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Veranstaltungen
Algebraic Topology: Methods, Computation, and Science (ATMCS 12)

Lettering: "Algebraic Topology: Methods, Computation, and Science"

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

26.01.26 30.01.26

Algebraic Topology: Methods, Computation, and Science (ATMCS 12)

with a special day dedicated to Sayan Mukherjee

conference

26.01.26 30.01.26

Algebraic Topology: Methods, Computation, and Science (ATMCS 12)

with a special day dedicated to Sayan Mukherjee

The <a href="https://atmcs.appliedtopology.org/">ATMCS conference series</a> focuses on Algebraic Topology and its role and applications in Computation and Science. The Max Planck Institute for Mathematics in the Sciences in Leipzig is delighted to host the 12th conference in this series.

A key goal is to further develop topological techniques for practical applications and to explore new areas where they can be used. The conference will bring together leading researchers in this emerging field while also providing a valuable opportunity for young mathematicians to get involved.

This conference is closely associated with the name of Sayan Mukherjee, who was instrumental in the initial idea and development of the event and who sadly passed away in 2025. In his honor, the 28th of January will be a special day of remembrance integrated into the conference program.

group picture

Confirmed Plenary Speakers

Ulrich Bauer

TU Munich

Paweł Dłotko

Dioscuri Centre in Topological Data Analysis

Herbert Edelsbrunner

Institute of Science and Technology Austria

Marzieh Eidi

ScaDS.AI Dresden/Leipzig

Jonathan C. Mattingly

Duke University

Facundo Mémoli

Rutgers

Ezra Miller

Duke University

Bernadette J. Stolz

Max Planck Institute of Biochemistry

Katharine Turner

Australian National University

Bei Wang

University of Utah

Program

12:30 - 13:00	Welcome Coffee and Registration
13:00 - 13:15	Welcome Address
13:15 - 14:15	Facundo Mémoli Grassmannian Persistence Diagrams We introduce Orthogonal Möbius Inversion, a concept analogous to Möbius inversion on finite posets, applicable to order-preserving functions from a finite poset to the Grassmannian $\mathrm{Gr}(V)$ of an inner product space $V$. This notion relies critically on the inner product structure on $V$, enabling it to capture finer information than standard integer-valued persistence diagrams.Orthogonal inversion is a special case of the broader concept of orthomodular inversion, in which the target is an arbitrary orthomodular lattice. We apply orthogonal inversion to construct a “nonnegative” persistence diagram for any given multiparameter filtration $F$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$, by applying it to the birth–death spaces of $F$.Analogously to classical one-parameter persistence diagrams, these multiparameter Grassmannian persistence diagrams admit a straightforward interpretation. Specifically, for each segment $(b,d)\in \mathrm{Seg}(P)$:1. the Grassmannian persistence diagram canonically assigns a vector subspace of degree-$*$ cycles in $K$ that are born at $b$ and become boundaries at $d$, and2. this assignment is exhaustive at the homology level.
14:15 - 14:40	Evgeniya Lagoda Persistent Homology for Labeled Datasets: Stability and Generalized Landscapes In many applications, point cloud data comes with a finite set of labels. Modeling this situation, we introduce variants of the Gromov–Hausdorff distance for labeled metric spaces. We consider variants with equal and unequal numbers of classes, and give a lower bound for the case of an equal number of classes. Next, we examine the case of generalized persistence, where the simplicial complexes are parametrized by a product of the real line and a partially ordered set induced by the inclusion structure of the classes in a dataset. We discuss the appropriate notion of the interleaving distance for such persistence modules and show stability under small perturbations of the points. To facilitate computations, we propose a notion of a landscape function suitable for this type of persistence, and discuss its theoretical properties and computational aspects. This is a joint work with Yaoying Fu, Lander Ver Hoef, Shiying Li, Tom Needham, and Morgan Weiler.
14:40 - 15:10	Coffee Break
15:10 - 15:35	Won Seong Interleaving Distance as an Edit distance
15:35 - 16:00	Ines Garcia Redondo Effective Resistance in Simplicial Complexes: Generalizations and Properties
16:00 - 16:30	Coffee Break
16:30 - 17:15	Flash Talks 1 Presenter of Flash Talk: Mathijs Wintraecken Antoine Commaret Manuel Soriano-Trigueros Julian Brüggemann Otto Sumray Jacob Skarby Andrea Guidolin Uzu Lim Justyna Signerska Overview Flash Talks
17:15 - 17:45	Reception
17:45 - 18:30	Flash Talks 2 Presenter of Flash Talk: Michael Bleher Xavier Crean Mu'izz Asif Freya Jensen Mikael Vejdemo-Johansson Elena Xinyi Wang Jānis Lazovskis Jenifer Steffi J

09:00 - 10:00	Herbert Edelsbrunner The expected length of a Euclidean minimum spanning tree and the expected 1-norms of chromatic persistence diagrams A classic result on Euclidean minimum spanning trees (EMSTs) is the existence of an asymptotic constant, c, such that the expected length of the EMST of n points sampled uniformly at random in the unit square is c times the square root of n, in the limit when n goes to infinity. However, the value of c is not known. Prior to this work, the known bounds were 0.6008 < c < 0.7072, and we improve the lower bound to 0.6289 < c. The motivation for this work is the stochastic analysis of chromatic persistence diagrams. In particular, we show that similar asymptotic constants exist for the expected 1-norms of all diagrams in the 6-pack of a randomly 2-colored set of points in the unit square, in which we study the inclusion of the disjoint union of the sublevel sets of the two monochromatic distance functions into the sublevel set of the bichromatic distance function. This is joint work with Ondrej Draganov, Sophie Rosenmeier, and Morteza Saghafian.
10:00 - 10:25	Yuliy Baryshnikov Counting Holes in Brownian Bubbles
10:25 - 11:00	Coffee Break
11:00 - 11:25	Nikola Sadovek Quantifying discontinuity
11:25 - 11:50	Kelly Maggs Cohomology classes in the RNA transcriptome
11:50 - 12:15	Maria Jose Jimenez TDA approaches for spatial biology problems. Spatial biology studies how cells and tissues are arranged and interact in their natural 2D/3D environment. At single-cell resolution, data can be modeled as chromatic (labeled) point clouds, where each point represents a cell and its color denotes cell type. The challenge is capturing not only the overall geometry of the cloud, but also relationships between labeled subsets, which standard persistent homology often misses. Recent TDA tools such as Dowker persistence or mix-up barcodes, address the analysis of pairs of labels. For multiple labels, the chromatic alpha filtration generalizes the classical alpha filtration; inclusion maps between labeled datasets induce maps between filtrations and derive a set of persistence diagrams encoding spatial relations. However, computing chromatic alpha values can be costly, motivating the use of chromatic Delaunay–Čech filtration as a cheaper alternative (Natarajan et al.). Using Morse-theoretic techniques, these two filtrations are linked by simplicial collapses, extending a Bauer–Edelsbrunner result to the chromatic setting. Within this framework, we describe some spatial biology problems of interest: (1) quantifying cancer–immune interactions in multiplex-imaged tumors, (2) probing stem-cell differentiation controlled by spatial signaling such as BMP, and (3) analyzing evolving 3D tissue organization during embryonic development using simulations like EmbryoMaker. Maria Jose Jimenez is funded by the Spanish government, MCIU, under grant PID2024-155867NB-I00.
12:15 - 14:15	Lunch Break
14:15 - 15:15	Bernadette J. Stolz Topological learning for spatial and dynamic biomedical data Topological data analysis (TDA) offers powerful tools for studying biological phenomena. In this talk, I will present recent applications to spatial and dynamic biomedical data. First, I will discuss topological model selection in tumour-induced angiogenesis, where TDA combined with approximate Bayesian computation enables parameter inference and objective comparison of spatial models. Second, I will present two relational TDA techniques based on Dowker and Witness complexes that encode spatial relation in multispecies data, i.e. datasets with multiple subtypes of data points. Our relational TDA features can extract biological insight and integrate naturally with popular machine learning approaches for spatial data, such as graph neural networks. Finally, I will show how we can apply path signatures to capture underlying structural relations in time series of multivariate dynamical processes, such as neural recordings.
15:15 - 15:40	Nicolò Zava Gromov-Hausdorff distance between metric graphs and their subspaces
15:40 - 16:10	Coffee Break
16:10 - 16:35	Jacek Brodzki Insights into the geometry of Deep Neural Networks through the estimates of decision boundary volume.
16:35 - 17:00	Jeffrey Giansiracusa Topological data analysis and the nature of nuclear matter

09:00 - 10:00	Marzieh Eidi From manifolds to data: towards an integrated mathematical lens for revealing shape and structure Many ideas in modern mathematics—geometry, topology, spectral theory, and stochastic processes—are often presented as separate theories, even though they arise from the same underlying structure viewed through different mathematical lenses. In this talk, I will present several results that illustrate how concrete connections between these perspectives can be established. In particular, we will see: (i) how curvature is used to detect equivalences in graphs and hypergraphs (joint with Nina Otter); (ii) how graphs can be used to characterize bipartiteness in simplicial complexes; and (iii) what geometric flows can tell us about the topology of manifolds and simplicial complexes (joint with Juergen Jost). Several of the ideas and results presented here are joint work with, inspired by, or shaped through the support of Sayan, in whose memory this talk is given.
10:00 - 11:00	Ezra Miller Statistics with bars in persistent homology Sayan and his collaborators pioneered the formalization of statistics in persistent homology by answering fundamental questions like whether there are spaces of persistence diagrams that admit reasonable probability distributions, and how to compute basic summaries like Fréchet means in such situations. These advances rest on the meaning, stability, and completeness of bars as features in one parameter. What happens in multiple parameters? Various ways of thinking about "bars" diverge for nonlinear posets, giving rise to a multitude of options for how to attempt statistics. All of the options are complicated by the much richer algebra of modules over posets and, crucially, how the algebra interacts with statistical considerations such as metrics and variation. This talk is an opportunity to take stock of where we have been and how we might move forward. It is dedicated to Sayan's memory, influence, and friendship.
11:00 - 11:30	Coffee Break
11:30 - 12:30	Katharine Turner How to Write a Statistics Theorem for Topologists One of Sayan’s legacies was his development of, and advocacy for, more statistical rigour in the world of TDA. Inspired by this goal I will give a crude (and idiosyncratic) introduction to writing a statistics theorem suitable for TDA. We will see some examples of Sayan’s work alongside two case studies I am currently working on. Case study one is kernel regression using persistence diagrams (joint with Saksham Aggarwal and James Nichols). Case study two is on learning Betti numbers of algebraic varieties (joint with Thomas Burnett and Martin Helmer).
12:30 - 14:30	Lunch Break
14:30 - 15:30	Bei Wang Topological and Geometric Approaches to Stratification Learning Given potentially high-dimensional point cloud data, a fundamental question is whether one can infer the structure of the underlying space from which the data are sampled. Manifold learning approaches typically assume that the data lie on or near a single low-dimensional manifold. In practice, however, many datasets exhibit substantially richer behavior, including mixed dimensionality, intersections, and singularities. To model such phenomena, we consider data sampled from a mixture of possibly intersecting manifolds and aim to recover the constituent strata pieces corresponding to manifolds of varying dimensions. This task falls within the area of stratification learning, which we use here in a broad sense to denote an unsupervised, exploratory paradigm for decomposing data into disjoint subsets that capture meaningful geometric and topological structure. In this talk, we discuss topological and geometric approaches to stratification learning, emphasizing tools from local homology, sheaf theory, and discrete stratified Morse theory.
15:30 - 16:00	Coffee Break
16:00 - 17:00	Jonathan C. Mattingly Centeral Limit Theorems on Singular Spaces I will describe a series of papers jointly with Ezra Miller and Do Tran that provide Central Limit Theorems for random variables taking values in a wide class of stratified spaces. We see unexpected new phenomena that do not arise in the smooth setting.
18:00 -	Conference Dinner with talk by Jonathan Mattingly

10:00 - 10:25	Ka Man Yim Stable and consistent spectral features of vertices
10:25 - 11:00	Coffee Break
11:00 - 11:25	Jan Jendrysiak On the Algebraic Structure of Multiparameter Persistence Modules
11:25 - 11:50	Woojin Kim Barcoding invariants and their comparison
11:50 - 12:15	Matthias Soels Fast free resolutions of bigraded chain complexes
12:15 - 14:15	Lunch Break
14:15 - 15:15	Paweł Dłotko Shape, Dynamics, and Biology: Topological Data Analysis in Action Topology interfaces with data science in two complementary ways: it provides robust, deformation-invariant descriptors that quantify the shape and structure of data, and it enables visualizations of high-dimensional data that preserve topological features. Grounded in algebraic and combinatorial topology, these ideas yield practical tools for data analysis when paired with efficient computation. In the first part of this talk, I will survey topological methods developed in my group for analyzing complex and multi-valued data, including constructions based on the Euler characteristic and techniques inspired by phylogenetics. I will emphasize non-stationary settings where data come with dynamics- either as trajectories (ordered point clouds) or as vector fields- and discuss how TDA-based invariants can capture dynamical structure and support reliable classification of different dynamics. I will also touch on recent developments in using topological summaries for statistical hypothesis testing, particularly nonparametric goodness-of-fit testing in high dimensions, and on interpretable visualization strategies based on Mapper and its generalizations. In the second part, I will present two applications of these tools to multiscale biological organization. First, I will show how persistent-homology descriptors of super-resolution microscopy quantify differentiation-dependent reorganization of CTCF nuclear clusters, revealing how RNA-binding proteins and the lncRNA Pantr1 contribute to chromatin-topology maturation and strengthening of architectural loops. Second, I will discuss Topological Malignant Regions (TopMR), a persistent-homology framework capturing multiscale malignant morphology while embedding malignant–immune spatial interactions into a unified topological interaction space. Together, these case studies illustrate how TDA links structural features to gene-regulatory stability and clinically relevant tumor–microenvironmental patterns.
15:15 - 15:40	Jayati Kaushik Multipersistence homology for predicting molecular properties
15:40 - 16:10	Coffee Break
16:10 - 16:35	Francesca Tombari Stratified Morse theory for cell complexes
16:35 - 17:00	Lukas Waas A persistent version of Latschev's theorem (joint with S. Oudot)

09:00 - 10:00	Ulrich Bauer On the additive image of 0th persistent homology I will present joint work with M. Botnan, S. Oppermann, and J. Steen on multiparameter persistent homology in degree zero. It is known that arbitrary diagrams of vector spaces and linear maps can be realized as homology of diagrams of simplicial complexes in some positive degree. We study the more restrictive case of degree zero, which corresponds to diagrams freely generated from sets and set maps. Despite their seemingly simple combinatorial nature, a full understanding of the structure of these representations remains elusive. I will summarize our findings and discuss some conjectures.
10:00 - 10:25	Hana Dal Poz Kourimska A free lunch: manifolds of positive reach can be smoothed without decreasing the reach
10:25 - 11:00	Coffee Break
11:00 - 11:25	Kang-Ju Lee G-Mapper: Learning a Cover in the Mapper Construction
11:25 - 12:00	Lunch Break
12:00 - 12:30	Closing Remarks

Local Organizers

Daniela Egas Santander

Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG)

Bernd Sturmfels

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences

Chairs of the Scientific Committee

Heather Harrington

Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG)

Ulrike Tillmann

University of Oxford

Administrative Contact

Katharina Matschke

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail