Erik Insko

On ( t , r ) broadcast domination numbers of grids

The domination number of a graph G = ( V , E ) is the minimum cardinality of any subset S ? V such that every vertex in V is in S or adjacent to an element of S . Finding the domination numbers of m by n grids was an open problem for nearly 30?years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomasse. Many variants of domination number on graphs have been defined and studied, but exact values have not yet been obtained for grids. We will define a family of domination theories parameterized by pairs of positive integers ( t , r ) where 1 ? r ? t which generalize domination and distance domination theories for graphs. We call these domination numbers the ( t , r ) broadcast domination numbers. We give the exact values of ( t , r ) broadcast domination numbers for small grids, and we identify upper bounds for the ( t , r ) broadcast domination numbers for large grids and conjecture that these bounds are tight for sufficiently large grids.

A proof of the peak polynomial positivity conjecture

We say that a permutation

Peak sets of classical Coxeter groups

\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n

Computing weight q-multiplicities for the representations of the simple Lie algebras

has a peak at index

Patch ideals and Peterson varieties

i

Affine pavings of regular nilpotent Hessenberg varieties and intersection theory of the Peterson variety

if

Intersection theory of the Peterson variety and certain singularities of Schubert varieties

\pi_{i-1} \pi_{i+1}

Schubert Calculus and the Homology of the Peterson Variety

. Let

The adjoint representation of a classical Lie algebra and the support of Kostant’s weight multiplicity formula

\mathcal{P}(\pi)

The

denote the set of indices where