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Topology and Computing
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Submissions closed on 2025-06-15 11:59PM [Central Time (US & Canada)].

This special session will feature the applications of computing for topology, and vice versa, broadly construed. Organized by Jim Fowler and Ziqin Feng.

This special session will feature the applications of computing for topology, and vice versa, broadly construed. Any research related to both topology and computing is welcome, including but not limited to, topological data analysis, formalization of topology, applications of artificial intelligence and machine learning on topology, and more. Organized by Jim Fowler and Ziqin Feng.

Accepted Submissions:

A mechanized characterization of coherent 2-groups — Perry Hart Icon: submission_accepted

In this talk, we will outline an Agda implementation of a higher-dimensional piece of the homotopy hypothesis, which provides a general correspondence between groupoids and homotopy types. The groupoids we look at are monoidal groupoids where every element has the structure of an adjoint equivalence, called coherent 2-groups. The correspondence for such groupoids takes the form of a biequivalence between the (2,1)-category of coherent 2-groups and the (2,1)-category of pointed connected (homotopy) 2-types. We build this biequivalence in homotopy type theory (HoTT) and use Agda to verify the construction (see https://github.com/PHart3/2-groups-agda). Thanks to the univalence axiom, we also obtain a verified identity between the two (2,1)-categories in question. This biequivalence has been suggested at a few places in the literature. Inside HoTT, Buchholtz, van Doorn, and Rijke (2018) propose it as a 2-dimensional generalization of the equivalence they construct between Grp and pointed connected 1-types. It also was suggested in the classical setting by Baez and Lauda (2004). Indeed, the biequivalence we construct generalizes the 1-dimensional equivalence. It consists of two broad steps. First, we construct the classifying space of a coherent 2-group G as a higher inductive type by generalizing the first Eilenberg-MacLane space of a group. This defines a function from the type of coherent 2-groups to the type of pointed connected 2-types. Second, we equip this function with the structure of a pseudofunctor and prove that it forms a biequivalence with the loop space pseudofunctor, which takes a pointed connected 2-type to its fundamental 2-group. Each step is purely algebraic, and all our proofs are constructive. Notably, combined with recent work by Owen Milner (unpublished), our biequivalence computes the Sinh invariant of a coherent 2-group within a constructive system, whereas the traditional method relies on the axiom of choice. Our formalization, however, involves several huge computations, and we will discuss our experience managing its memory requirements and its arduous type-checking.

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Comparison of Precinct and District Voting Data Using Persistent Homology to Identify Gerrymandering in North Carolina — Ananya Shah Icon: submission_accepted

We present an extension of Feng & Porter’s 2019 paper on the use of the level-set method for the construction of a filtered simplicial complex from geospatial election data, by applying their method to identify gerrymandering. Using the fact that precincts are regarded to be too small to be gerrymandered, we identify discrepancies between precinct and district level voting data to quantify gerrymandering. Comparing the persistent homologies of democratic voting areas on the precinct and district level shows when areas have been ‘cracked’ or ‘packed’ for partisan gain. This analysis was done for North Carolina House of Representatives elections (2012-2024). NC has been redistricted 4 times in the past 10 years, whereas most states redistrict decennially, allowing us to understand how and when redistricted maps deviate from precinct-level voting data, and when gerrymandering occurs. Comparing persistence barcodes at the precinct and district levels (using the bottleneck distance) shows that precinct-level voting patterns do not significantly fluctuate biannually, while district level patterns do, suggesting that shifts are likely a result of redistricting rather than voter behavior, providing strong evidence of gerrymandering. NC Election data was collected from the public domain. Composite shapefiles were created using QGIS and R, and rasterized using Python. The level-set method was employed to generate filtered similar complexes. Persistence barcodes were produced using GUDHI and PHAT libraries. Additionally, we compare our results with traditional measures such as Polsby-Popper and Reock scores (gerrymandering identification measures). This research presents a novel application of topological data analysis in analyzing gerrymandering.

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Computing path homology chains by inductive construction — Matthew Burfitt Icon: submission_accepted

The path homology introduced by Grigor’yan, Lin, Muranov and Yau plays a central role in digraph topology and the emerging field of digraph homotopy theory more generally. Unfortunately, the computation of the path homology of a digraph is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In particular, our understanding of the path chains is the primary obstruction to the development of fast path homology algorithms, which in turn would enable the practicality of a wide range of applications to directed networks. I will introduce an inductive method of constructing elements of the path homology chain modules from elements in the proceeding two dimensions. When the coefficient ring has prime characteristic the inductive elements generate the path chains. Moreover, in low dimensions the inductive elements coincide with naturally occurring generating sets up to sign, making them excellent candidates to reduce to a basis. Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph. During the talk I will demonstrate how inductive elements yield the explicit structure of the dimension 3 path chains and enable the construction of a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of coefficients.

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Efficient evader detection in mobile sensor networks — William Ott Icon: submission_accepted

Suppose one wants to monitor a domain with sensors, each sensing a small ball-shaped region, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly placed sensors could be required to obtain static coverage. Instead, one can use fewer sensors by providing mobile coverage, a generalization of the static setup wherein every possible evader is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors’ locations, only their connectivity information. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Our detailed simulations show, for example, that collective motion can enhance performance: The expected time until the mobile sensor network achieves mobile coverage is lower for the D’Orsogna collective motion model than for the billiard motion model. Further, we show that even when the probability of static coverage is low, all possible evaders can nevertheless be detected relatively quickly by the mobile sensor network.

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Facets in the Vietoris--Rips complexes of hypercubes — Ziqin Feng Icon: submission_accepted

In this talk, we'll discuss the facets (maximal simplices) of the Vietoris--Rips complex VR(Qn;r) where Qn denotes the n-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension n. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale r, assuming that n is sufficiently large. We show also that the (2r−1)-th dimensional homology of the complex VR(Qn;r) is non-trivial when n is large enough, provided that the Hadamard matrix of order 2r exists.

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Homotopy connectivity of Cech complexes of spheres. — Sucharita Mallick Icon: submission_accepted

Let Sn be the n-sphere with the geodesic metric and of diameter π. The intrinsic \v{C}ech complex of Sn at scale r is the nerve of all open balls of radius r in Sn. In this talk, we will show how to control the homotopy connectivity of \v{C}ech complexes of spheres at each scale between 0 and π in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case n=1, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \v{C}ech complexes of the sufficiently dense, finite subsets of Sn. Our bounds imply the new result that for n≥1, the homotopy type of the \v{C}ech complex of Sn at scale r changes infinitely many times as r varies over (0,π). Additionally, we lower bound the homological dimension of \v{C}ech complexes of finite subsets of Sn in terms of their packings. This is joint work with Henry Adams and Ekansh Jauhari.

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Persistence-Augmented Neural Networks — Elena Wang Icon: submission_accepted

Topological Data Analysis (TDA) provides tools to describe the shape of data, but integrating topological features into deep learning pipelines remains challenging, especially when preserving local geometric structure rather than summarizing it globally. We propose a persistence-based data augmentation framework that encodes local gradient flow regions and their hierarchical evolution using the Morse–Smale complex. This representation, compatible with both convolutional and graph neural networks, retains spatially localized topological information across multiple scales. Importantly, the augmentation procedure itself is efficient, with computational complexity O(nlog⁡n), making it practical for large datasets. We evaluate our method on histopathology image classification and 3D porous material regression, where it consistently outperforms baselines and global TDA descriptors such as persistence images and landscapes. We also show that pruning the base level of the hierarchy reduces memory usage while maintaining competitive performance. These results highlight the potential of local, structured topological augmentation for scalable and interpretable learning across data modalities.

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Representations of Micrograph Geometry for Machine Learning — Benjamin Schweinhart Icon: submission_accepted

Two-dimensional micrographs are a common data format in several important machine learning applications. One example is histopathological classification: the detection of disease-related abnormalities in 2D scans of biological tissue. Another, from materials science, is the classification of polycrystalline materials and the prediction of their physical properties based on images of their microstructures. In both applications, the local topology and geometry --- namely the shape and arrangement of cells/grains --- are thought to be essential. However, information about these features may be lost in traditional machine learning pipelines such as those involving convolutional neural nets (CNNs). In this talk, I will discuss two methods to represent the geometry of micrographs in formats amenable to machine learning. The first augments images of biological tissue with additional fields representing the persistent homology of local windows. The second represents the grain structure of a polycrystal as a metric measure space of local configurations.

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Sequential topological complexity of symmetric products of surfaces — Ekansh Jauhari Icon: submission_accepted

Sequential topological complexities (denoted TCm for each m≥2) are numerical homotopy invariants of topological spaces motivated by the motion planning problem in robotics. Given a space X, TCm(X) measures the discontinuity of planning a motion between any given sequence of m points in X for any robot whose configuration space is X. Usually, cohomological data of a space helps estimate its TCm values. In this talk, we focus on the case when our space X is a symmetric product of a closed orientable surface. Using Macdonald's description of the cohomology ring of these spaces, we completely determine all sequential topological complexities of all symmetric products of closed orientable surfaces. Our methods involve explicit computations of their Lusternik--Schnirelmann category and rational zero-divisor cup-lengths. Using our computations, we also verify the “TC-rationality conjecture" of Farber and Oprea for these spaces.

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Sheaf Cohomology and the Algebraic Path Problem — Kaelyn Willingham Icon: submission_accepted

Routing problems in computer science often involve computing efficient routes for moving entities between different points in some defined space. Given a semi-ring R and a graph G, the Algebraic Path Problem provides a unifying framework for analyzing various routing problems mathematically by abstracting the notion of combining weighted paths on G under the additive operation defined on R. Routing problems defined on planar graphs are fairly understood, but these same problems remain elusive when defined on more-complex topological structures. In this talk, I will discuss current work that utilizes sheaf cohomology to understand the nature of routing problems defined on cellular complexes. In so doing, we will find a nice generalization of the Algebraic Path Problem. This is joint work with Russell Funk and Thomas Gebhart.

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Squares inscribed in ray compactifications — Ulises Morales-Fuentes Icon: submission_accepted

In this talk we will prove that ray compactifications in the plane admit inscribed squares provided that their remaider is a piecewise linear simple triod. Also we will show visualizations of sections of Vaughan's function implemented in python and visualized with Ipyvolume. We will discuss how this visualizations have proven to be very useful in finding squares inscribed in plane sets.

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Topological Feature Selection for Time Series Data — Johnathan Bush Icon: submission_accepted

I will describe how tools from applied topology may be used to identify components of time series most responsible for cyclic dynamics observed in orbits of an underlying dynamical system. In this setting, I will show that derivatives of the persistent homology may be computed explicitly and describe a simple algorithm for gradient descent. As an example, we will consider neuronal data from the model organism C. elegans and identify subsets of neurons driving global cyclic brain dynamics in the spirit of dimensionality reduction.

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Unmapped Territory for Topological Data Analysis: Mathematics Education — Devin Hensley Icon: submission_accepted

Topological data analysis has been used to analyze social systems, disease spread, and polling locations, but it is not typically used in mathematics education research. In this study, university calculus students created concept maps–visual representations of connections–which were analyzed using homology groups. We argue that homology is an innovative and useful tool for analyzing concept maps, complementing previous analyses conducted via qualitative techniques or scoring systems. We further argue that topological data analysis can be a valuable tool for mathematics education research.

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Unveiling Topological Structures in Text & Speech — Adaku Uchendu Icon: submission_accepted

The surge of data available on the internet has led to the adoption of various computational methods to analyze and extract valuable insights from this wealth of information. Among these, the field of Machine Learning (ML) has thrived by leveraging data to extract meaningful insights. However, ML techniques face notable challenges when dealing with real-world data, often due to issues of imbalance, noise, insufficient labeling, and high dimensionality. To address these limitations, some researchers advocate for the adoption of Topological Data Analysis (TDA), a statistical approach that discerningly captures the intrinsic shape of data despite noise. Despite its potential, TDA has not gained as much traction within the Natural Language Processing (NLP) domain compared to structurally distinct areas like computer vision. Nevertheless, a dedicated community of researchers has been exploring the application of TDA in NLP, yielding 93 papers we comprehensively survey in this paper. Our findings categorize these efforts into theoretical and non-theoretical approaches. Theoretical approaches aim to explain linguistic phenomena from a topological viewpoint, while non-theoretical approaches merge TDA with ML features, utilizing diverse numerical representation techniques. We conclude by exploring the challenges and unresolved questions that persist in this niche field.

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