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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 whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset., Comment: 18 pages, 6 figures. Preprint submitted to NeurIPS 2023

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 affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations., Comment: 32 pages, 5 figures. Comments welcome

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 setting is given and we apply this theory to finite CW complexes. An explicit construction of the harmonic chain within a homology class is described when applicable. We show how the coefficients of usual discrete harmonic chains due to Eckmann can be reduced to localizations of the integers, allowing us to compare classical harmonicity with the notion introduced here. We focus on applications throughout, including CW decompositions of orientable surfaces and examples of spaces arising from sampled data sets., Comment: 20 pages

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 stable. We show that this construction can produce indecomposable persistence modules with arbitrarily large dimension. In the case of smooth functions on a compact manifold, we apply cobordism theory and Cerf theory to study the resulting persistence modules. We give examples in which we obtain a complete description of the persistence module as a direct sum of indecomposable summands and provide a corresponding visualization., Comment: 21 pages, added proof of stability, added statistical motivation, improvements in exposition thanks to referee's comments

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 admits a global bound on its degree and rates. Our approach is elementary. Our main intention is to fill a gap in the literature., Comment: Fixed a zoo of typos

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, where the labels are required to vary continuously. If the space is a one-dimensional manifold, then a protocol determines a continuous time Markov chain. When a homological gap condition is present, we associate to each protocol a 'characteristic' cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the low temperature limit. We also exhibit examples of protocols having nontrivial hypercurrent., Comment: This version fixes some references

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 design with a new motivation from computational topology. We are interested in designing and visualizing vector fields to aid the study of Morse functions, Morse vector fields, and Morse-Smale complexes. To achieve such a goal, we present MVF Designer, a new interactive design system that provides fine-grained control over vector field geometry, enables the editing of vector field topology, and supports a design process in a simple and efficient way using elementary moves, which are actions that initiate or advance our design process. Our system allows mathematicians to explore the complex configuration spaces of Morse functions, their gradients, and their associated Morse-Smale complexes. Understanding these spaces will help us expand further their applicability in topological data analysis and visualization.

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 classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in $\mathbf R^3$. Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode., Comment: 30 pages, 16 figures. Amended with reviewer suggestions, text is as in published version. Comments welcome

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 uniformly over $\mathbb{F}_q$, conditioned on ensuring that the composition of consecutive boundary maps is the zero map. We then investigate the combinatorial and homological properties of this random chain complex., Comment: 9 pages, comments welcome

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 called the average current. The latter measures the average flux of the prob- ability in the process. In the low temperature, adiabatic limit, we prove that the average current fractionally quantizes, in which the denominators are combinatorial invariants of the CW complex., Comment: Added an example showing that the average current in the low temperature adiabatic limit can be non-trivial. Corrected typos

Attachment of chemical substituents (such as polar moieties) constitutes an efficient and convenient way to modify physical and chemical properties of conjugated polymers and oligomers. Associated modifications in the molecular electronic states can be comprehensively described by examining scattering of excitons in the polymer's backbone at the scattering center representing the chemical substituent. Here, we implement effective tight-binding models as a tool to examine the analytical properties of the exciton scattering matrices in semi-infinite polymer chains with substitutions. We demonstrate that chemical interactions between the substitution and attached polymer are adequately described by the analytical properties of the scattering matrices. In particular, resonant and bound electronic excitations are expressed via the positions of zeros and poles of the scattering amplitude, analytically continued to complex values of exciton quasi-momenta. We exemplify the formulated concepts by analyzing excited states in conjugated phenylacetylenes substituted by perylene.

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 excitations. The quasi-particle properties at branching centers are described by the corresponding scattering matrices. Here we identify the topological invariant of scattering center, referred to as its winding number, and apply topological intersection theory to count the number of quantum states in a quasi-one-dimensional system.

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 turns out to be the most common possibility, especially as N grows. Other spaces include 2-simplices, tetrahedra boundaries, and the harmonic strip (in both its cylinder and Mobius band variants). The final and most exotic space we find is something we call a `circle of tetrahedra boundaries'. These are the Tonnetze for spaces of triads which contain a tritone. They are closely related to Peck's Klein bottle Tonnetz.

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 measure of how far a stochastic system is driven out of equilibrium. By viewing a Langevin process on a compact oriented manifold of arbitrary dimension m as a theory of a random vector field associated with the environment, we are able to consider stochastic motion of higher-dimensional objects, which allow new observables, called higher-dimensional currents, to be introduced. These higher dimensional currents arise by counting intersections of a k-dimensional trajectory, produced by a evolving (k-1)-dimensional cycle, with a reference cross section, represented by a cycle of complimentary dimension (m - k). We further express the mean fluxes in terms of the solutions of the Supersymmetric Fokker-Planck (SFP), thus generalizing the corresponding well-known expressions for the conventional currents.

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 and fluxes in these systems provide a convenient set of observables to describe their non-equilibrium steady states. Here we consider stochastic motion of a (k - 1)-dimensional object, which sweeps out a k-dimensional trajectory, and gives rise to a higher k-dimensional current. By employing the low-temperature (low-noise) limit, we reduce the problem to a discrete Markov chain model on a CW complex, a topological construction which generalizes the notion of a graph. This reduction allows the mean fluxes and currents of the process to be expressed in terms of solutions to the discrete Supersymmetric Fokker-Planck (SFP) equation. Taking the adiabatic limit, we show that generic driving leads to rational quantization of the generated higher dimensional current. The latter is achieved by implementing the recently developed tools, coined the higher-dimensional Kirchhoff tree and co-tree theorems. This extends the study of motion of extended objects in the continuous setting performed in the prequel to this manuscript., Comment: To appear in Chemical Physics

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 substituting for the graph a CW complex and a degree d, we are able to define, by direct analogy, a higher dimensional Boltzmann distribution as a certain (d-1)-cycle on the real cellular chain complex which is characterized by appropriate constraints. We then give an explicit formula for this cycle. We explain how this circle of ideas relates to the authors' Higher Kirchhoff Network Theorem. We also give an improved version of the authors' Higher Matrix-Tree Theorems., Comment: In this revision, we have fixed the definition of the constants a_L appearing in Theorems A and C

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 is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data. We provide a lower bound for this number. The bound is shown to be sharp in a limiting case., Comment: Final version; to appear in the Journal of Topology and Analysis

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 flat unitary line bundle., Comment: Added improvements to the proof of Theorem C. Added some additional figures

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 application, we give a formula expressing Reidemeister torsion as an enumeration of higher dimensional spanning trees., Comment: Fixed several typos