Finite topological spaces offer a natural combinatorial setting for investigating topological structures and phenomena. In this work, we introduce a novel distance-like notion, called furtherness, for finite topological spaces. We show that every finite space can be endowed with the structure of an asymmetric pseudometric space, and particularly, in the T_0 case, this structure is an asymmetric metric. Moreover, the topology generated by the forward balls agrees with the original topology, whereas the backward balls generate its opposite topology. We further encode the resulting asymmetric metric structure in a furtherness matrix, which provides a useful representation of the underlying topological structure. 

Building on this framework, we associate persistence diagrams with individual elements through the furtherness function, providing a new method for classifying elements according to their topological evolution. We conclude by establishing a bottleneck stability theorem: the bottleneck distance between the persistence diagrams of two elements is bounded above by their mutual furtherness.

Topological Data Analysis (TDA) is a relatively recent area of applied mathematics to comprehensively describe shapes and patterns based on results from algebraic topology. TDA provides a powerful framework to quantify the global structure of a dataset by transforming it into a topological object of interest such as a simplicial complex. TDA then describes such a dataset as a collection of topological features that persist across a given filtration function, capturing both global and local salient shape features. Traditional, distance-based TDA constructions of simplicial complexes often disregard adjacency relationships between neighboring data points. However, when the data consists of plant cells across leaf or sepal tissue, neighbor interactions are essential for accurately describing intracellular communication and function. In this work, we focus on the geospatial arrangement and distribution of cells since it preserves geometric information necessary to describe global structure. We apply an adjacency filtration to compute the persistent homology of Arabidopsis thaliana leaf and sepal cells based on microscopy images. We then analyze the resulting persistence objects using persistence diagram distance metrics and dimensionality reduction techniques. Our results characterize distinctions between the spatial arrangement of real and synthetic leaf and sepal cells. Joint work with: Erik Amezquita, Tyler Mitts.

Birdsong is a model for studying the acquisition and development of language. Manually parsing hours of song into its basic units, syllables, is a major bottleneck that slows down birdsong research and can introduce bias. To address this issue, we present CHURP, a fully unsupervised framework for the segmentation and classification of birdsong syllables. CHURP utilizes a hybrid convolutional-LSTM autoencoder to encode audio spectrograms into latent trajectories that exhibit loop-like structures corresponding to discrete syllables. The trajectories are segmented based on their spatial deviation from a quiet ambient baseline and are subsequently encoded using time-augmented path signatures, which capture the geometric and temporal dynamics of each vocalization. The embedded syllables are then clustered with HDBSCAN to identify unique groupings and infer a dictionary of syllable types. Compared to the current state-of-the-art unsupervised model, TweetyBERT, CHURP distinctly isolates syllables from background noise, achieving significantly lower frame error rates and higher Jaccard Index—indicating closer agreement with ground-truth syllable durations. Furthermore, CHURP demonstrates strong accuracy in recovering the underlying syntactic structure of the songs by inferring syllable transition probability matrices based on the relative frequencies of ordered pairs of syllables occurring in sequence. This is joint work with Levi Svaren (UW-Madison), Samuel Brudner (Yale) and Dhananjay Bhaskar (UW-Madison).

Estimating the dynamic aperture of circular particle accelerators typically requires expensive long-term tracking. Using classical homology, we construct a two-dimensional theoretical classification of regular and chaotic orbits and use persistent homology for its numerical implementation on short orbit segments. In four dimensions, we introduce topological chaos indicators based on persistent homology and use normal-form theory to apply the classification to two symplectic-plane projections. The methods reproduce the expected organization of regular and chaotic motion in several canonical Hamiltonian systems. Applied to symplectified one-turn maps of a nonlinear lattice model of the Integrable Optics Test Accelerator, the results obtained from the first 200 turns identify the central connected stability region and its surrounding transition to chaos. They are consistent with the short-term dynamic aperture of the electron configuration obtained from $10^4$ turns and the long-term dynamic aperture of the proton configuration obtained from $10^8$ turns. For the configurations studied, persistent homology of short orbit segments predicts the dynamic aperture obtained by substantially longer tracking, potentially reducing the need for exhaustive simulations in studies of existing accelerators and the design of future facilities. The methods could also complement classical chaos indicators and modern approaches based on artificial intelligence.

Topological data analysis is a growing field of methodologies for analyzing data with shape structure, including point clouds, graphs, and images. In recent years, great strides have been made in the subfield of vectorization for topological data, wherein shape data is embedded in Euclidean space for comparison or machine learning. Vector summaries that distinguish topological structures effectively are inherently useful for shape classification models and unsupervised learning procedures like clustering. We consider the potential of vectorization for clustering topological data through the popular method known as k-means, which groups observations into a fixed number of classes by minimizing a squared-error loss function. We demonstrate that by exploiting the iterative structure of the k-means algorithm to interpose a step which updates element-wise weights on the vector summaries, we are able to attain improved clustering performance across a wide range of both simulated and real shape-structured data. We show that finding such weights is a least-squares problem, which poses a minimal computational burden and allows for the easy implementation of coefficient constraints. We also substantially improve the (negative) correlation between the loss value and clustering accuracy as measured through true cluster labels, improving the optimality of the k-means strategy which selects one of several possible clustering assignments based on lowest achieved loss. These results remain robust under changes to vector length and number of initializations. We further show that element-wise weighted topological vector summaries remain stable under small structural differences between data points, especially when constraints on the weights are employed. Our method can be recommended as an all-purpose alternative to clustering topological data through the usual algorithms, with flexibility in the type of k-means performed, choice of vectorization method, and implementation of weight coefficient constraints.

The study of persistent homology is largely built upon the structure theorem for persistence modules over a field, which guarantees a decomposition into simple interval modules. However, such a decomposition may fail to exist under more general coefficients. In this talk we show that, given a pointwise free and finitely-generated persistence module over a principal ideal domain, an interval decomposition exists if and only if the cokernel of every structure map is a free module. We first show the existence of an interval decomposition in the setting of finitely-indexed persistence modules, where we iteratively construct an interval decomposition one module at a time. We also formulate a polynomial-time algorithm that determines computes an interval decomposition or certifies that no decomposition exists. We then extend our work to the setting of infinitely-indexed persistence modules. 

This is joint work with Gregory Henselman-Petrusek.

In a recent paper, Adams et al. compare the Gromov-Hausdorff distance between a Riemann manifold and a subset to the Hausdorff distance between the two. They show that the Gromov-Hausdorff distance between the circle (equipped with the arc-length metric) and a subset is equal to the Hausdorff distance between the two, if the subset is dense enough. A follow-up paper on metric graphs includes an upgraded version of this result, together with a more elementary proof. In this poster, we replace the arc-length metric on the circle with the ambient Euclidean metric. We show that the Gromov-Hausdorff distance between the circle and a subset is strictly less than the Hausdorff distance between the two when one uses the ambient metric.

Given a manifold $\mathcal{M} \subset \mathbb{R}^2$ without boundary, the Radial Persistence Transform (RPT) assigns to each point $x \in \mathbb{R}^2$ the persistence diagram of the squared Euclidean distance function $d^2_{\mathbb{E}}(\cdot, x)$ restricted to $\mathcal{M}$. As $x$ moves along a closed loop $\gamma$, these diagrams form a periodic vineyard. The RPT is a variant of the persistent homology transform (PHT) introduced by Turner and collaborators, which is more sensitive, see Onus et al. Both the RPT and PHT can exhibit monodromy; after a full traversal of $\gamma$, points of the persistence diagram return permuted.

We are exploring the relation between the monodromy in a vineyard and the topology of the vineyard for the RPT on the one hand and the symmetry and focal set (also known as the bifurcation set) on the other. Classical results of Bruce, Giblin, and Gibson, and of Porteous, show that the Morse decomposition of the distance function changes only when $x$ crosses the symmetry set or the focal set of $\mathcal{M}$.

Here we present Bouquet2D, an interactive web-based tool for exploring these phenomena. The user defines $\mathcal{M}$ as a spline curve and draws a loop $\gamma$, and the tool displays the symmetry and focal sets of $\mathcal{M}$ together with the vineyard traced out as $x$ moves along $\gamma$. Deforming $\mathcal{M}$ updates all of these in real time, so one can directly observe how bifurcations of the symmetry and focal sets appear as events in the vineyard, and where monodromy arises. Bouquet2D is intended as an exploratory aid for studying the geometric origins of vineyard topology.

Understanding the three-dimensional organization of the genome through high-throughput chromosome conformation capture (Hi-C) is critical for studying gene regulation. However, identifying topologically associating domains (TADs) from Hi-C contact maps can be hindered by experimental noise, resolution dependence, and arbitrary scale selection. HICTDA, an open-source computational framework, applies persistent homology to the detection of TADs and higher-order chromatin features. Normalized contact matrices are transformed into distance representations to construct Vietoris–Rips filtrations across spatial scales. Persistent H₀ and H₁ features are then used to characterize genomic connectivity and one-dimensional topological features arising from TAD-associated contact structure. Persistence diagrams and barcodes quantify the lifetime of these topological features across the filtration, reducing reliance on predefined domain boundaries and a single fixed window size. HICTDA provides a flexible, mathematically principled framework for extracting persistent structural signals from Hi-C contact maps and identifying chromatin domains through topological invariants.

This poster presents a research program that connects topological structures, intuitionistic logic, and computational reasoning. The first component develops the idea of treating truth not as an isolated scalar label, but as a geometric and relational structure. Within this perspective, uncertainty, partial information, and inference are studied through topological organization, transformations, and higher-order relations.

The second component is ongoing work on intuitionistic simplicial semantics. Abstract simplicial complexes are considered as structured information spaces, while faces and inclusions represent relations between partial states and possible extensions of information. The project investigates when geometric entailment can preserve the monotone information growth characteristic of intuitionistic reasoning. Soundness, completeness, the finite model property, and structural limitations are treated as questions under investigation rather than established results.

The third component is VeriLogos, an open-source Python research framework for structural dynamics and verification. It combines simplicial complexes, temporal filtrations, persistent homology, geometric logic, and modal reasoning. Its current application focus is topological market-regime detection, with an architecture extensible to other structural-reasoning tasks. Together, these components propose a common language linking formal constructive reasoning with applied algebraic topology and interpretable computational systems.

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces. 

In this poster, we focus on the construction of non-degenerate simplicial maps of degree $d\in \mathbb{Z}$ on $n$-spheres for $n\geq 2$. We develop a general method, based on connected sums and facet orientations, to construct simplicial maps of any prescribed degree $d \in \mathbb{Z}$ between triangulated spheres. 

We investigate the asymptotic behavior of $\Lambda(n,d)$, defined as the minimum number of vertices required for a triangulated $n$-sphere to admit a simplicial map of degree $d$ to $\Sp^n_{n+2}$, for $n \geq 3$ and $d \geq 1$. Consequently, we answer the question posed by Ryabichev: is $\limsup_{d \to \infty} \frac{(\lambda(n,d))^{\lfloor \frac{n+1}{2} \rfloor}}{d} \neq 0, for $n \geq 3$? In addition to vertex-minimal constructions, we obtain facet-minimal degree maps for large degrees. Specifically, for each $d \geq n^2 + 1$, we construct a simplicial map of degree $d$ from a triangulated $n$-sphere with $d(n+2)$ facets to $\Sp^n_{n+2}$, for $n \geq 3$. 

As an application of the constructions, we derive improved bounds on the covering type of Moore spaces. Finally, we conclude with several open questions that may be of independent interest.

Inverse folding seeks an amino acid sequence that will adopt a given protein structure. Most existing methods operate on static structures or conformational ensembles, without explicitly modeling protein motion over time. Because these dynamics are closely tied to function, incorporating them into inverse folding could enable the design of sequences with targeted dynamic and functional properties. Here we introduce SPINET, a spatiotemporal sheaf neural network that infers protein sequences from molecular dynamics trajectories. At each time step, SPINET represents the protein as a cellular sheaf over a residue interaction graph. Learnable restriction maps, conditioned on node and edge features, project neighboring residue representations into shared edge stalks, where their discrepancies drive sheaf-based message passing. A recurrent neural network then aggregates these spatial representations across time before predicting the amino acid sequence. On proteins from the mdCATH and ATLAS datasets, SPINET outperforms existing methods in sequence recovery and structural similarity, evaluated by folding predicted sequences with ESMFold and comparing them with ground-truth structures. These results highlight cellular sheaves as a promising framework for modeling dynamic geometric relationships in protein design.

Authors: Jens Lundsgaard*, Colin Mikulski*, Jenny Yan, Dhananjay Bhaskar. All affiliated with UW-Madison.  *Equal contribution

Persistence diagrams summarize topological features of data and are used in machine learning via kernel methods. Kernel methods require a conditionally negative semidefinite (CSDN) function to build valid kernels. We study projected and reflected sliced transport distances between persistence diagrams. The projected diagram distance is a metric for every order. The reflected diagram distance is a metric only for order one. The sliced distances of order one are metrics and CSDN, so their exponentials define valid kernels. For higher orders, both sliced distances fail the triangle inequality; we prove this with an anchor method and an exact ratio between two limiting values. We also show that the projected sliced distance of order two is not CSDN, certified by interval arithmetic. Finally, on bounded diagrams, the sliced distances are bi-Lipschitz equivalent to the diagram distances.

Path homology is a homological invariant for directed graphs that retains information about edge orientation. More generally, it can be defined for path complexes, which provide a common language for digraphs, hypergraphs, and sequence hypergraphs. Persistent path homology studies how these invariants evolve across a filtration.

We prove a general stability theorem for the persistent path homology of filtered path complexes. Using correspondences between vertex sets, we introduce a Gromov–Hausdorff-type pseudometric d_path. For filtered path complexes P and Q, our main result states that, in every homological degree k, d_B(Dgm_k(P), Dgm_k(Q)) ≤ 2d_path(P,Q).

We also develop a covering framework that transfers this theorem to weighted combinatorial structures. As applications, we recover the known stability result for weighted digraphs and derive stability results for hypergraphs and sequence hypergraphs.

“The stability of Multiparameter persistent homology of Hypergraphs” is inspired by the paper of Jie Wu and Shiquan Ren on ""The stability of Persistent Homology of Hypergraphs"" in which embedded homology is defined to capture the homological feature of hypergraphs. As the name ""embedded homology"" suggests the homology of Hypergraph H is embedded between the homology of associated and lower associated simplicial complex of hypergraphs. This gave us the idea that to define the persistent homology of hypergraphs this embedded structure should travel “together” through the poset 𝑃 index to define the embedded homology of the hypergraph at each point. We think of this structure as the “𝐼-shaped diagram” in the category of hypergraphs HypGph and form the category of such functors HypGphI . We then define our filtration functor 𝐹:𝑃→HypGphI and the Persistent module 

𝑴:𝑷→𝑽𝒆𝒄𝒕𝑰

, where 𝑽𝒆𝒄𝒕𝑰 is the category of 𝐼-shaped diagrams in the category 𝑉𝑒𝑐𝑡. We then define the interleaving distance between such persistent modules, inspired by Michael Lesnick's paper ""The Theory of the Interleaving Distance on Multidimensional Persistence Modules,"" and prove the stability of multiparameter sublevel filtration and the multiparameter stability of morphism between hypergraphs. The second main result proves that the persistent maps of a morphism between two hypergraphs is stable with respect to perturbations of the filtrations on both the source hypergraph and the target hypergraph.

Characterizing the transition from regular to chaotic dynamics remains a fundamental challenge in nonlinear dynamical systems theory. This study introduces a topological machine learning framework based on Mapper graph representations that captures the geometric organization of dynamical trajectories directly from time series data. Unlike classical approaches that rely on Lyapunov exponents and bifurcation diagrams, our methodology transforms trajectories into topological graphs whose structural descriptors, including connectivity, clustering, and global complexity measures, systematically encode the underlying dynamical regime. The framework is validated across multiple systems: the numerical Rössler and Lorenz models, as well as experimental data from an erbium-doped fiber laser. In all cases, the resulting Mapper graphs exhibit progressive structural reorganization along the period-doubling route to chaos, with metrics such as the number of edges, Estrada index, and clustering coefficient increasing monotonically while graph diameter and average path length decrease. A key contribution is the identification of a universal threshold in the normalized Estrada index that distinguishes periodic from chaotic dynamics across both numerical and experimental systems. The Mapper-based framework consistently captures essential dynamical signatures across numerical and experimental systems, offering a robust and interpretable approach for analyzing structural transitions and complexity in nonlinear systems, even in the absence of conventional indicators.

Vietoris–Rips homology is a fundamental tool for extracting topological information from metric data, but its simplicial-complex formulation can make direct chain-level arguments difficult. Skeletal homology provides a more metric and hands-on chain model at a fixed scale \varepsilon, built from simplices whose images have diameter less than \varepsilon. We prove that, for every metric space X, K_n^\varepsilon(X)\cong H_n^\Delta(\mathrm{VR}_\varepsilon(X)), 

extending Goldfarb’s result from finite metric spaces to the general setting. This chain-level viewpoint also gives a direct route to classical topology-recovery results. Hausmann’s theorem shows that, for a Riemannian manifold M, \mathrm{VR}_\varepsilon(M) has the homotopy type of M for sufficiently small \varepsilon, while Latschev’s theorem gives a corresponding recovery result for sufficiently dense finite approximations of M. By adapting classical methods from singular homology, we obtain short, direct homological proofs of strong generalizations of both results. Our arguments work directly with chains and homology, rather than by constructing a homotopy inverse, and apply in settings where the corresponding homotopy-type statement is not known.

The Euler Characteristic Transform (ECT) is a popular topological transform that encodes “shapes” (well-behaved subsets of euclidean space) as a collection of integer-valued curves. In this poster, we present some links between the ECT and classical constructions in convex geometry, namely, the support function and the Steiner point of a convex shape. By exploiting this, it is possible, for instance, to easily compute the perimeter of a 2D shape given its ECT. Moreover, these connections serve as inspiration for new ECT-induced pseudodistances, which present different advantages and drawbacks than classical ECT-induced distances. The poster is based on the author's arxiv preprint of the same name: https://arxiv.org/abs/2607.28021

Repeated patterns occur all over in biological sequences. The same motifs recur within and across DNA, RNA, and proteins as tandem repeats, microsatellites, transposable elements, and reused regulatory and structural elements. Accurately modeling such repetition is central to sequence analysis, but remains challenging for modern learning methods. While transformer architectures can capture long-range dependencies, they do not explicitly account for recurrence or cyclic structure, which are key features in many sequences. Here we introduce a topological framework for learning on sequences with repetition using zigzag persistent homology. We view sequence traversal as a walk on a k-mer graph, where a sliding window moves through the sequence, drawing edges between the consecutive k-mers. This construction detects and tracks repeated patterns through the appearance and persistence of loops. Building on this idea, we develop ZZFormer, a transformer architecture that incorporates zigzag topological structure directly into token embeddings, enabling end-to-end learning on variable-length sequences via cross-attention. This mechanism allows ZZFormer to reason directly about recurrence and repetition, rather than inferring such structure implicitly. This framework applies broadly to problems involving repeated or cyclic patterns. We demonstrate its effectiveness on tasks involving repetitive DNA, including classification of transposable element superfamilies and orders. This is joint work with Kritika Kumari* (UW-Madison), and Dhananjay Bhaskar (UW-Madison). * indicates co-first author. 

Face-number invariants are fundamental in combinatorial topology, with the invariant $g_2$ playing a central role. For a normal $d$-pseudomanifold $K$ ($d \ge 3$), it is known that $g_2(K) \ge g_2(\lk(v,K))$ for every vertex $v$. Moreover, if $K$ has at most two singular vertices and satisfies $g_2(K)=g_2(\lk(t,K))$ at a singular vertex $t$, then $g_3(K) \ge g_3(\lk(t,K))$. We call $K$ \emph{$g_2$- and $g_3$-optimal} if

$g_2(K)=g_2(\lk(t,K)) \quad \text{and} \quad g_3(K)=g_3(\lk(t,K)),$

for some singular vertex $t$.

In this article, we establish structural results for normal $4$-pseudomanifolds under $g_2$- and $g_3$-optimality conditions. We prove that if $K$ is a normal $4$-pseudomanifold with exactly one singular vertex $t$ and is $g_2$- and $g_3$-optimal at $t$, then $K$ can be obtained from the boundary complexes of $5$-simplices via a sequence of vertex foldings and connected sum operations.

If $K$ has exactly two singular vertices and is $g_2$- and $g_3$-optimal at one of them, then $K$ can be derived from the boundary complexes of $4$-simplices through a sequence of one-vertex suspensions, vertex foldings, and connected sums. 

Moreover, in this two-singularity case, we provide an alternative characterization. We prove that such a complex $K$ can also be obtained from the boundary complexes of $5$-simplices via a sequence of vertex foldings, edge foldings, and connected sums.

Standard pathway analysis methods rank genes by differential expression but ignore how those genes are connected and how that connectivity changes in disease. GenPath-PPH takes a different route: it encodes gene co-expression within directed KEGG pathway networks as a filtered point cloud, then tracks the evolution of connected components and directed cycles using persistent path homology. Applied to peripheral blood mononuclear cell RNA-seq data from hepatocellular carcinoma patients and healthy controls, the pipeline recovers known HCC-associated pathways and identifies biologically relevant ones, including NF-κB and sphingolipid signaling, that enrichment-based methods miss entirely. The gain comes from reading pathway topology rather than gene lists.

Persistent discrete homology (PDH) was recently suggested  as a noise-resistant alternative to simplicial homology, in particular for non-metric data. In degree one, discrete homology can be computed as the singular homology of the CW complex obtained by gluing in a 2-cell along every simple 3- and 4-cycle. This means that it only witnesses cycles of length at least 5, whereas simplicial homology of the clique complex starts at length 4. For the Vietoris-Rips graph, this represents most of the noise, especially on non-metric data where simple 4-cycles can persist arbitrarily long. However, since a graph has many more 4-cycles than 3-cycles, it is more computationally demanding, and no existing software was able to efficiently compute it.

I will present a new algorithm called RedZeD_PDH: a modification of RedZeD_VR to compute the PDH of a Vietoris–Rips filtration. RedZeD_VR was introduced by Kapulkin and Kershaw to compute simplicial homology of a Vietoris–Rips filtration via reduction to zero differentials. It boasts an improvement to previous algorithms by using active enumeration, which greatly reduces the number of enumerated birth simplices, often to under 1%. By applying this to the discrete setting, RedZeD_PDH avoids enumerating almost every square: the worst of our tests at n = 1000 enumerates 1.2% of all squares, with many other tests under 0.02%. This improvement allows PDH to become computationally viable for the first time. Furthermore, RedZeD_PDH performs particularly well on noisy and non-metric datasets, where discrete homology has the greatest benefits.

This is based off joint work with Chris Kapulkin and Nathan Kershaw.

Missing data imputation is commonly evaluated using pointwise reconstruction measures such as mean squared error, but these measures provide limited information about whether the broader structure of the data is preserved after imputation. We propose a complementary evaluation framework based on Topological Data Analysis (TDA) to assess structural preservation following missing-data imputation. The framework constructs Mapper representations of complete-case and imputed data, summarizes their topological features using extended persistence diagrams, and quantifies structural differences using bottleneck distance. A permutation-based inference procedure is then used to determine whether the observed topological discrepancy exceeds that expected under random group assignment across a predefined Mapper parameter grid. The framework was evaluated across datasets with different structural characteristics, under multiple missingness mechanisms, missingness levels, and imputation methods. The resulting topological assessments broadly agreed with conventional reconstruction-error measures in distinguishing weaker from better-performing imputation methods, while also showing that reconstruction accuracy and structural preservation do not always yield identical rankings, particularly under MNAR. These findings suggest that topological preservation provides a complementary dimension for evaluating missing-data imputation quality.