An essential question for democracy is how to rigorously determine the likelihood that a congressional map has been gerrymandered. A number of state constitutions require districting plans to be "compact," yet no technical legal definition of compactness exists in this context, leaving us to contend with the oft-cited sentiment that "you know it when you see it." In this talk, I will introduce a novel compactness measure based on a geometric structure known as the medial axis. This skeleton-like structure has been shown to have strong ties to the science of how the human brain perceives and processes complex shapes, thus offering a mathematically rigorous version of “the eye test.” I will explain the construction of this metric in detail, and then compare it to a recent machine-learning-based compactness metric introduced by Kaufman, King, and Komisarchik (2021). Specifically, in this work we examine the performance of our measure and theirs in several case studies, including two states whose districting plans were especially contentious and the entire 2016 congressional district map. This is joint work with Ellen Gasparovic at Union College, and Jason D’Amico and Mushan Zhong, who were undergraduates at Union College at the time of their contributions.
View Submission
To help combine statistics and machine learning with multiparameter persistence, we would like to map signed barcodes to a Banach space or Hilbert space. We use iteratively refined triangulations to define a Schauder basis of compactly supported Lipschitz functionals. We prove that evaluation of these functionals embeds signed barcodes into sequence space via a map which is both linear, and Lipschitz with respect to the 1-Wasserstein distance. I will illustrate these results with examples for one-parameter persistence, two-parameter persistence, and the variant called mixup barcodes.
View Submission
In differential geometry, a metric surface $M$ is said to be an isometric filling of a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for all $x,y\in C$. Gromov's filling area conjecture from 1983 asserts that among all isometric fillings of the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov's conjecture has been verified if, say, $M$ is homeomorphic to the disk and in a few other cases, but it is still open in general. Admittedly, I'm not a differential geometer, so we consider instead a particular discrete version of Gromov's conjecture which is likely fairly natural to anyone who studies graph embeddings on arbitrary surfaces. We obtain reasonable asymptotic bounds on this discrete variant by applying standard graph theoretic results, such as Menger's theorem. These bounds can then be translated to the continuous setting to show that any isometric filling of the Riemannian circle of length $2\pi$ has surface-area at least $1.36\pi$ (the hemisphere has area $2\pi$). This appears to be the first quantitative lower-bound on Gromov's problem that applies to an arbitrary isometric fillings. (Based on joint work with Joe Briggs)
View Submission
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.
View Submission
Topological data analysis (TDA) has achieved remarkable success in molecular sciences over the past decade. Integrating TDA with deep learning has led to advances in drug design, materials discovery, protein engineering, and COVID‑19 research. However, many TDA tools rely heavily on persistent homology, which captures only limited aspects of the underlying algebraic and geometric structures. To move beyond these limitations, we develop new mathematical foundations that bridge pure mathematics with modern AI. Recently, we introduced a multiscale commutative algebra embedding that captures intrinsic physical and chemical interactions in molecular systems for the first time. Using Persistent Stanley–Reisner Theory, we extract algebraic invariants—including facet ideals and $f$‑vectors—to construct a Commutative Algebra Neural Network (CANet). Our approach integrates deep learning with rich algebraic information, producing AI models that are mechanistic, interpretable, and highly generalizable. I will present the mathematical framework of CANet and show how these descriptors reveal structural patterns underlying genetic disease–causing mutations, pushing TDA beyond persistent homology.
View Submission
Topological Data Analysis (TDA) has emerged as a powerful framework for extracting meaningful structure from complex, high-dimensional data. In particular, persistent homology is widely used for its ability to quantify multiscale topological features while exhibiting robustness to noise. In this work, we apply persistent homology to hyperspectral images of retinal tissue in order to further investigate Spaceflight Associated Neuro-ocular Syndrome. Hyperspectral imaging captures vast spectral information, but its high dimensionality poses challenges for analysis and interpretation. For each spectral band, we treat pixel intensity as a scalar function and construct a sublevel set filtration of cubical complexes, which provide a natural cell-complex structure for image data. From the resulting persistence diagrams, we derive summary statistics including total persistence and feature counts in dimension 0. Preliminary results indicate that these persistence-based summaries distinguish between pigmented and albino retinal tissue. Ongoing work focuses on further interpretation of the detected topological structure and the implementation of additional persistence-based methods.
View Submission
As Alzheimer’s disease pathology begins decades before clinical symptoms, early functional brain changes in asymptomatic Alzheimer’s disease (AsymAD) remain poorly characterized, particularly from a longitudinal perspective. Although AsymAD individuals are biomarker-positive for Alzheimer’s pathology, they remain cognitively unimpaired, representing a preclinical stage that is clinically silent yet biologically active. At this stage, conventional MRI markers and cross-sectional analysis often lack sensitivity to detect the subtle functional alterations that precede symptom onset. Consequently, compared to symptomatic Alzheimer’s disease, AsymAD is substantially more difficult to identify using imaging markers alone, necessitating more sensitive, network-level, and longitudinal approaches. Resting-state functional MRI provides a noninvasive framework for probing intrinsic functional brain networks and detecting early network-level disruptions. Since brain networks exhibit a small-world topology, defined by high local clustering (segregation) and short average path lengths (integration), graph-theoretical metrics sensitive to subtle perturbations related to these properties may provide early network-level signatures of AsymAD. In this study, we examine longitudinal changes in functional connectivity in AsymAD compared with cognitively normal controls using complementary connectivity approaches, including region-to-region connectivity (RRC), graph-theoretical metrics, and seed-based connectivity (SBC). General linear models are used to assess between-subject and repeated-measure effects, with primary emphasis on group-by-session interactions.
View Submission
We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups and discuss topological consequences. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. Of particular focus is the case of a toric arrangement: a finite collection of codimension-one subtori in a complex torus. If the intersection pattern of the subtori satisfies the combinatorial condition of supersolvability, the complement of the toric arrangement sits atop a tower of fiber bundles. This structure provides insight into topological invariants of these toric arrangement complements, including the homotopy groups, cohomology, and topological complexity. Based on joint work with Daniel C. Cohen and Emanuele Delucchi.
View Submission
View Submission
Advancements in sequencing technologies have produced a wealth of genomic data. In parallel, the development of artificial intelligence has enabled novel folding models that predict molecular structures from sequences. These advancements have resulted in a myriad of biomolecular structure data. Analytics of structure data offers more accurate approaches to genotype-to-phenotype analyses, as biomolecular structures are more evolutionarily conserved than sequences and more directly linked to biological functions. A major challenge of structure data analytics is the lack of efficient and accurate structure encodings. In this talk, we introduce encodings of RNA secondary structures using polynomial invariants of graphs. We show that the graph polynomial encodings enable efficient, accurate and interpretable RNA secondary structure analyses using modern data analytics tools.
View Submission
Gromov-Wasserstein (GW) distances provide a method for comparing probability measures defined on different metric spaces, thereby giving an optimal transport-inspired variant of the well-known Gromov-Hausdorff distance. As GW distances admit computationally tractable approximations, they have become popular in machine learning applications where one wishes to learn trends in a dataset consisting of incomparable spaces, such as ensembles of graphs. In this talk, I will overview recent advances in the theory of GW distances. In particular, I will discuss a certain approximation technique which relies on comparing the distributions of pairwise distances between metric measure spaces. This approach naturally gives rise to fascinating questions about the geometrical and topological features that are encoded in this distributional information, and I will explain some partial answers to these questions.
View Submission
The hyperoctahedral group is the group of symmetries of the hypercube graph. It acts on the Vietoris-Rips complexes of the hypercube graph with the Hamming distance and, therefore, on their homology groups. I will present a method to understand this action and show how it can be used as an alternative approach to compute homology. This is joint work with Jonathan Montaño and Zoe Wellner.
View Submission
We lower bound the Gromov-Hausdorff distance between Euclidean unit balls of different dimensions, $d_{GH}(B^m,B^n)$ for $m>n$. This is significant because the standard persistent homology lower bound is zero, since all balls possess trivial persistent homology. Our most powerful approach to lower bound the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions leverages the Borsuk-Ulam theorem. We exploit the fact that any continuous map between a sphere and a ball of appropriate dimensions must identify antipodal points. This yields a positive metric distortion and a computable lower bound.
View Submission
In "A Multicover Nerve for Geometric Inference" paper, Sheehy shows that filtering the barycentric decomposition of a Čech complex by the cardinality of the vertices recovers exactly the topology of k-covered regions among a collection of balls. We describe this construction and present ideas related to this.
View Submission
Skin cancer is a common and potentially fatal disease where early detection is crucial, especially for melanoma. Current deep learning systems classify skin lesions well, but they primarily rely on appearance cues and may miss deeper structural patterns in lesions. We present TopoCon-MP, a method that extracts multiparameter topological signatures from dermoscopic images to capture multiscale lesion structure, and fuses these signatures with Vision Transformers using a supervised contrastive objective. Across three public datasets, TopoCon-MP improves in-distribution performance over strong pretrained CNN and ViT baselines, and in cross-dataset transfer, it maintains competitive performance. Ablations show that both multiparameter topology and contrastive fusion contribute to these gains. The resulting topological channels also provide an interpretable view of lesion organization that aligns with clinically meaningful structures. Overall, TopoCon-MP demonstrates that multipersistence-based topology can serve as a complementary modality for more robust skin cancer detection.
View Submission
Sheaf Laplacians generalize graph Laplacians to vector-valued node signals, enabling richer relational models but increasing computational cost. We present a spectral sparsification method for the $0$-dimensional sheaf Laplacian using leverage-style edge sampling from trace effective resistance with reweighting. The resulting sparse operator preserves the original quadratic form on $(\ker L_{\mathcal F})^\perp$ with high probability: for $\varepsilon\in(0,1)$ and $p_{\mathrm{fail}}\in(0,1)$, we obtain a $(1\pm\varepsilon)$ approximation with probability at least $1-p_{\mathrm{fail}}$. This gives a principled path to faster sheaf diffusion and scalable sheaf-based learning, and supports empirical study of the sparsity--accuracy tradeoff through tunable sampling.
View Submission
The shadow of an abstract simplicial complex $\mathcal{K}$, whose vertices are in $\mathbb{R}^{N}$, is defined as the union of the convex hulls of its simplices. For a metric space $(S,d)$ at scale $\beta$, the Vietoris–Rips complex $\mathcal{R}_{\beta}(S)$ is the abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points in $S$ with a diameter at most $\beta$. Latschev's theorem provides a qualitative guarantee for manifold reconstruction: for any closed Riemannian manifold $X$, there exists a scale $\epsilon_{0}>0$ such that for any $0<\beta \leq \epsilon_{0}$, there is a $\delta >0$ where any metric space $S$ within Gromov–Hausdorff distance $\delta$ of $X$ yields a Vietoris–Rips complex homotopy equivalent to $X$. Recently, Latschev's theorem has been quantified, allowing $X$ to be a more general geodesic space. When $X\subset \mathbb{R}^{N}$ is a Euclidean geodesic space (e.g., submanifold, graph), we address the theorem's geometric analog: under what conditions is the shadow of the Vietoris–Rips complex of a Hausdorff-close Euclidean sample $S\subset\mathbb{R}^N$ both homotopy equivalent and Hausdorff-close to $X$? Unlike the abstract complex, the shadow provides a geometric embedding within the host space, which is essential for the practical reconstruction of Euclidean shapes. In this talk, we discuss recent developments in answering this question and explore their implications for faithful reconstruction of low-dimensional submanifolds and Euclidean-embedded graphs.
View Submission
For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $sh(K)$, is the union of the convex hulls of simplices of $K$. We consider the homotopy properties of the shadow of Vietoris--Rips complexes $K=Rips_\beta(X)$ with vertices from $\mathbb{R}^N$, along with the canonical projection map $p\colon Rips_\beta (X) \to sh(Rips_\beta(X))$. The study of the geometric/topological behavior of $p$ is a natural yet non-trivial problem. The map $p$ may have many "singularities", which have been partially resolved only in low dimensions $N\leq 3$. The obstacle naturally leads us to study systems of these complexes {$sh(Rips_{\beta}(S)) \mid \beta > 0, S\subset X$}. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from Shape Theory, we show that in the limit "$\beta \to 0$ and $S \to X$", the limit map "$\lim p$" behaves well with respect to homotopy/homology groups when $X$ is an ANR (Absolute Neighborhood Retract) and admits a metric that satisfies some regularity conditions. This results in limit theorems concerning the homotopy properties of systems of these complexes as the proximity scale parameter approaches zero and the sample set approaches the underlying space (e.g., a submanifold or Euclidean graph). This is joint work with Kazuhiro Kawamura and Sushovan Majhi.
View Submission
Topological data analysis (TDA) provides powerful tools for understanding the structure of complex, high-dimensional data, yet most existing methods focus on points, graphs, or simplicial complexes. In this talk, I will present our recently developed de Rham–Hodge–based frameworks for analyzing data on manifolds. These methods provide effective and efficient ways to capture both the topological and geometric information of data and are well-suited for integration with machine learning tasks. I will demonstrate their usefulness through applications in mathematical biology, including protein–ligand binding affinity prediction, single-cell RNA velocity analysis, medical image classification, and B-factor analysis.
View Submission
The Vietoris--Rips complex is a construction central to applied topology, including applications to geometric group theory, topological data analysis, and more. However, even for simple spaces such as spheres, their homotopy types are yet to be characterized. In this talk, I will present connectivity bounds for the Vietoris--Rips complexes of spheres $S^n$ in terms of covering properties of $\mathbb{R}P^n$. We leverage the connection to neighborhoods of $S^n$ in the tight span $E(S^n)$ (a.k.a hyperconvex hull) and tools from equivariant topology. These techniques generalize to the study of Vietoris--Rips complexes of antipodal metric spaces. This is joint work with Florian Frick.
View Submission