17 citations · 48 across the 16 of their papers we have counts for
19 papers
Topological Parallax: A Geometric Specification for Deep Perception Models
Abraham D. Smith, Michael J. Catanzaro, Gabrielle Angeloro +2
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine…
The Persistence Landscapes of Affine Fractals
Michael J. Catanzaro, Lee Przybylski, Eric S. Weber
We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of af…
Harmonic Representatives in Homology over Arbitrary Fields
Michael J. Catanzaro, Brantley Vose
We introduce a notion of harmonic chain for chain complexes over fields of positive characteristic. A list of conditions for when a Hodge decomposition theorem holds in this settin…
Multiparameter persistent homology via generalized Morse theory
Peter Bubenik, Michael J. Catanzaro
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stab…
Probability measures on graph trajectories
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
The aim of this note is to construct a probability measure on the space of trajectories in a continuous time Markov chain having a finite state diagram, or more generally which adm…
Hypercurrents
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, w…
MVF Designer: Design and Visualization of Morse Vector Fields
Youjia Zhou, Janis Lazovskis, Michael J. Catanzaro +2
Vector field design on surfaces was originally motivated by applications in graphics such as texture synthesis and rendering. In this paper, we consider the idea of vector field de…
Moduli Spaces of Morse Functions for Persistence
Michael J. Catanzaro, Justin Curry, Brittany Terese Fasy +5
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classe…
A Model for Random Chain Complexes
Michael J. Catanzaro, Matthew J. Zabka
We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps…
On fluctuations of cycles in a finite CW complex
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
We use algebraic topology to study the stochastic mo- tion of cellular cycles in a finite CW complex. Inspired by statis- tical mechanics, we introduce a homological observable cal…
Excited-State Structure Modifications Due to Molecular Substituents and Exciton Scattering in Conjugated Molecules
Hao Li, Michael J. Catanzaro, Sergei Tretiak +1
Attachment of chemical substituents (such as polar moieties) constitutes an efficient and convenient way to modify physical and chemical properties of conjugated polymers and oligo…
Generalized Tonnetze
Michael J. Catanzaro
We study a generalization of the classical Riemannian Tonnetz to N-tone equally tempered scales (for all N) and arbitrary triads. We classify all the spaces that result. The torus…
Counting the number of excited states in organic semiconductor systems using topology
Michael J. Catanzaro, Tian Shi, Sergei Tretiak +1
Exciton Scattering (ES) theory attributes excited electronic states to standing waves in quasi-one-dimensional molecular materials by assuming a quasi-particle picture of optical e…
Stochastic Dynamics of Extended Objects in Driven Systems: I. Higher-Dimensional Currents in the Continuous Setting
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
The probability distributions, as well as the mean values of stochastic currents and fluxes, associated with a driven Langevin process, provide a good and topologically protected m…
Stochastic Dynamics of Extended Objects in Driven Systems II: Current Quantization in the Low-Temperature Limit
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
Driven Langevin processes have appeared in a variety of fields due to the relevance of natural phenomena having both deterministic and stochastic effects. The stochastic currents a…
A higher Boltzmann distribution
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
We characterize the classical Boltzmann distribution as the unique solution to a certain combinatorial Hodge theory problem in homological degree zero on a finite graph. By substit…
Exciton Scattering via Algebraic Topology
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target…
Kirchhoff's theorems in higher dimensions and Reidemeister torsion
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff's network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an appli…
On Kirchhoff's theorems with coefficients in a line bundle
Michael J. Catanzaro, Vladimir Y. Chernyak, John R. Klein
We prove `twisted' versions of Kirchhoff's network theorem and Kirchhoff's matrix-tree theorem on connected finite graphs. Twisting here refers to chains with coefficients in a fla…