Summary

Office: 204 Vincent Hall, University of Minnesota
Curriculum Vita


Research Interests: combinatorics, convex geometry, higher representation theory, quantum field theory and string theory.

The foundation for my thesis work was conceived through countless hours of discussions with my co-adviser Adrian Ocneanu, who spent the past more than 15 years developing a new higher representation theory.

The preprint Canonical Bases for Permutohedral Plates gives a new derivation of the basic results about plates and introduces a new basis for the space of plates which contains new geometrical insights: the basis consists of a set of characteristic functions of permutohedral cones which can be understood intuitively to be "maximally distributed across the various dimensions."  The second installment, on the representation theory of hypersimplex dilations, as announced in the bottom preprint below, is in preparation.

Here is my Ph.D. Thesis.

Preprints
 
The picture is animated!  It's the degeneration of a permutohedron to a "period solid" or fundamental root parallelepiped, part of the development of a quantum analog of plates, in my Ph.D. thesis.   The middle rows of the matrices give the destinations of the vertices in the degeneration.  Click on the gif for the animation...


Posters and Talks:
  • UC Davis (Center for Quantum Mathematics and Physics (QMAP) ), Amplitudes 2018 Summer School
  • Harvard University, Mathematical Physics Seminar, February 2018
  • University of Minnesota, Combinatorics Seminar, December, 2017
  • University of Minnesota, Combinatorics Seminar, April, 2017
  • University of Sao Paulo, Seminar on Lie and Jordan Algebras and their Representations
  • University of Montreal, (Scattering Amplitudes and the Positive Grassmannian)
  • University of Michigan, Combinatorics Seminar
  • Symmetry and Geometric Structure for the Worpitzky identity. ( Grassmannian Geometry of Scattering Amplitudes)
    • Abstract: The classical Worpitzky identity for the symmetric group $S_n$ decomposes a cubical lattice into $n!$ simplices of different sizes, each with a multiplicity encoded by the number of permutations of $n$ with a fixed number of descents.  Traditionally, this is understood to be a purely combinatorial identity.  We unpack its full symmetry, observing that the Worpitzky identity amounts to the decomposition of a polyhedral representation of the symmetric group into vector spaces spanned by localized polyhedral cones.  It is well-known that the Eulerian numbers become volumes of hypersimplices; on the other hand, it is not known that the hypersimplices are actually vector spaces of dimensions the Eulerian numbers.  We show how the Worpitzky identity encodes localization data for our polyhedral cones, and finally we introduce the Character Polyhedron, whose volume equals the number of fixed points of a given permutation.
  • Permutohedral Arrangements, Simplicial Decompositions and a Geometrization of the Classical Worpitzky Identity (Applied Algebra and Network Theory Seminar, October 1, 2014)
    • Abstract: In forthcoming joint work with Adrian Ocneanu, we prove a symmetric group character formula conjectured several years ago by the latter as he was studying certain polyhedra emerging from hyperplane arrangements in simplices. We describe geometric content which lies behind the classical Worpitzky identity, which expands a cubical number $r^{n-1}$ in terms of the Eulerian numbers. Our proof suggests a new geometric interpretation of combinatorial results of Sagan-Shareshian-Wachs on Eulerian quasi-symmetric functions.
  • Hives and Multiparameter Deformations of the Vandermonde Determinant (March, 2013, Tensor Networks and Applications Seminar, )
    • Abstract: In some on-going work, I introduce a multi-parameter deformation of the Vandermonde determinant in an attempt eventually to relate the so-called hive model for computing tensor product invariants to the geometry of the configuration space of points on the real line.  With the key motivating examples as guide, I will outline the construction of a space $F$ of polynomials which emerges from the deformation of the Vandermonde determinant.  I will conclude by presenting computational results in which I give the decomposition into irreducible representations of the action of symmetric group on the space $F$. 
  • Specht Modules and the Plucker Algebra (Oral Comprehensive Exam, August, 2012)
    • Abstract: We summarize on-going work with L. Oeding on certain subalgebras $\mathcal{M}_{2}$ of the Plucker algebra, that are generated by compound determinants, in the special case where the Plucker variables are $3 \times 3$ minors of a $3\times 6$ matrix. It is known that two of these generators appear as "special" variables in the cluster algebra of the Grassmannian Gr$_3(6)$. The problem of classifying these special cluster variables for Gr$_k(n)$ for arbitrary $k$ and $n$ remains open. I show how these variables give a new basis for the Specht module associated to the $3\times 2$ Young tabloid. With L. Oeding, we conjecture a presentation for the algebra $\mathcal{M}_2$.  Our generators satisfy one polynomial relation; this relation defines the Igusa quartic.  A similar construction yields the Segre cubic.
Subpages (1): Teaching