Conference Schedule
Legend
Room 1 (Strickland Hall 114) hosts all keynote and invited talks; Room 2 (Strickland Hall 117) is used only for parallel sessions. Poster session takes place in Math & Sciences Building 306.
Keynote Speaker
▶ Facundo Mémoli (Rutgers University) — The Gromov-Hausdorff distance between spheres
Saturday, November 22, 9:00 – 10:00
Abstract: A very natural question in geometry and in many applications is: How can we quantify how different two shapes are? One way to do this is by invoking distances between metric spaces, such as the Gromov–Hausdorff distance, which is used in areas like Riemannian geometry and data analysis. Nevertheless, exact values are known only in a few special cases. For finite metric spaces, the exact computation of this distance (and closely related distances) is NP-hard. Despite this hardness, practitioners develop heuristic algorithms to estimate it, which in turn motivates the search for theoretically determined benchmarks against which to evaluate such methods.
In this talk, I will focus on the case of unit round spheres equipped with the geodesic metric. I will describe how to obtain lower bounds for the Gromov–Hausdorff distance between spheres and, in certain cases, establish sharpness by constructing optimal correspondences, thereby determining the exact value of dGH for the corresponding pairs of spheres.
Many challenges remain: for most pairs of spheres, the exact value of the distance is still unknown, and even the right techniques for tackling these cases are not yet clear. This makes the problem a rich source of open questions at the intersection of geometry, topology, and data analysis. Interestingly, these results connect with a classical topological result called the Borsuk–Ulam theorem—but in the unusual setting where we must deal with discontinuous functions.
▶ Facundo Mémoli (Rutgers University) — The G-Gromov-Hausdorff distance and Equivariant Topology
Sunday, November 23, 9:00 – 10:00
Abstract: In this talk, for a finite group G, we consider G-metric spaces: metric spaces equipped with an isometric G-action. We introduce a G-equivariant Gromov–Hausdorff distance for compact G-metric spaces and derive lower bounds using equivariant persistent invariants and related constructions in equivariant topology. To analyze and compare these bounds, we further develop two complementary G-equivariant distances—the homotopy-type and interleaving distances—and establish stability relations linking them to the G-Gromov–Hausdorff distance.
As applications: (1) we analyze how the G-actions descend to and enrich persistence modules and obtain lower bounds via the G-interleaving distance, comparing these to those induced by equivariant topology; (2) we prove equivariant rigidity and finiteness theorems; (3) we obtain sharp bounds on the Gromov–Hausdorff distance between spheres; and (4) we obtain a G-equivariant quantitative Borsuk–Ulam theorem.
Invited Speakers
▶ Saugata Basu (Purdue University) — Harmonic persistent homology
Sunday, November 23, 10:30 – 11:30 (50 min)
Abstract: I will define harmonic persistent homology, which is a means to decorate the barcode of a filtration of a finite simplicial complex with additional data.
I will discuss our stability results which use properties of singular values of real matrices.
Finally, I will also mention an important property of harmonic representatives related to “essential simplices” and state some conjectures for future work.
(Joint work with Nathanael Cox.)
▶ Ryu Hayakawa (Kyoto University) — Computational complexity of the homology problem with orientable filtration: MA-completeness
Sunday, November 23, 15:00 – 15:30 (30 min)
Abstract: We show the existence of an MA-complete homology problem for a certain subclass of simplicial complexes. The problem is defined through a new concept of orientability of simplicial complexes that we call a "uniform orientable filtration", which is related to sign-problem freeness in homology. The containment in MA is achieved through the design of new, higher-order random walks on simplicial complexes associated with the filtration. For the MA-hardness, we design a new gadget with which we can reduce from an MA-hard stoquastic satisfiability problem. Therefore, our result provides the first natural MA-complete problem for higher-order random walks on simplicial complexes, combining the concepts of topology, persistent homology, and quantum computing. (This is based on https://arxiv.org/abs/2510.07014)
▶ Vasileios Maroulas (University of Tennessee Knoxville) — Simplicial convolutional recurrent neural network
Saturday, November 22, 11:30 – 12:00 (30 min)
Abstract: The brain's spatial orientation system relies on specific neuron groups for navigation, such as head direction cells for orientation and grid cells for mapping environments. These neurons work together in patterns, firing simultaneously to create directional and positional signals. To better understand and decode these patterns, we developed a new topological deep learning model that goes beyond traditional graph-based approaches. Our model, a simplicial convolutional recurrent neural network, uses topological structures to capture complex neural relationships. This method allows us to predict head direction and location from neural data without needing prior similarity measures, proving effective in head direction and trajectory prediction.
▶ Erik Amezquita Morataya (University of Missouri-Columbia) — Exploring the mathematical shape of plants
Monday, November 24, 9:00 – 10:00 (50 min)
Abstract: Shape is foundational to biology. Observing and documenting shape has fueled biological understanding as the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. To comprehensively quantify the vast morphological diversity in biology, we focus thus on Topological Data Analysis (TDA). TDA is an emerging mathematical discipline that uses principles from algebraic topology to comprehensively measure shape in a broad scope of different datasets. As a proof of concept, we explore a series of applications of TDA in plant biology with a variety of data inputs.
With the Euler Characteristic Transform and X-ray CT scans of a wide array of barley varieties, we can predict genomic markers based solely on shape characteristics of their seeds alone. With persistent homology and Molecular Cartography technology, we can model patterns of mRNA spatial localization for different genes, cell types, and organs of the soybean root and nodule. With mapper and RNAseq data, we can uncover novel lung cancer subtypes. With adjacency complex filtrations, we can describe the intricate patterns made by pavement cells in arabidopsis leaves.
The vision of TDA, that data is shape and shape is data, will be relevant as biology transitions into a data-driven era where meaningful interpretation of large datasets is a limiting factor.
▶ Laxmi Parida (IBM) — Harmonic Persistent Homology on genomic data
Sunday, November 23, 11:30 – 12:00 (30 min)
Abstract: I will discuss Harmonic persistent homology and discuss its application to omics data. I will share our results of this exercise that uncovered hidden patterns highlighting the relationships between different omic profiles.
▶ Emilie Purvine (Pacific Northwest National Laboratory) — A Topological Approach to Fuse Structure and Similarity in Clustering
Sunday, November 23, 14:00 – 15:00 (50 min)
Abstract: While network science based approaches have proven to be surprisingly effective across a variety of application domains, they are fundamentally limited by the inability to incorporate domain information and metadata in a mathematically principled way. This is especially apparent when attempting to cluster real-world networks, where oftentimes the failure to identify the correct cluster membership can be traced to a misalignment between the combinatorial structure and metadata relationships influencing the desired clusters.
Our team has recently introduced the inner product Laplacian (IPL) which provides a mathematically grounded approach to address this lack and inject domain-awareness into spectral methods by incorporating inner product spaces (similarity measures) on both the nodes and edges that are informed by additional metadata. The construction of the IPL relies on the topological (chain complex) definition of the graph Laplacian. We construct a chain complex on the node and edge inner product spaces whose chain maps incorporate the graph structure boundary maps from the graph chain complex. The traditional graph Laplacians (combinatorial, normalized, and their weighted variants) all become special cases of the IPL with appropriate choices of inner products. Moreover, since this construction is fundamentally chain complex-based it can be generalized to arbitrary simplicial complexes.
This talk will cover IPL theory, generalizations of some traditional spectral graph results to the IPL context, and an example use case in the cyber domain where multiple log types that are traditionally analyzed separately can be brought together into a single IPL.
▶ Yusu Wang (University of California San Diego) — Discrete Morse based Graph Reconstruction: Algorithms and Applications
Saturday, November 22, 10:30 – 11:30 (50 min)
Abstract: Recent years have witnessed a surge in the use of topological objects and methods in various applications. Many such applications leverage either the summarization (e.g, persistent homology) or the characterization power of topological objects. In this talk, we will talk about our graph skeletonization algorithm based on discrete-Morse theory, both for 2D / 3D images and for (high-dimensional) points data. We will then describe various applications of the resulting algorithms: from automatic road network reconstruction, to neuronal processes segmentation from whole brain images, to the analysis of high dimensional single-cell RNASeq data. This is joint work with many collaborators which we will acknowledge in the talk.
Parallel Talks Details
Slot 1 (14:00 – 14:30, Saturday)
▶ Benjamin Jones (Michigan State University) — Efficient Computation of Persistent Topological Laplacians: Toward Broader Applications in TDA
Abstract: Persistent Topological Laplacians (PTLs) are both the extension of the graph Laplacian to filtered simplicial complexes and a discrete version of the Hodge Laplacian. Their spectra provide multiscale geometric and topological information that cannot be detected by other methods, such as persistent homology. This additional information comes at a significant computational cost, making large scale data analysis challenging. To improve the scale and broaden the impact of PTL applications, we introduce several computational techniques as part of the open-source Persistent Topological Laplacian Software (PETLS), including a novel algorithm using the topology of the underlying complex.
▶ Minh Quang Le (Ho Chi Minh city Open University) — Topological Data Analysis of multiplex Markov chains encoding brain activity
Abstract: Markov chains (MCs) are widely used to construct models in the social, physical and biological sciences and engineer technologies for computing and communications. In recent work, we utilized tools from topological data analysis and persistent homology to automate the detection and summary of convection cycles that can arise for irreversible MCs. In the present work, we extend these methods to study convection cycles arising for multiplex Markov chains (MMCs) that are constructed by coupling together sets of MCs. MMCs are closely related to multilayer networks and multiplex networks, whereby network "layers" are interconnected. In the context of MMCs, convection cycles can arise within and/or across MC "layers" and that the strength ω of coupling between MCs acts as a homological "regularizer" for these cyclic flows. Motivated by applications in neuro-AI, we utilize this mathematical framework to develop homological characterizations for fMRI-recorded human brain activity and reinforcement learning.
Slot 2 (14:30 – 15:00, Saturday)
▶ Manuel M. Cuerno (CUNEF Universidad) — On the Gromov--Hausdorff distance for metric pairs and tuples
Abstract: Gromov–Hausdorff-type distances play a key role in computational settings such as Topological Data Analysis. In this talk, we will introduce the Gromov–Hausdorff distance for metric pairs and tuples, outlining its definition, key properties, and relevant results. We will also discuss concrete scenarios in which this distance naturally arises. The presentation is based on the following preprint: arXiv:2505.12735.
▶ Mao Li (Donald Danforth Plant Science Center) — Application of Topological Data Analysis in Plant Science
Abstract: With advances in imaging technologies, we can now generate vast amounts of image data every day. While 2D imaging remains highly accessible, 3D imaging enables the capture of more intact structures. We are making efforts to extract and utilize as much meaningful information from this data as possible. Topological data analysis (TDA) has emerged as a flexible and powerful tool and has been widely applied in plant science. It can be tailored to quantify diverse plant morphology. In this talk, I will present two examples of TDA application in plant science, with a focus on investigating grass morphological evolution. Grasses are crucial crop plants that provide food, building materials, and biofuels. Pollen and phytolith (microscopic silica body) are commonly used to reconstruct past vegetation. By applying TDA to microscope images, we measure the surface patterns of pollen in 2D and the complex shapes of phytolith in 3D. These highlight the value of TDA in uncovering novel insights into plant form and evolution.
Slot 3 (15:30 – 16:00, Saturday)
▶ Cagatay Ayhan (Florida State University) — Equivalence of Landscape and Erosion Distances for Persistence Diagrams
Abstract: This work establishes connections between three of the most prominent metrics used in the analysis of persistence diagrams in topological data analysis: the bottleneck distance, Patel's erosion distance, and Bubenik's landscape distance. Our main result shows that the erosion and landscape distances are equal, thereby bridging the former's natural category-theoretic interpretation with the latter's computationally convenient structure. The proof utilizes the category with a flow framework of de Silva et al., and leads to additional insights into the structure of persistence landscapes. Our equivalence result is applied to prove several results on the geometry of the erosion distance. We show that the erosion distance is not a length metric, and that its intrinsic metric is the bottleneck distance. We also show that the erosion distance does not coarsely embed into any Hilbert space, even when restricted to persistence diagrams arising from degree-0 persistent homology. Moreover, we show that erosion distance agrees with bottleneck distance on this subspace, so that our non-embeddability theorem generalizes several results in the recent literature.
▶ Guangyu Meng (University of Notre Dame) — A Stable Algorithmic Framework for Reeb Graph Comparison via Gromov-Wasserstein Distance
Abstract: Reeb graphs and Mapper graphs are a commonly used tool in shape analysis. Essentially, these allow for a simplified representation of scalar field data as a labeled graph which captures the topology of the original structure, allowing for better computational tools and visualization. As a result, comparing Reeb graphs is a fundamental problem in computational topology and geometry. The Gromov-Wasserstein (GW) distance offers a powerful approach for this task, but a critical challenge remains: the method's reliability depends entirely on the choice of a metric and a measure for the graph nodes. Existing heuristics for defining this structure lack formal stability guarantees, making any comparison based on them potentially unstable and unreliable for noisy data. This paper directly addresses this stability problem. We introduce a novel framework that equips Reeb graphs with a provably stable metric-measure structure for GW-based comparison. Our framework is built on two fundamental pieces: (1) a Symmetric Reeb Radius, which defines a robust intra-graph metric, and (2) a Persistence Image-based probability measure, which provides a stable, topologically significant weighting of the nodes. Our main theoretical contribution is a proof that the resulting distance is stable with respect to the L∞-norm of the generating scalar fields, i.e., RGWₚ(Rf*, Rg*) ≤ C₁ ‖f - g‖∞^(1/p) + C₂ ε^(1/p). This guarantee, previously missing in the literature, certifies the robustness of our approach. Computational experiments validate our theoretical predictions and demonstrate the practical advantages of principled metric-measure design for graph comparison algorithms.
Slot 4 (16:00 – 16:30, Saturday)
▶ Wanchen Zhao (University of Florida) — Wasserstein Stability of Barcodes and Persistence Landscapes
Abstract: Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the p-Wasserstein distances. However, the p-Wasserstein distances depend on a choice of a metric on the set of interval modules, and there is no canonical choice. One convention is to use the length of the symmetric difference between 2 intervals, which is equivalent to the L¹ norm of the difference between their Hilbert functions. We propose a new metric for interval modules based on the rank invariants instead of the dimension invariants. Our metric is topologically equivalent to the metrics induced by ℓᵖ on ℝ² for p ∈ [1,∞]. We establish stability results from filtered CW complexes to barcodes, as well as from barcodes to persistence landscapes. In particular, we show that vectorization via persistence landscapes is 1-Lipschitz with a sharp bound, with respect to the 1-Wasserstein distance on barcodes and the L¹ norm on landscapes.
▶ Collin Kovacs (University of Tennessee Knoxville) — A Topological Framework for Quantifying Order in Complex Physical Systems
Abstract: Twisted bilayer materials exhibit rich interference patterns whose periodicity heavily depends on twist angle. We present a topological framework for quantifying this periodic order using persistent homology. By leveraging Wasserstein distances between persistence diagrams (PDs) of untwisted and twisted lattices, we measure topological deviations across angles. We introduce δ-separated PDs, a compressed representation that removes redundant local motifs while preserving key topological features. δ-separation sharpens the topological signal, revealing sharp minima in referential Wasserstein curves at commensurate angles. To connect theory and simulation, we implement a warm-start Gaussian Process (GP) model that extrapolates topological trends from theoretical lattices to sparse and noisy experimental data, yielding uncertainty-aware predictions. Together, δ-separation and warm-start GP models establish a unified topological and statistical framework, linking theoretical periodicity to experimental structure in moiré materials.
Slot 5 (10:30 – 11:00, Monday)
▶ Ling Zhou (Duke University) — Persistent Cup-Length: A New Invariant for Topological Structure Detection
Abstract: Persistent cohomology enhances persistent homology by incorporating the cup product, introducing a graded ring structure that encodes higher-order topological information. We define the persistent cup-length, an invariant that tracks the evolution of nontrivial cup products across a filtration. This captures interactions between cohomology classes, offering insights beyond what homology alone can provide. We present a polynomial-time algorithm for computing this invariant from representative cocycles and prove its stability under interleaving-type distances. We also provide the first practical implementation, optimized through integration with Ripser and landmark subsampling. Applied to neural population data, our method successfully detects toroidal structure in neural manifolds, which persistent homology alone fails to reliably identify.
▶ Hayden Everett (The University of Tennessee Knoxville) — Bayesian Topological Convolutional Neural Nets
Abstract: Convolutional neural networks (CNNs) have been established as the main workhorse in image data processing; nonetheless, they require large amounts of data to train, often produce overconfident predictions, and frequently lack the ability to quantify the uncertainty of their predictions. To address these concerns, we propose a new Bayesian topological CNN that promotes a novel interplay between topology-aware learning and Bayesian sampling. Specifically, it utilizes information from important manifolds to accelerate training while reducing calibration error by placing prior distributions on network parameters and properly learning appropriate posteriors. One important contribution of our work is the inclusion of a consistency condition in the learning cost, which can effectively modify the prior distributions to improve the performance of our novel network architecture. We evaluate the model on benchmark image classification datasets and demonstrate its superiority over conventional CNNs, Bayesian neural networks (BNNs), and topological CNNs. In particular, we supply evidence that our method provides an advantage in situations where training data is limited or corrupted. Furthermore, we show that the new model allows for better uncertainty quantification than standard BNNs since it can more readily identify examples of out-of-distribution data on which it has not been trained. Our results highlight the potential of our novel hybrid approach for more efficient and robust image classification.
Slot 6 (11:00 – 11:30, Monday)
▶ Cheng Xin (Rutgers University) — TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration
Abstract: Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields, yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently, intrinsically interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TOPING: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TOPING employs a rationale filtration learning approach to model an autoregressive generation process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TOPING's effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-art methods on both predictive accuracy and interpretation quality.
▶ Layal Bou Hamdan (University of Tennessee Knoxville) — Bayesian Sheaf Neural Networks
Abstract: Equipping graph neural networks with a convolution operation defined in terms of a cellular sheaf offers advantages for learning expressive representations of heterophilic graph data. The most flexible approach to constructing the sheaf is to learn it as part of the network as a function of the node features. However, this leaves the network potentially overly sensitive to the learned sheaf. As a counter-measure, we propose a variational approach to learning cellular sheaves within sheaf neural networks, yielding an architecture we refer to as a Bayesian sheaf neural network. As part of this work, we define a novel family of reparameterizable probability distributions on the rotation group SO(n) using the Cayley transform. We evaluate the Bayesian sheaf neural network on several graph datasets, and show that our Bayesian sheaf models achieve leading performance compared to baseline models and are less sensitive to the choice of hyperparameters under limited training data settings.