffman skein algebra of an oriented surface to surfaces with marked points on the boundary. When there are enough marked points to admit a triangulation, this skein algebra is naturally a quantum cluster algebra, in which each triangulation determines a cluster. The central technique is a quantum analog of the theory of locally acyclic cluster algebras. - Locally acyclic cluster algebras (arXiv)
This paper introduces locally acyclic cluster algebras, a class of cluster algebras which can be covered by certain elementary cluster algebras. This implies the cluster algebra is finitely generated, normal, locally a complete intersection, and equal to their own upper cluster algebra. Thus, locally acyclic cluster algebras avoid the many pathologies which may be found in general cluster algebras. It is then shown that this class includes the cluster algebra of any marked surface with at least two marked points on the boundary. - Character algebras of decorated SL_2(C)-local systems (arXiv)
Joint with Peter Samuelson.
A decorated SL_2(C)-local system on a marked surface is a local system with SL_2(C) monodromy, together with a distinguished section over each marked point. We consider the algebra of invariants of decorated SL_2(C)-local systems, and show that it corresponds to a oriented analog of the Kauffman skein algebra. - The Weil-Petersson form on an acyclic cluster variety (arXiv)
The variety associated to a finitely generated cluster algebra admits a canonical 2-form on certain smooth open patches, called the Weil-Petersson form, which encodes the exchange matrix of any cluster. This paper demonstrates that, when the cluster algebra is acyclic, the Weil-Petersson form extends to the entire variety. - 2D Locus configurations and the charged trigonometric Calogero-Moser system (arXiv)
This paper considers Schrödinger operators with a potential with double poles along a hyperplane arrangement. The existence of a Baker-Akhiezer function for this operator can be reduced to a system of equations on the hyperplane arrangement. This paper demonstrates that these equations are satisfied in 2 dimensions when the angles of the hyperplanes are in equilibrium for repulsion proportional to the inverse cube of their separation. - The Beilinson equivalence for differential operators and Lie algebroids (arXiv)
The category of algebraic D-modules on a smooth affine variety may be studied by considering a closely related category; the quotient category of graded D-modules by the subcategory of modules supported in finitely many degrees. This paper demonstrates this category has many of the same properties of a bundle of projective spaces over the variety. This generalizes work of Ben-Zvi-Nevins, and Berest-Wilson.
A longer version of this work with many more results may be found in my doctoral dissertation. - Computing a generating set of arithmetic Kleinian groups (arXiv)
This short note considers certain arithmetically defined subgroups of SL_2(C). A generating set of such a group may be produced by considering the walls of a fundamental domain for the induced action of the group on the hyperbolic 2-space. An example is provided.