Research Papers

On the LusternikSchnirelmann category of a simplicial map (with Willie Swei)
Topology and its applications (to appear)In this paper, we study the LusternikSchnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to simplicial products and fibrations. We relate this category of a map to the classical LusternikSchnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the category weight of Y. Rudyak and J. Strom.

Cosheaf Theoretical Constructions in Networks and Persistent Homology (with Karthik Yegnesh)
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally, (co)sheaves have proven to be powerful instruments in tracking locally defined data across global systems, resulting in innovative applications to network science. In this paper, we combine the topological results of persistent homology and the quantitative data tracking capabilities of cosheaf theory to develop novel techniques in network data flow analysis. Specifically, we use cosheaf theory to construct persistent homology in a framework geared towards assessing data flow stability in hierarchical recurrent networks (HRNs). We use cosheaves to link topological information about a filtered network encoded in persistence diagrams with data associated locally to the network. From this construction, we use the homology of cosheaves as a framework to study the notion of ``persistent data flow errors." That is, we generalize aspects of persistent homology to analyze the lifetime of local data flow malfunctions. We proceed with several constructions motivation by the persistent homology of filtered topological spaces to fit our network theoretical environment. We conclude with an algorithmic construction of persistence diagrams parameterizing network data flow errors, thus enabling novel applications of statistical methods to study data flow malfunctions. Our results can be applied to analyze data flows in complex systems such as financial, social, and biological networks.

Homology of Boolean functions and the complexity of simplicial homology (with Erick Chastain)
We study the topology of Boolean functions from the perspective of Simplicial Homology, and characterize Simplicial Homology in turn by using Monotone Boolean functions. In so doing, we analyze to what extent topological invariants of Boolean functions (for instance the Euler characteristic) change under binary operations over Boolean functions and other operations, such as permutations over the input variables. We apply these tools to proving the $\Delta^p_2$ hardness of calculating the Euler characteristic of general Boolean functions (as defined by Kulkarni and Santha) and the coNPhardness of calculating the Euler characteristic for a simplicial complex of arbitrary dimension. We also show that calculating the Betti numbers for Simplicial Homology is coNP hard.

Estimating the discrete LusternikSchnirelmann category (with Mimi Tsuruga and Brian Green)
Topological Methods in Nonlinear Analysis, 45, No. 1 (2015), 103116Let $K$ be a simplicial complex and suppose that $K$ collapses onto $L$. Dene $n$ to be $1$ minus the minimum number of collapsible sets it takes to cover $L$. Then the discrete LusternikSchnirelmann category of $K$ is the smallest $n$ taken over all such $L$. In this paper, we give an algorithm which yields an upper bound for the discrete category. We show our algorithm is correct and give several bounds for the discrete category of wellknown simplicial complexes. We show that the discrete category of the dunce cap is $2$, implying that the dunce cap is ``further" from being collapsible than Bing's house.

Graph Isomorphisms in Discrete Morse Theory (with Seth Aaronson, Marie Meyer, Mitchell T. Smith, and Laura M. Stibich)
AKCE International Journal of Graphs and Combinatorics 11 (2014), no. 2, 163176.A discrete Morse function $f$ on a graph $G$ induces a sequence of subgraphs of $G$. In a paper, the authors introduce a notion of equivalence between discrete Morse functions based on a sequence of homology groups induced by the subgraphs of $G$. In this paper, we use the homology sequence to study a new notion of equivalence between discrete Morse functions. This equivalence is based on the isomorphism type of the subgraphs of $G$. We count the number of equivalence classes on a particular collection of graphs, and deduce an upper bound for the number of equivalence classes for a large collection of graphs.

Metric Structures for CW Complexes
Topology Proceedings, Volume 44 (2014) 117131We investigate the discrete LusternikSchnirelmann category of a $1$dimensional simplicial complex $G$. In this case, the invariant is shown to be the minimum number of subtrees of $G$ whose union is equal to $G$. After some preliminary computations and bounds, we study the discrete category of the product of $1$complexes. We prove a general upper bound, analogous to the classical upper bound for products in the smooth case, and prove a Ganealike conjecture for some classes of spaces. We conclude by proposing a Ganea conjecture for all $1$dimensional complexes.

A Distributed Greedy Algorithm for Constructing Connected Dominating Sets in Wireless Sensor Networks(with Akshaye Dhawan and Michelle Tanco)
3rd International Conference on Sensor Networks (SENSORNETS), Lisbon, Portugal, January, 2014A Connected Dominating Set (CDS) of the graph representing a Wireless Sensor Network can be used as a virtual backbone for routing in the network. Since sensor nodes are constrained by limited onboard batteries, it is desirable to have a small CDS for the network. However, constructing a minimum size CDS has been shown to be a NPhard problem. In this paper we present a distributed greedy algorithm for constructing a CDS that we call Greedy Connect. Our algorithm operates in two phases, first constructing a dominating set and then connecting the nodes in this set. We evaluate our algorithm using simulations and compare it to the twohop K2 algorithm in the literature. Depending on the network topology, our algorithm generally constructs a CDS that is up to 30% smaller in size than K2.

LusternikSchnirelmann category for simplicial complexes (with Seth Aaronson)
Illinois J. Math. 57 (2013), no. 3, 743753.The discrete version of Morse theory due to Robin Forman is a powerful tool utilized in the study of topology, combinatorics, and mathematics involving the overlap of these fields. Inspired by the success of discrete Morse theory, we take the first steps in defining a discrete version of the LusternikSchnirelmann category suitable for simplicial complexes. This invariant is based on collapsibility as opposed to contractibility, and is defined in the spirit of the geometric category of a topological space. We prove some basic results of this theory, showing where it agrees and differs from that of the smooth case. Our work culminates in a discrete version of the LusternikSchnirelmann theorem relating the number of critical points of a discrete Morse function to its discrete category.

Homological sequences in discrete Morse theory(with Mike Agiorgousis, Brian Green, Alex Onderdonk, and Kim Rich)
A theory of homological equivalence of discrete Morse functions is developed in this paper, extending the work of Ayala et al. We define the homological sequence associated with a discrete Morse function on any finite simplicial complex. This sequence is shown to satisfy specified desirable properties. These properties allow us to show that homological sequences may be viewed as lattice walks satisfying certain parameters. We count the number of discrete morse functions up to homological equivalence on any collapsible $2$dimensional simplex by constructing discrete Morse functions satisfying certain properties. The paper concludes with an example to illustrate our construction.

LusternikSchnirelmann Category and the Connectivity of $X$
Algebraic & Geometric Topology 12 (2012) 435448In this paper, we define and study a homotopy invariant called the connectivity weight to compute the weighted length between spaces $X$ and $Y$. This is an invariant based on the connectivity of $A_i$, where $A_i$ is a space attached in a mapping cone sequence from $X$ to $Y$. We use the LusternikSchnirelmann category to prove a theorem concerning the connectivity of all spaces attached in any decomposition from $X$ to $Y$. This theorem is used to prove that for any positive rational number $q$, there is a space $X$ such that $q=\mathrm{cl}^{\omega}(X)$, the connectivity weighted conelength of $X$. We compute $\mathrm{cl}^{\omega}(X)$ and $\mathrm{kl}^{\omega}(X)$ for many spaces and give several examples.

Georg Cantor at the Dawn of PointSet Topology
Loci (March 2012), DOI: 10.4169/loci003861A first course in pointset topology can be challenging for the student because of the abstract level of the material. In an attempt to mitigate this problem, we use the history of pointset topology to obtain natural motivation for the study of some key concepts. In this paper, we study an 1872 paper by Georg Cantor. We will look at what Cantor was attempting to accomplish and see how the now familiar concepts of a pointset and derived set are natural answers to his problem. We emphasize ways to utilize Cantor's methods in order to introduce pointset topology to the student.

Mapping Cone Sequences and a Generalized Notion of Cone Length
JP Journal of Geometry and Topology, Volume 11, Issue 3, November 2011, 209233We introduce a weighted length between spaces. This is accomplished by using the numerical invariants of cone length and killing length as a framework and by considering other topological invariants to determine the complexity of spaces. This leads us to the denition of a weighted length between spaces. We estimate the weighted length amongst certain maps and spaces for pushouts, pullbacks, and brations. Examples of spe cic weights are given to show that hypotheses in theorems are necessary.

Categorical Sequences (with Rob Nendorf and Jeff Strom)
Algebraic & Geometric Topology 6 (2006) 809838We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the LusternikSchnirelmann category of a space $X$ by induction on its CW skeleta. The $k^{th}$ term in the categorical sequence of a CW complex $X$, $\sigma_X(k)$, is the least integer $n$ for which $\mathrm{cat}_X(X_n) = k$. We show that $\sigma_X$ is a welldefined homotopy invariant of $X$. We prove that $\sigma_X(k + l) = \sigma_X(k) + \sigma_X(l)$, which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if $X$ is a member of the Mislin genus of the Lie group $S_p(3), then $\mathrm{cat}_(X) = \mathrm{cat}(Sp(3)).$
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Last modified on January 29, 2016