Current support: Simons Foundation Grant, Award Number 711639 [2020-2025]. Past support: NSA grant H98230-15-1-0022 [2015-2017] and NSF grant DMS-1101383 [2011-2015].

My research interests lie at the interface of representations theory of finite-dimensional algebras and geometric invariant theory (``GIT''). The papers below focus on theoretical aspects of quiver invariant theory, as well as applications to Brascamp-Lieb theory in harmonic analysis, Edmonds problem in algebraic complexity, and simultaneous robust subspace recovery in machine learning.

    I. Applications of semi-stability of quiver representations

    Background: The motivation in this project goes back to the celebrated Brascamp-Lieb (BL) inequalities in harmonic analysis. The optimal constants in the BL inequalities, also known as BL constants, permeate many areas of mathematics and computer science. The fundamental work of J. Bennett, A. Carbery, M. Christ, and T. Tao hints at a relationship between BL constants and semi-stability of quiver representations.

    In joint work with H. Derksen, we have developed a quiver invariant theoretic approach to BL theory by introducing the capacity of a quiver datum (V,σ), denoted by cap(V, σ), where V is a representation and σ is a weight (i.e. an assignment of integers to the vertices) of a quiver Q. When Q is the m-subspace quiver, the capacity of quiver data is intimately related to the BL constants that occur in the m-multilinear BL inequalities.

    Our first result states that cap(V, σ) > 0 if and only if V is σ-semi-stable. This has important consequences for quiver theory since it yields a deterministic polynomial time algorithm for checking semi-stability of quiver representations. The importance of the existence of such an algorithm stems from the fact that, in general, quiver semi-stability requires one check a number of linear homogeneous inequalities that can grow exponentially.

    We have obtained a string of other structural results, including a character formula for the capacity/BL constants of quiver data. Our main tool is a quiver version of a celebrated result of Kempf-Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties, denoted by G(σ,V), that carry information relevant to our main objects of study. It allows us to express cap(V, σ) in terms of the character induced by σ, denoted by χ, and any point A of G(σ,V) as follows:

    cap(V, σ)=χ(A)2.

    These results have already found applications to Edmonds problem in algebraic complexity and robust subspace recovery in machine learning.

  1. Edmonds' problem and the membership problem for orbit semigroups of quiver representations
    with Dan Kline
    Abstract  |  pdf  |  arxiv
  2. Simultaneous robust subspace recovery and semi-stability of quiver representations
    with Dan Kline
    Abstract  |  pdf  |  arxiv
  3. The capacity of quiver representations and Brascamp-Lieb constants
    with Harm Derksen
    Abstract  |  pdf  |  arxiv

    II. Moduli spaces for finite-dimensional algebras and their representation type

    Background: One of the fundamental problems in the representation theory of algebras is to classify their indecomposable representations. Based on the complexity of these representations, algebras come in three flavors: representation-finite, tame, and wild. The classification of all indecomposable representations for wild algebras is regarded as a hopeless problem. Therefore, in order to make progress, especially in the presence of wild algebras which occur very frequently in representation theory and other areas, one is naturally led to consider special but large classes of representations. A common theme throughout my research has been to use objects and methods from invariant theory to provide a geometric framework for parametrizing large classes of representations. The invariant-theoretic objects that I have been studying in the papers listed below are:

    • moduli spaces of representations , constructed by King via Geometric Invariant Theory (GIT)
    • -on the algebraic side, they parametrize finite direct sums of (rather special) Schur representations

      -on the geometric side, they are projective varieties which, as shown by Hille and Huisgen-Zimmermann, can be arbitrarily complicated in the sense that any projective variety can be realized as such a moduli space; in particular, no special features of moduli spaces of representations can be expected in general unless one imposes conditions on either the algebra in question or the classes of representations to be parametrized

    • fields of rational invariants for bound quiver algebras; these are function fields of rational quotients (in the sense of Rosenlicht) that parametrize generic representations.

    The overarching goal of this project is to find characterizations of the GIT-tameness of an algebra in terms of its moduli spaces of representations. This research will lead to an analog of Drozd’s theorem at the level of moduli spaces of representations of algebras. It will also provide solutions (of a geometric nature) to classification problems for wild GIT-tame algebras. To achieve this goal, several general reduction techniques for dealing with these objects have been developed. These reduction results enable us to:

    • decompose a moduli space of representations or a field of rational invariants into ``smaller'' pieces which are easier to handle, especially when the algebra in question is either tame or wild but GIT-tame;

    • study the behavior of moduli spaces of representations under tilting functors; this allows one to study moduli spaces of representations for an algebra by reducing the considerations to an algebra of smaller global dimension.

    Work-in-progress/Submitted papers/Preprints

  4. Decomposing moduli of representations of finite-dimensional algebras
    with Ryan Kinser
    Mathematische Annalen, 372 (1), 555-580, 2018
    Abstract  |  pdf  |  arxiv
  5. Moduli spaces of representations of special biserial algebras
    with Andrew Carroll, Ryan Kinser, and Jerzy Weyman
    International Mathematics Research Notices 2018
    Abstract  |  pdf  |  doi  |  arxiv
  6. On locally semi-simple quiver representations
    with Dan Kline
    Journal of Algebra, 467 (2016), 284-306.
    Abstract  |  pdf  |  arxiv
  7. Moduli spaces of modules of Schur-tame algebras
    with Andrew Carroll
    Algebras and Representation Theory, (2015) 18:961-976.
    Abstract  |  pdf  |  arxiv
  8. On the invariant theory for acyclic gentle algebras
    with Andrew Carroll
    Transactions of the American Mathematical Society, 367 (2015), no. 5, 3481-3508.
    Abstract  |  pdf  |  arxiv
  9. Module varieties and representation type of finite dimensional algebras
    with Ryan Kinser and Jerzy Weyman
    International Mathematics Research Notices 2015, no 3, 631-650.
    Abstract  |  pdf  |  arxiv
  10. On the invariant theory for tame tilted algebras
    Algebra & Number Theory, Vol. 7 (2013), No. 1, 193-214.
    Abstract  |  pdf  |  arxiv
  11. On the geometry of orbit closures for representation-infinite algebras
    Glasgow Mathematical Journal, 54 (2012), 629-636.
    Abstract  |  pdf  |  arxiv
  12. Geometric characterizations of the representation type of hereditary algebras and of canonical algebras
    Advances in Mathematics, 228 (2011), no. 3, 1405-1434.
    Abstract  |  pdf  |  arxiv
  13. Orbit semigroups and the representation type of quivers
    Journal of Pure and Applied Algebra, 213 (2009), pp. 1418-1429.
    Abstract  |  pdf  |  arxiv

  14. III. Cones of effective weights for quivers, Littlewood-Richardson coefficients, and eigenvalue problems

    Background: Let Q be an acyclic quiver with n vertices and d a dimension vector of Q. The cone Eff(Q,d) of effective weights associated to (Q,d) is the rational convex polyhedral cone in ℜn whose lattice points are precisely those integral vectors θ (also called integral weights) for which the corresponding moduli space of θ-semi-stable d-dimensional representations of Q is non-empty. In the general context of GIT, these effective cones are some of the fundamental objects of study.

    In the quiver set-up, Derksen and Weyman found a beautiful description of the faces of Eff(Q,d) in terms of the root system of Q. Furthermore, they also showed that the lattice points of this cone are precisely those integral weights θ for which the corresponding weight spaces of semi-invariants SI(Q, d)θ are non-zero. These tools turn out to be very powerful in the study of:

    (i) Littlewood-Richardson coefficients (and of more general structure constants) since these coefficients can be realized as dimensions of weight spaces of semi-invariants of (star-shaped) quivers;

    (ii) eigenvalue problems for Hermitian matrices since Hermitian matrices with prescribed spectra and satisfying certain matrix equations arise from the symplectic description of moduli spaces of quiver representations.

    In the papers listed below, we have made essential use of the rich combinatorics and geometry of quiver representations to address a series of questions on: (1) the possible interactions between semi-stable (sub)categories of quiver representations; (2) finite stability weights and cluster fans; (3) Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients; and (4) generalized Littlewood-Richardson coefficients, long exact sequences of finite abelian p-groups, and eigenvalue problems.

  15. GIT-equivalence and semi-stable subcategories of quiver representations
    with Valerie Granger
    Journal of Pure and Applied Algebra, 223(8), 3499-3514, 2019
    Abstract  |  pdf  |  arxiv
  16. Cluster fans, stability conditions, and domains of semi-invariants
    Transactions of the American Mathematical Society, 363 (2011), no. 4, 2171-2190.
    Abstract  |  pdf  |  arxiv
  17. Quivers, long exact sequences and Horn type inequalities II
    Glasgow Mathematical Journal, 51 (2009), 1-17.
    Abstract  |  pdf  |  arxiv
  18. Quivers, long exact sequences and Horn type inequalities
    Journal of Algebra, 320 (2008), no. 1, 128-157.
    Abstract  |  pdf  |  arxiv
  19. Counterexamples to Okounkov's log-concavity conjecture
    with Harm Derksen and Jerzy Weyman
    Compositio Mathematica, 143 (2007), 1545-1557.
    Abstract  |  arxiv
  20. Eigenvalues of Hermitian matrices and cones arising from quivers
    International Mathematics Research Notices, 2006, Art. ID 59457, 27 pp.
    Abstract  |  pdf
  21. Notes on GIT fans for quivers
    Abstract  |  pdf  |  arxiv

  22. IV. Quiver representations of constant Jordan type and vector bundles

    Background: Let Q be an acyclic quiver and KQ its path algebra. In a series of papers, Benson, Carlson, Friedlander, Pevtsova, and Suslin have introduced and studied the class of modules of constant Jordan type for finite group schemes. Inspired by their work, Andrew Carroll, Zongzhu Lin, and myself have introduced the class of KQ-modules of (relative) constant Jordan type. On the algebraic side, these modules have constant decomposition into indecomposable modules when pulled-back to truncated polynomial rings in one variable. On the geometric side, we set up a process that assigns to any KQ-module a sequence of coherent sheaves over moduli spaces of (thin) KQ-modules. Under this correspondence, a KQ-module has constant Jordan type if and only if its corresponding sheaves are locally-free.

    In our approach to building these KQ-modules of constant Jordan type, we use GIT, more precisely moduli spaces of thin modules, instead of the Friedlander-Pevtsova's Π-point schemes, which are not available in the context of modules over path algebras.

  23. Quiver representations of constant Jordan type and vector bundles
    with Andrew Carroll and Zongzhu Lin
    Abstract  |  pdf  |  arxiv