arXiv subjects

Karthik Yegnesh

Publications and source records attributed to Karthik Yegnesh.

4 recordsLinked to original sources

arXiv2017

Braid Groups on Triangulated Surfaces and Singular Homology

Let $Σ_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $Σ_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on $Σ_g$ to the first singular homology group of $Σ_g$ with integral coefficients. In particular, we show that the kernel of this homomorphism is generated by canonical braids which arise from the triangulation of $Σ_g$. This provides a simple description of natural subgroups of surface braid groups which are closely tied to the homology groups of the surfaces $Σ_g$.

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arXiv2017

Persistence and Sheaves

We study a variation of Carlsson and Zomorodian's persistent homology. As an application, we analyze the persistent cellular sheaf cohomology of network coding sheaves on deteriorating networks.

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arXiv2017

The Fundamental Infinity-Groupoid of a Parametrized Family

Given an infinity-category C, one can naturally construct an infinity-category Fam(C) of families of objects in C indexed by infinity-groupoids. An ordinary categorical version of this construction was used by Borceux and Janelidze in the study of generalized covering maps in categorical Galois theory. In this paper, we develop the homotopy theory of such "parametrized families" as generalization of the classical homotopy theory of spaces. In particular, we study homotopy-theoretical constructions that arise from the fundamental infinity-groupoids of families in an infinity-category. In the same spirit, we show that Fam(C) admits a Grothendieck topology which generalizes the canonical/epimorphism topology on the infinity-topos of infinity-groupoids in the sense of Carchedi.

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arXiv2016

Cosheaf Theoretical Constructions in Networks and Persistent Homology

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 study an algorithmic construction of persistence diagrams parameterizing network data flow errors, thus enabling novel applications of statistical methods to study data flow malfunctions. We conclude with an application to network packet delivery systems.

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