Summary




Office: 212 McAllister Building
Curriculum Vita



Research Interests: Geometry, Combinatorics, Higher representation theory, Quantum field theory

In preparation: Early, N.  Combinatorics and Representation Theory for Generalized Permutohedra II.

In preparation (with Adrian Ocneanu): Symmetry and Geometric Structure for the Worpitzky identity.

Preprints
 
The picture is animated!  It's the degeneration of a permutohedron to a "period solid" or fundamental root parallelepiped.   The middle rows of the matrices give the destinations of the vertices in the degeneration.  The distance function for the faces is interesting and will be discussed in a forthcoming paper.  Click on the gif for the animation...


Posters and Talks:
  • University of Minnesota, Combinatorics Seminar
  • 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