arXiv subjects

Steven Scheirer

Publications and source records attributed to Steven Scheirer.

5 recordsLinked to original sources

arXiv2026

Grouped Stirling complexes

Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call Grouped Stirling complexes, denoted by $S_{\vec r}(G)$, in which we place robots in groups subject to two constraints. First, there must be at least one robot on each vertex of $G$, and second, any two robots from the same group must be "separated by at least one full open edge" of $G$. The space $S_{\vec r}(G)$ has a closed cell structure, which means it can be built out of cells of various dimensions. Our main results show $S_{\vec r}(G)$ is path-connected, provided there are at least three groups, and determine the number of cells of $S_{\vec r}(G)$ in certain cases.

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arXiv2025

On minimal bases in homotopical combinatorics

We present a development in the computational suite for the study of $N_\infty$ operads for a finite group $G$. This progress is achieved using the simple yet powerful observation that Rubin's generation algorithm can be interpreted as a closure operator. Leveraging this perspective, we establish the existence of minimal bases for $N_\infty$ operads. By investigating these bases for certain families of groups we are led to introduce and analyze several novel combinatorial invariants for finite groups.

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arXiv2021

Relative Topological Complexity and Configuration Spaces

Given a space $X$, the topological complexity of $X$, denoted by $TC(X)$, can be viewed as the minimum number of "continuous rules" needed to describe how to move between any two points in $X$. Given subspaces $Y_1$ and $Y_2$ of $X$, there is a "relative" version of topological complexity, denoted by $TC_X(Y_1\times Y_2)$, in which one only considers paths starting at a point $y_1\in Y_1$ and ending at a point $y_2\in Y_2$, but the path from $y_1$ to $y_2$ can pass through any point in $X$. We discuss general results that provide relative analogues of well-known results concerning $TC(X)$ before focusing on the case in which we have $Y_1=Y_2=C^n(Y)$, the configuration space of $n$ points in some space $Y$, and $X=C^n(Y\times I)$, the configuration space of $n$ points in $Y\times I$, where $I$ denotes the interval $[0,1]$. Our main result shows $TC_{C^n(Y\times I)}(C^n(Y)\times C^n(Y))$ is bounded above by $TC(Y^n)$ and under certain hypotheses is bounded below by $TC(Y)$.

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arXiv2019

Topological complexity of unordered configuration spaces of certain graphs

The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$ unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph $\Gamma.$ The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by $m(\Gamma).$ We discuss a combinatorial approach to compute the topological complexity of a "discretized" version of this space, $UD^n(\Gamma),$ and give results for certain classes of graphs. In the first case, we show that for a large class of graphs, as long as the number of robots is at least $2m(\Gamma)$, then $TC(UD^n(\Gamma))=2m(\Gamma)+1.$ In the second, we show that as long as the number of robots is at most half the number of vertex-disjoint cycles in $\Gamma,$ we have $TC(UD^n(\Gamma))=2n+1.$

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arXiv2016

Topological complexity of n points on a tree

The topological complexity of a path-connected space $X,$ denoted $TC(X),$ can be thought of as the minimum number of continuous rules needed to describe how to move from one point in $X$ to another. The space $X$ is often interpreted as a configuration space in some real-life context. Here, we consider the case where $X$ is the space of configurations of $n$ points on a tree $\Gamma.$ We will be interested in two such configuration spaces. In the, first, denoted $C^n(\Gamma),$ the points are distinguishable, while in the second, $UC^n(\Gamma),$ the points are indistinguishable. We determine $TC(UC^n(\Gamma))$ for any tree $\Gamma$ and many values of $n,$ and consequently determine $TC(C^n(\Gamma))$ for the same values of $n$ (provided the configuration spaces are path-connected).

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