About

Professor Peter Bubenik leads a research group at the University of Florida focused on topological data analysis (TDA). His group studies a wide range of problems in TDA, emphasizing the development of mathematical tools and their theoretical foundations at the graduate level and beyond, while focusing on applications and computations at the undergraduate level. The research integrates algebraic topology with various mathematical disciplines such as linear algebra, general topology, metric geometry, commutative algebra, category theory, representation theory, functional analysis, measure theory, and probability. Additionally, the work connects crucially with statistics, algorithms, and machine learning, and extends to applications in biology, computer science, material science, physics, and other fields. Professor Bubenik's group has a strong record of mentoring graduate students and postdoctoral researchers who work on topics including metric thickenings, multi-parameter persistent homology, deep neural networks, stochastic topology, and mathematical biology. His former students and postdocs have gone on to positions in academia, industry, and research institutions, reflecting the group's impact on both theoretical and applied aspects of topological data analysis. The group actively encourages prospective students to engage with their research and contribute to advancing the theory and applications of TDA under Professor Bubenik's leadership.

Research topics

• Computer Science
• Pure mathematics
• Mathematics
• Mathematical analysis

Selected publications

• Stabilizing Localization of Representative cycles

  ArXiv.org · 2025-10-14

  preprintOpen accessSenior author

  We introduce the persistence heatmap, a parametrized summary based on representative cycles in persistence diagrams, designed to enhance stability and explainability in topological data analysis. Algorithms to compute persistence diagrams produce representative cycles and boundaries. These chains are difficult to use because they are unstable to perturbations of the input. Instead, we average to produce chains with real-valued coefficients. We prove Lipschitz stability and uniform continuity of our heatmap. Moreover, we use machine learning to learn a task-specific parametrization of the heatmap.

  PublisherOA PDFDOI

• Wasserstein Stability of Persistence Landscapes and Barcodes

  ArXiv.org · 2025-09-25

  preprintOpen accessSenior author

  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 equals to the 1-norm of the difference between their Hilbert functions. We propose a new metric for interval modules based on the rank invariant instead of the dimension invariant. Our metric is topologically equivalent to the metrics induced by the $p-$norms on $\R^2$. 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 1-norm on persistence landscapes.

  PublisherOA PDFDOI

• Continuous persistence landscapes

  ArXiv.org · 2025-11-28

  preprintOpen accessSenior author

  As the size of data increase, persistence diagrams often exhibit structured asymptotic behavior, converging weakly to a Radon measure. However, conventional vector summaries such as persistence landscapes are not well-behaved in this setting, particularly for diagrams with high point multiplicities. We introduce continuous persistence landscapes, a new vectorization defined on a special class of Borel measures, which we call q-tame measures. It includes both the persistence diagrams and their weak limits. Our construction generalizes persistence landscapes to a measure-theoretic setting, preserving the intrinsic structure of persistence measures. We show that this vector summary is bijective and L^1-stable under mild assumptions, and that the original measure can be uniquely reconstructed. This approach gives a more faithful description of the shape of data in the limit and provides a stable, invertible way to analyze topological features in large systems.

  PublisherOA PDFDOI

• Relative cell complexes in closure spaces

  Canadian Mathematical Bulletin · 2025-06-19 · 1 citations

  articleOpen access1st authorCorresponding

  Abstract We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree.

  PublisherOA PDFDOI

• A Schauder Basis for Multiparameter Persistence

  ArXiv.org · 2025-10-11

  preprintOpen access1st authorCorresponding

  Certain classes of multiparameter persistence modules may be encoded as signed barcodes, represented as points in a polyhedral subset of Euclidean space, we refer to as signed persistence diagrams. These signed persistence diagrams exist in the dual space of compactly supported, Lipschitz functionals on a polyhedral pair. In the interest of statistics and machine learning on multiparameter persistence modules, we aim to embed these signed persistence diagrams into Banach or Hilbert space. We use iteratively refined triangulations to define a Schauder Basis of compactly supported Lipschitz functionals. Evaluation of these functionals embeds signed persistence diagrams into the space of real-valued sequences. Furthermore, we show that in the larger space of relative Radon measures, the Schauder basis we have defined is minimal to induce an embedding.

  PublisherOA PDFDOI

• Exact weights and path metrics for triangulated categories and the derived category of persistence modules

  Journal of Pure and Applied Algebra · 2025-02-01

  article1st author
  PublisherDOI

• Topological data analysis of pattern formation of human induced pluripotent stem cell colonies

  Scientific Reports · 2025-04-04 · 2 citations

  articleOpen access

  Understanding the multicellular organization of stem cells is vital for determining the mechanisms that coordinate cell fate decision-making during differentiation; these mechanisms range from neighbor-to-neighbor communication to tissue-level biochemical gradients. Current methods for quantifying multicellular patterning tend to capture the spatial properties of cell colonies at a fixed scale and typically rely on human annotation. We present a computational pipeline that utilizes topological data analysis to generate quantitative, multiscale descriptors which capture the shape of data extracted from 2D multichannel microscopy images. By applying our pipeline to certain stem cell colonies, we detected subtle differences in patterning that reflect distinct spatial organization associated with loss of pluripotency. These results yield insight into putative directed cellular organization and morphogen-mediated, neighbor-to-neighbor signaling. Because of its broad applicability to immunofluorescence microscopy images, our pipeline is well-positioned to serve as a general-purpose tool for the quantitative study of multicellular pattern formation.

  PublisherDOI

• What is missing from this picture? Persistent homology and mixup barcodes as a means of investigating negative embedding space

  ArXiv.org · 2025-10-16

  preprintOpen access

  Recent work in the information sciences, especially informetrics and scientometrics, has made substantial contributions to the development of new metrics that eschew the intrinsic biases of citation metrics. This work has tended to employ either network scientific (topological) approaches to quantifying the disruptiveness of peer-reviewed research, or topic modeling approaches to quantifying conceptual novelty. We propose a combination of these approaches, investigating the prospect of topological data analysis (TDA), specifically persistent homology and mixup barcodes, as a means of understanding the negative space among document embeddings generated by topic models. Using top2vec, we embed documents and topics in n-dimensional space, we use persistent homology to identify holes in the embedding distribution, and then use mixup barcodes to determine which holes are being filled by a set of unobserved publications. In this case, the unobserved publications represent research that was published before or after the data used to train top2vec. We investigate the extent that negative embedding space represents missing context (older research) versus innovation space (newer research), and the extend that the documents that occupy this space represents integrations of the research topics on the periphery. Potential applications for this metric are discussed.

  PublisherOA PDFDOI

• Homotopy, homology, and persistent homology using closure spaces

  Journal of Applied and Computational Topology · 2024-07-01 · 6 citations

  article1st authorCorresponding
  PublisherDOI

• Topological and metric properties of spaces of generalized persistence diagrams

  Journal of Applied and Computational Topology · 2024-01-23 · 1 citations

  article1st authorCorresponding
  PublisherDOI

Frequent coauthors

• Peter T. Kim

  University of Guelph

  20 shared

• Gunnar Carlsson

  Stanford University

  17 shared

• Zhi‐Ming Luo

  Keimyung University

  17 shared

• Michael Hull

  9 shared

• Benjamin Whittle

  University of North Carolina at Chapel Hill

  9 shared

• Dhruv Patel

  9 shared

• Nikola Milićević

  Pennsylvania State University

  7 shared

• Jonathan Scott

  Cleveland State University

  7 shared

Labs

• Peter Bubenik Research GroupPI

Education

• Ph.D.

  University of Toronto

  2003

• Other

  Ecole Polytechnique Federale de Lausanne (EPFL)

  2005

Awards & honors

• Invited Address at the Spring 2025 Southeastern Sectional Me…