Amit Kanubhai Patel

2024

1. 

   A Categorical Approach to Möbius Inversion via Derived Functors

   Alex Elchesen, and Amit Patel

   2024

   Abs Bib

   We develop a cohomological approach to Möbius inversion using derived functors in the enriched categorical setting. For a poset 𝑃 and a closed symmetric monoidal abelian category 𝒞, we define Möbius cohomology as the derived functors of an enriched hom functor on the category of 𝑃-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical Möbius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota’s Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra.

   @misc{elchesen2024categorical,
     title = {{A Categorical Approach to M\"obius Inversion via Derived Functors}},
     author = {Elchesen, Alex and Patel, Amit},
     journal = {arXiv preprint arXiv:2411.04362},
     year = {2024},
   }

2023

1. 

   Möbius Homology

   Amit Patel, and Primoz Skraba

   2023

   Abs Bib HTML

   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

   Abs Bib HTML

   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},
   }