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Research

Research interests: geometry and topology, quantum algebra, Schubert calculus, quivers

My research focuses on algebro-geometric problems which generalize classical enumerative geometry and Schubert calculus. Generally, I enjoy problems in which geometric and topological methods are used to establish combinatorial properties or algebraic relationships.

What is "enumerative geometry"? What is "Schubert calculus"?


The word "calculus" here should literally be interpreted as the art of "calculating". Newton's "The Calculus" was intended to "calculate", for example, instantaneous velocity or the work done by moving along a path. However, in this specific context the type of calculations we seek use geometry to count or "enumerate" objects: say points, lines, planes, conic sections, etc. The title "Schubert calculus" is one modern name for this pursuit.

For example, the image at the right depicts a solution to the Problem of Apollonius, a "Schubert-y" problem from antiquity. Given three circles in the plane, one must produce a fourth circle which is mutually tangent to each of the original three. In this example, the three given circles are filled. One of the possible solution circles appears in red. A prototypical problem in enumerative geometry asks: how many solution circles are there? 

My work

I have studied combinatorial properties of classes in cohomology or K-theory representing Schubert varieties or classes of degeneracy loci (a generalization of what is typically called a determinantal variety), which is the modern approach to enumerative geometry. In particular, I have studied such loci associated to quivers and the related representation theory, and I am currently interested in the formality of iterated residue techniques in these settings.

Recently, I am working on Donaldson–Thomas invariants for quivers. These are Rogers–Ramanujan type series identities which live in a non-commutative (or quantum algebra) associated to the quiver, with connections to counting states in quantum conformal field theories on the physical side, and with connections to cluster algebra combinatorics on the mathematical side. I am interested, however, in studying DT invariants through the lens of the geometry and topology of quvier loci.

Publications
  • With R. Rimanyi, Quantum dilogarithm identities for the square product of A-type Dynkin quivers. Math. Res. Lett., to appear (2017). arXiv:1702.04766
  • With R. Rimanyi, An iterated residue perspective on Grothendieck polynomials. preprint, (2014). arXiv:1408.1911
  • Grothendieck classes of quiver cycles as iterated residues. Michigan Math. J., 63 (2014) no. 4, 865-888. arXiv:1310.3548
  • With J. E. Grabowski, A quantum analogue of the dihedral action on Grassmannians. J. Algebra, 359 (2012), 49-68. arXiv:1102.0422
  • K-classes of quiver cycles, Grothendieck polynomials, and iterated residues. PhD thesis, UNC Chapel Hill, 75 pages (2014). [pdf via ProQuest]
  • Actions of finite dimensional non-commutative, non-cocommutative Hopf algebras on rings. MA thesis, Wake Forest University, 118 pages (2009). [pdf at WFU]
Talks

Invited talks at research seminars and colloquia

  • Grothendieck polynomials and iterated residues, Combinatorics-Algebra-Topology seminar at USNA (Sept 2017)
  • Generalizations of the pentagon identity and quiver representations, Lie Groups and Representation Theory seminar at U. of Maryland (Mar 2017) [remark: Yes, I work at the Naval Academy, but not that "Pentagon". Joke credit: Swarnava Mukhopadhyay]
  • Cohomology of Grassmannians and geometry, Combinatorics-Algebra-Topology seminar at USNA (Nov 2016)
  • Quantum dilogarithm series identities from topology and geometry, Algebra seminar at Virginia Tech (April 2016)
  • Dilogarithm series identities for quivers, Algebra and Discrete Mathematics seminar at Clemson U. (Dec 2015)
  • Formulas for quiver loci and Grothendieck polynomials as iterated residues, Algebra-Geometry-Combinatorics seminar at U. of Illinois (Sept 2015)
  • Geometric, Topological, and combinatorial methods in the representation theory of quivers, colloquium at Wake Forest U. (Nov 2014)
  • An overview of quiver loci, colloquium at Wake Forest U. (April 2014)
  • Iterated residues and K-classes of quiver loci, algebra seminar at Virginia Tech (Feb 2014)

Invited talks at conferences and meetings

  • Donaldson–Thomas invariants for A-type square product quivers, at Conference on Geometric Methods in Representation Theory, U. of Missouri (Nov 2016)
  • Grothendieck polynomials as iterated residues, at Southeastern Lie Theory Workshop, NC State U. (Oct 2015)
  • Positivity for K-theoretic Pieri rule via iterated residues, at AMS SE sectional meeting special session on Geometry and Combinatorics on Homogeneous Spaces, UNC Greensboro, (Nov 2014)
  • A generating sequence for K-theoretic quiver polynomials, at AMS SE sectional meeting special session on Algebraic Combinatorics, U. of Mississippi (Mar 2013)
  • A quantum analogue of the dihedral action on Grassmannians, at AMS SE sectional meeting special session on Noncommutative Algebra, Wake Forest U. (Sept 2011)
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