Amit Kanubhai Patel

2023

1. 

   Möbius Homology

   Amit Patel, and Primoz Skraba

   2023

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   This paper introduces Möbius homology, a homology theory for representations of finite posets into abelian categories. While the connection between poset topology and Möbius functions is classical, we establish a direct connection between poset topology and Möbius inversions. More precisely, the Möbius homology categorifies the Möbius inversion because its Euler characteristic is equal to the Möbius inversion of the dimension function of the representation. We also introduce a homological version of Rota’s Galois Connection Theorem which relates the Möbius homology over two posets connected by a Galois connection. Our main application is to persistent homology over general posets. We show that under one definition, the persistence diagram is an Euler characteristic over a poset of intervals and hence Möbius homology is a categorification of the persistence diagram. This provides a new invariant for persistent homology over general posets. Finally, we use our homological Rota’s Galois Connection Theorem to prove several results about the persistence diagram.

   @misc{patel2023mobius,
     title = {M\"obius Homology},
     author = {Patel, Amit and Skraba, Primoz},
     year = {2023},
     eprint = {2307.01040},
     archiveprefix = {arXiv},
     primaryclass = {math.AT},
   }

2021

1. Advances

   Persistent local systems

   Robert MacPherson, and Amit Patel

   Advances in Mathematics, 2021

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   In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions.

   @article{MACPHERSON2021107795,
     author = {MacPherson, Robert and Patel, Amit},
     journal = {Advances in Mathematics},
     keywords = {Persistent homology, Stability, Stratified spaces, Sheaves, Cosheaves},
     pages = {107795},
     title = {Persistent local systems},
     volume = {386},
     year = {2021},
   }

2018

1. JACT

   Generalized persistence diagrams

   Amit Patel

   Journal of Applied and Computational Topology, Jun 2018

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   We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the \emphtype \mathcalA persistence diagram of a persistence module. If the category is also abelian, then we define a second \emphtype \mathcalB persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module. The type \mathcalB persistence diagram carries less information than the type \mathcalA persistence diagram, but it enjoys a stronger stability theorem.

   @article{Patel2018,
     title = {Generalized persistence diagrams},
     author = {Patel, Amit},
     journal = {Journal of Applied and Computational Topology},
     publisher = {Springer International Publishing},
     volume = {1},
     number = {3},
     pages = {397--419},
     month = jun,
     year = {2018},
     language = {en},
     dimensions = {false},
   }