Discrete Laplacians 2026

Jun 22, 2026, 7:10 AM → Jun 25, 2026, 5:00 PM Europe/Berlin

Max Planck Institute of Molecular Cell Biology and Genetics

Discrete Laplacians 2026

Discrete Laplacians appear broadly throughout mathematics and the applied sciences. They are a key concept in connecting continuous methods and theories to their discrete analogues, and underlie many of the modern tools in applied mathematics, data analysis, with numerous applications in biology, physics, social sciences and other disciplines.

This workshop aims to bring together researchers from a variety of subjects to present and discuss recent advances in the theory and use of discrete Laplacians. The workshop will focus on discrete Laplacians arising in graph theory, topology, probability, numerics, and data science & machine learning.

Location: MPI-CBG, Dresden, Germany

Dates: 22 - 25 June 2026

Organisers: Karel Devriendt, Otto Sumray, Yu Tian, Giulio Zucal

Confirmed keynote speakers:

• Aida Abiad - TU Eindhoven, NL
• Francesca Arrigo - University of Strathclyde, UK
• Sebastian Engelke - University of Geneva, CH
• Kaibo Hu - University of Oxford, UK
• Jürgen Jost - MPI MiS, Leipzig, DE
• Adrien Kassel - ENS Lyon, FR
• Facundo Mémoli - Rutgers University, US
• Michael Schaub - RWTH Aachen, DE
• Melanie Weber - Harvard University, US

 

The timetable can be seen here.

NB: some of our speakers have received an email from gtravelexpertise.com regarding travel plans.
This is a scam/phishing email, please check with our organising team if you have any suspicions about any further emails. We will never ask for you to send money.

Mon, June 22

8:30 AM Welcome & registration

1 Opening

Speaker: Heather Harrington (CSBD)

Keynote

2 Persistent Laplacians and spectral refinements of persistent homology

Since Eckmann's combinatorial Hodge theory, combinatorial Laplacians have provided a spectral approach to topology: for a finite simplicial complex, the nullity of the q-th Laplacian recovers the q-th Betti number. Persistent homology, on the other hand, studies how homology classes appear and disappear along a filtration. In this talk I will discuss persistent Laplacians, a spectral construction which connects these two viewpoints.

Given an inclusion of simplicial complexes K into L, the q-th persistent Laplacian is an operator on q-chains of K whose kernel recovers the persistent homology group given by the image of the map from H_q(K) to H_q(L). Thus persistent Betti numbers can be obtained spectrally, while the nonzero eigenvalues provide additional quantitative information beyond ordinary persistent homology. I will explain the definition, its relationship with combinatorial Hodge theory, and its interpretation via Schur complements.

I will then describe how this viewpoint leads to refinements of persistent homology. In particular, Grassmannian persistence diagrams attach to intervals in a filtration not only multiplicities, but canonically selected subspaces of cycles encoding representatives of the corresponding persistent classes. This yields a strengthened persistence invariant naturally tied to the spectral theory of Laplacians.

Speaker: Facundo Mémoli (Rutgers University)

10:30 AM Coffee

Contributed

3 Graph limits and their connections to the discrete Laplacian

The goal of the talk is to present connections between graph limit theory, random graphs and matrices, and the discrete Laplacian. The main motivation of graph limit theory is to understand the structure of large networks and graphs by using various tools from analysis and probability theory. Starting from different distance notions for graphs, one can define convergence notions for graph sequences. We can also construct limit objects, which are not graphs, but continuous (or measurable) functions or operators. In the talk we present the main concepts of graph limit theory, its applications in spectral theory of random graphs and matrices, and the connections to the discrete Laplacian.

Speaker: Ágnes Backhausz (Eötvös Loránd University & HUN-REN Rényi Institute)

4 Effective Resistance in Simplicial Complexes: Generalizations and Properties

The concept of effective resistance, originally developed in electrical network theory, has become a powerful tool for studying the structure of graphs. Based on the discrete Laplacian of a graph, it captures both direct and indirect connections between vertices, relates to random walks and spanning trees, and underlies applications ranging from graph sparsification to community detection. In recent years, several matrix expressions have been introduced to extend the notion of effective resistance from graphs to simplicial complexes. In this work, existing approaches are unified via a basis-free definition of effective resistance that reveals new structural insights. More precisely, our framework introduces the effective resistance bilinear form, thereby extending the classical notion from edges to higher-dimensional simplices, chains, and cochains. This leads to generalizations of important properties of the effective resistance in graphs to simplicial complexes, such as Foster’s Theorem, Thomson's principle, connections to simplicial spanning trees, and beyond. To do so, we bridge concepts from algebraic topology, spectral graph theory, and network science, opening new directions for analyzing higher-order structures with tools inspired by the effective resistance. This is joint work with Inés Garcia Redondo, Sarah Percival, Anda Skeja, Bei Wang, and Ling Zhou.

Speaker: Claudia Landi (University of Modena and Reggio Emilia)

11:50 AM Coffee

Keynote

5 A Geometric Lens on Challenges in Graph Machine Learning: Insights and Remedies

Graph Neural Networks (GNNs) are a popular architecture for learning on graphs. While they achieved notable success in areas such as biochemistry, drug discovery, and material sciences, GNNs are not without challenges: Deeper GNNs exhibit instability due to the convergence of node representations (oversmoothing), which can reduce their effectiveness in learning long-range dependencies that are often crucial in applications. In addition, GNNs have limited expressivity in that there are fundamental function classes that they cannot learn. In this talk we will discuss both challenges from a geometric perspective. We propose and study unitary graph convolutions, which allow for deeper networks that provably avoid oversmoothing during training. Our experimental results confirm that Unitary GNNs achieve competitive performance on benchmark datasets. An effective remedy for limited expressivity are encodings, which augment the input graph with additional structural information. We propose novel encodings based on discrete Ricci curvature, which lead to significant gains in expressivity, as well as in empirical performance thanks to capturing higher-order relational information. As part of this discussion, we will also provide rationale for the use of curvature in graph-based geometric data analysis, by presenting discrete-to-continuum consistency results which show that discrete Ricci curvature can provably characterize the geometry of a manifold based on a finite sample.

Speaker: Melanie Weber (Harvard University)

1:10 PM Lunch

Keynote

6 Random abstract cell complexes and applications

Speaker: Michael Schaub (RWTH Aachen)

3:40 PM Coffee

Contributed

7 Generalized Persistent Laplacians and Their Spectral Properties

Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in computational and data science fields and application areas such as machine learning and network science. I will present a unifying operator-theoretic framework of generalized Laplacians that we introduced in our recent paper and that encompasses all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, I will introduce and study a generalized notion of persistent Laplacians. While the persistent Laplacians fails to satisfy the desirable properties of monotonicity and stability, I will demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, I will provide a condition for full monotonicity and show that the non-zero spectra of these separate components determine the non-zero spectra of the full Laplacians, making them not only preferable but sufficient for analysis.

Speaker: Arne Wolf (Imperial College London)

8 Cheeger Inequalities for the Persistent Laplacian

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $\Delta_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $\varphi_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $\Delta_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2024; arXiv:2302.01069).
We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski--Rosenthal--Tessler (Combinatorica, 2016; arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen--Klivans--Mukherjee (Adv. Appl. Math., 2014; arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case.

Speaker: Rui Dong (Vrije Universiteit Amsterdam)

Tue, June 23

8:30 AM Welcome & registration

Keynote

9 Beyond de Rham: Hodge Laplacians on BGG Complexes

The Hodge Laplacian associated with the de Rham complex plays a central role in geometry, topology, and numerical analysis. Many physical and geometric theories, however, are governed not by differential forms but by tensor fields with additional symmetry constraints.

This talk explores Hodge Laplacians arising from Bernstein–Gelfand–Gelfand (BGG) complexes, which extend the de Rham framework to a broad class of tensor-valued differential operators. Examples include the elasticity (Calabi, Kröner, Riemannian deformation) complex, describing generalized continua and geometric structures.

We discuss the analytical structure of these Laplacians, their associated Hodge decompositions and cohomologies, and their interpretation in continuum mechanics. We then describe compatible discretizations and discrete tensor complexes, providing connections to discrete geometric structures and leading to tensor decompositions for fields defined on triangulations.

Speaker: Kaibo Hu (University of Oxford)

10:30 AM Coffee

Contributed

10 Discrete Approximation of Differential Forms on Sampled Vietoris-Rips Complexes

In this talk, we study Vietoris–Rips (VR) complexes built from random samples of an embedded manifold. We define discrete approximations of differential forms as simplicial cochains, together with suitable inner products, and discuss their convergence in the large-sample limit. This provides a probabilistic convergence framework for discrete models of differential forms, based on inner products on VR simplicial cochains that are compatible with the continuum geometry. As an application, we consider the approximation of harmonic 1-forms and the resulting discrete evaluation of natural pairings with differential 1-forms. Based on joint work with Kelly Maggs.

Speaker: Darrick Lee (University of Edinburgh)

11 The Discrete Laplacian from Polyhedral Meshes to Graphs

The Laplacian appears in the eikonal equation through the vanishing viscosity method, where it regularizes the problem to select the unique viscosity solution. How this regularization is realized computationally, however, depends fundamentally on the choice of discretization. This talk examines three formulations in which the Laplacian plays distinct but related roles. On polyhedral meshes, we discretize the Laplacian regularized eikonal equation directly by a cell-centered finite volume method. The discrete Laplacian is constructed from the mesh connectivity and geometry, and combined with the Soner boundary condition, it ensures convergence to the viscosity solution with second-order accuracy. Here the quality of the discrete Laplacian on general polyhedral cells is a central numerical challenge. In a variational formulation, the vanishing viscosity problem is recast through a direct method of calculus of variations. The Laplacian no longer appears as an operator to be discretized; instead, its effect is absorbed into a variational energy whose minimizer approximates the viscosity solution as the viscosity parameter is reduced. Variable splitting and normalized neural representations handle the nonlinearity, while the mesh-free nature of deep learning bypasses the singular perturbation difficulty. In a stochastic formulation based on the Derivative-Free Loss Method, the Laplacian is encoded implicitly through the diffusion of stochastic walkers along gradient trajectories. The computational cost becomes independent of the viscosity parameter, and non-convex domains with obstacles are handled without explicit boundary conditions. These three perspectives, discrete operator, variational energy, stochastic diffusion, illustrate different computational manifestations of the same mathematical mechanism. We conclude by discussing how this viewpoint extends to graph Laplacians, where analogous questions arise in Reeb graph analysis and diffusion-based classification on data.

Speaker: Jooyoung Hahn (Czech Technical University in Prague)

11:50 AM Coffee

Contributed

12 Regularization for the inverse conductance problem on networks

Calderon’s inverse conductivity problem consists in determining whether it is possible to determine the electrical conductivity of a medium from voltage and current measurements at its boundary, (see [1]). The problem is formulated in terms of an elliptic Dirichlet problem and a Dirichlet-to-Neumann map, and its resolution has applications in non-invasive imaging. The discrete version of the problem on networks, the inverse conductance problem, consists in determining the conductance of a network from electrical measurements at certain nodes. Using a version of discrete vector calculus on weighted networks, we show that the discrete analogues of the elliptic Dirichlet problem and the Dirichlet-to-Neumann map involve the Laplacian of the network and one of its (Laplacian) Schur complements. Both the discrete and continuous problems are severely ill-posed. Based on a result in [2] of the stability of the Calderón problem and on the analogy between the discrete and continuous problems, we propose reformulating the inverse conductance problem as a polynomial optimization problem with regularization. The regularization term penalizes deviations from the hypothesis that the conductance is piecewise constant on a partition of the edge set with few subsets, (see [3]). We discuss the stability and resolution methods of our approach, and we show numerical results of its application.

[1] Calderón, A.P., 2006. On an inverse boundary value problem. Comput. Appl. Math. 25, 133–138. (Reprint of the original work in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasil. Mat. Rio de Janeiro 65-73, 1980).
[2] Alessandrini, G., Vessella, S., 2005. Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35, 207–241.
[3] Samperio, Á., 2025. Some inverse problems on finite networks. PhD Thesis, University of Valladolid.
The author was supported by the grant PID2022-138906NB-C21 funded by MICIU/AEI/ 10.13039/501100011033 and by ERDF/EU.

Speaker: Álvaro Samperio (CUNEF Universidad)

13 Approximating Ricci Flow on Point Clouds via Optimal Transport

The Ricci flow is a geometric evolution equation central to Perelman's proof of the Poincaré conjecture, but computing it on discrete data remains a fundamental challenge. We present an algorithm that approximates Ricci flow on finite point clouds equipped with a distance matrix, based on the Gigli–Mantegazza reformulation: each point is embedded into the space of probability measures via its heat kernel measure, and a new metric is defined through pairwise Wasserstein-2 distances. These distances are computed efficiently using entropy-regularized optimal transport via the Sinkhorn algorithm on a nearest-neighbor graph, with approximation guarantees derived from classical heat kernel and volume comparison theorems. We validate the algorithm on point clouds sampled from manifolds of varying dimension and geometry, observing convergence behavior consistent with the theory of Ricci flow.

Speaker: Dongwoo Gang (Seoul National University)

1:10 PM Lunch + Group photo @14:20

Please gather at 14:20 at the Atrium for a group photo

Keynote

14 Cheeger inequalities for Laplacians on simplicial complexes

Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis. The natural problem of extending such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. Before proving any inequality, however, one needs to identify the right Cheeger-type constant for which such an inequality can hold. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2-Laplacians and the analytically more difficult 1-Laplacians, for which, however, the inequalities become equalities.
The talk represents joint work with Dong Zhang.

Speaker: Jürgen Jost (MPI MiS Leipzig)

3:40 PM Coffee

Contributed

15 Point-Level Topological Representation Learning at the Intersection of Topology and Geometry using the Hodge Laplacian

The discrete Hodge Laplacian offers a way to extract network topology and geometry from higher-order networks. The operator is inspired by concepts from algebraic topology and differential geometry and generalises the graph Laplacian. In particular, it allows to relate global structure of networks to the local properties of nodes. In my talk/poster, I will talk about some general behaviour of the Hodge Laplacian and then continue to show how to use the extracted information to a) to use trajectory data infer the topology of the underlying network while simultaneously classifying the trajectories and should time allow b) to extract cell differentiation trees from single-cell data, an exciting new application in computational genomics.

This takes excerpts from:
[1] Grande, V. & Schaub, M.T. (2024), "Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal", In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2024)., March, 2024. , pp. 9896-9900.
[2] Cheng, M.; Jansen, J.; Reimer, K.C.; Grande, V.P.; Nagai, J.S.; Li, Z.; Kießling, P.; Grasshoff, M.; Kuppe, C.; Schaub, M.T.; Kramann, R. & Costa, I.G. (2025), "PHLOWER leverages single-cell multimodal data to infer complex, multi-branching cell differentiation trajectories", Nature Methods., October, 2025. , pp. 1-9. Nature Publishing Group US New York.
[3] Grande, V.P.; Hoppe, J.; Frantzen, F. & Schaub, M.T. (2024), "Topological Trajectory Classification and Landmark Inference on Simplicial Complexes", In 58th Annual Asilomar Conference on Signals, Systems, and Computers., October, 2024. , pp. 44-48.

Speaker: Vincent Grande (RWTH Aachen University)

16 Discrete approximation of the Laplacian on a covering manifold

By a result of Dodziuk and Patodi, the spectrum of the Hodge-de Rham Laplacian on a closed Riemannian manifold can be approximated using the so-called Whitney forms, which provide a natural discretization of differential forms. There is also a corresponding convergence result for eigenforms. After introducing Whitney forms and their approximation properties, I will discuss the tools and methods necessary to obtain similar results for the case of non-compact coverings of closed Riemannian manifolds. Here, continuous spectrum typically arises, which warrants a careful consideration of the modes of convergence we can expect. A key tool is the von Neumann trace associated to the group of deck transformations, which provides a natural way to measure spectral invariants. If time permits, I will discuss the relation of von Neumann trace methods to more commonly used methods like the integrated density of states and Bloch-Floquet theory.

Speaker: Erik Babuschkin (Universität Göttingen)

Wed, June 24

8:30 AM Welcome & registration

Keynote

17 A spectral approach to Kemeny’s constant

Kemeny’s constant, a fundamental parameter in the theory of Markov chains, has recently received significant attention within the graph theory community. Originally defined for a discrete, finite, time-homogeneous, and irreducible Markov chain based on its stationary vector and mean first passage times, Kemeny’s constant finds special relevance in the study of random walks on graphs. Kemeny’s constant gives a measure of how quickly a random walker can move around a graph and is thus a good measure of the connectivity of a graph. Kemeny's constant has many useful interpretations, including the spread of infectious diseases (how quickly a disease will reach epidemic levels), molecular conformation dynamics (presence or absence of metastable sets), and urban road networks (how well connected a network is). In general, a lower Kemeny's constant means that a graph is more connected, and a higher Kemeny's constant means that a graph is less connected. For these and other applications, the main question is: how do changes in the network lead to changes in Kemeny's constant? In this talk we investigate the effect of the network structure on Kemeny's constant. We do so by showing several new approximations for Kemeny’s constant, which we derive using spectral graph theory techniques.

Speaker: Aida Abiad (TU Eindhoven)

10:30 AM Coffee

Contributed

18 Diameter, Algebraic Connectivity, and Maximum Relaxation Time in Regular Graphs

Aldous and Fill conjectured that the maximum relaxation time of the random walk on a connected regular graph with $n$ vertices is $(1+o(1))\frac{3n^2}{2\pi^2}$. The algebraic connectivity of a graph (the second-smallest eigenvalue of its Laplacian matrix) is inversely related to the relaxation time of the random walk. A conjecture of Mohar and Guiduli predicts the structure of graphs with minimum degree $d$ that minimize algebraic connectivity. We prove that, among all $d$-regular graphs on a fixed number of vertices, those with maximum diameter attain the smallest algebraic connectivity and therefore the largest relaxation time. As a consequence, the Guiduli--Mohar conjecture implies the Aldous--Fill conjecture for odd values of $d$. For cubic and quartic graphs, we carry out a more detailed analysis and characterize the graphs that achieve the maximum relaxation time (equivalently, the minimum algebraic connectivity). This talk is based on joint work with Maryam Abdi.

Speaker: Ebrahim Ghorbani (Hamburg University of Technology, Institute for Algorithms and Complexity)

19 A Ricci flow for resistance curvature

Various notions of curvature on graphs exist and have been studied extensively. Recently, some notions of curvature for graphs have been defined by way of the effective resistance in the graph. I will discuss a Ricci flow on graphs based on the recently defined Ricci-Foster curvature, which is a variant on these curvatures defined from effective resistance. I will discuss the existence and preservation of positive curvature for this Ricci flow.

Speaker: Mark Kempton (Brigham Young University)

11:50 AM Coffee

Contributed

20 How To Control Smoothing in Graph Neural Networks

Graph neural networks (GNNs) have emerged as powerful tools for processing relational data in applications. However, GNNs suffer from the problem of oversmoothing, the property that features of all nodes exponentially converge to the same vector over layers, prohibiting the design of deep GNNs. In this work we study oversmoothing in graph convolutional networks (GCNs) by using their Gaussian process (GP) equivalence in the limit of infinitely many hidden features. By generalizing methods from conventional deep neural networks (DNNs), we can describe the distribution of features at the output layer of deep GCNs in terms of a GP: as expected, we find that typical parameter choices from the literature lead to oversmoothing. The theory, however, allows us to identify a new, non-oversmoothing phase: if the initial weights of the network have sufficiently large variance, GCNs do not oversmooth, and node features remain informative even at large depth. We demonstrate the validity of this prediction in finite-size GCNs by training a linear classifier on their output. Moreover, using the linearization of the GCN GP, we generalize the concept of propagation depth of information from DNNs to GCNs. This propagation depth diverges at the transition between the oversmoothing and non-oversmoothing phase. We test the predictions of our approach and find good agreement with finite-size GCNs. Initializing GCNs near the transition to the non-oversmoothing phase, we obtain networks which are both deep and expressive.

Speaker: Bastian Epping (RWTH Aachen University)

21 Affine-Invariant Geometry of Laplacians for Fast Graph Comparison

Graph similarity measures are central in network science, yet many existing measures are too computationally expensive for large graphs. We study graph geodesic distance (GGD), a recently proposed spectral distance that can be efficiently computed for large node-labeled graphs of the same size. GGD compares their Laplacians by viewing them as points on the cone of symmetric positive definite (SPD) matrices equipped with the affine-invariant Riemannian metric (AIRM). To make the Laplacians positive definite, the original formulation adds a small multiple of the identity. We instead consider connected, possibly weighted, graphs and obtain SPD matrices by restricting the Laplacians to the subspace orthogonal to the common zero eigendirection, thereby realizing GGD as the limit of the original construction as the regularization parameter tends to zero. We show that when two Laplacians have similar quadratic forms, for instance when one is a spectral sparsifier of the other, their GGD is small, which naturally leads to normalization by the square root of the number of nodes. Using the spectral theory of AIRM, we derive lower and upper bounds on GGD that are independent of node labeling. Finally, for large unweighted graphs, we obtain general upper and lower bounds on GGD between any pair of graphs: the upper bound is attained by the complete and path graphs, while the lower bound is estimated by comparing a graph with itself after removing a single edge.

Speaker: Boris Stupovski (Institute of Physics Belgrade)

1:10 PM Lunch

Keynote

22 Walk-based Laplacians for complex networks

In this talk I will discuss a novel framework for modeling diffusion on complex networks by constructing Laplacian-like operators based on walks around a graph. This approach introduces a parametric family of walk-based Laplacians that naturally incorporate memory effects by excluding or downweighting backtracking trajectories, where walkers immediately revisit nodes. The framework includes: (i) walk-based Laplacians that count all traversals in the network; (ii) nonbacktracking variants that eliminate immediate reversals; and (iii) backtrack-downweighted variants that provide a continuous interpolation between these two regimes. We establish that these operators extend the definition of the standard Laplacian and also preserve some of its properties. I will briefly mention numerical strategies for their computation, which ensure scalability of the proposed techniques. Numerical experiments will be discussed to showcase the properties of these operators and to compare their performance against the standard and path-based Laplacians. This work is in collaboration with Prof. Fabio Durastante (Universita' Statale di Pisa, Italy).

Speaker: Francesca Arrigo (University of Strathclyde)

3:40 PM Coffee

Contributed

23 Experimental methods for the automatic estimation of real cohomology

We show how spectral data analysis methods may be used to construct a completely automated pipeline for data clustering and the direct estimation of real cohomology (and not persistent real cohomology). In particular, we introduce several new spectral heuristics for the data-driven selection of the radius of the Vietoris-Rips complex given data sampled from a metric-measure space, we give experimental evidence for the the conjecture that the resulting Vietoris-Rips complex given by these methods recovers the correct cohomology of the underlying space with high probability.

Speaker: Antonio Rieser (SECIHTI-CIMAT)

24 Neural Feature Geometry Evolves as Discrete Ricci Flow

Deep neural networks learn feature representations through geometric transformations of the input data manifold. We study this process using geometric graphs that approximate local similarity structure in feature space. Our theoretical results show that nonlinear activations play a central role in shaping these graphs during training. Empirically, across over 20,000 feedforward networks trained on synthetic and real-world binary classification tasks, we find that feature-geometry evolution resembles a discrete Ricci flow: class separability emerges together with community structure in the associated graph representations. Based on this connection, we propose a local framework for comparing neural feature transformations with discrete Ricci flow dynamics, yielding practical design principles such as geometry-informed early stopping and depth selection.

Joint work with Moritz Hehl (Leipzig) and Melanie Weber (Harvard)

Speaker: Max-Konstantin von Renesse

6:30 PM Workshop dinner

Thu, June 25

8:35 AM Welcome & registration

Keynote

25 Laplacians in Extreme Value Theory

Graphical models provide a powerful framework for understanding complex dependence structures, but extending them to extreme events requires new probabilistic tools. In this talk, we introduce the theory of extremal graphical models and highlight the central role played by Laplacian matrices. After reviewing the foundations of multivariate extreme value theory, we discuss a general notion of conditional independence for infinite measures that unifies several existing approaches and yields a coherent graphical framework for extremes. We then show how Laplacians characterize dependence structures in these models and facilitate statistical inference. The talk concludes with an overview of probabilistic properties, parameter estimation techniques, and graph learning methods for extremal graphical models.

Speaker: Sebastian Engelke (University of Geneva)

10:30 AM Coffee

Contributed

26 Martingales and Asymptotic Phase

The interpretations of the graph Laplacians in terms of electric networks and (reversible) Markov processes has been one of the most fruitful connections between geometry and topology and discrete mathematics, in particular through Hodge decomposition, a representation of 1-cohomology classes in terms of harmonic forms. What happens when the Markov process is not reversible? This situation arises in many applications, such as biochemical networks or molecular machines. Nontrivial fluxes indicate non-reversibility, and the standard machinery of Hodge theory fails. To account for those fluxes I replace 1-harmonic forms by the stochastic phase. (Stochastic phases were introduced by physicists to describe Winfree/Guckenheimer asymptotic phase for randomly perturbed dynamical systems.) In our context stochastic phase is best described as a section of a certain sheaf of functions which is local martingale when composed with the random walk. I prove the existence of such stochastic phases for broad classes of random walks with nontrivial fluxes, and derive from it an analogue of the Hodge decomposition of the space of stochastic phases.

Speaker: Yuliy Baryshnikov (University of Illinois, Urbana Champaign)

27 Gremban Expansion for Signed Networks: Algebraic and Combinatorial Foundations for Community-Faction Detection

Graph Laplacians form the backbone of many structural and spectral methods in network science, yet their extension to networks with positive and negative interactions remains fragmented. We revisit the Gremban expansion as a principled construction that maps a network with signs to a larger unsigned network and analyze the properties of the associated combinatorial Laplacian. We show that the Laplacian of the expanded graph simultaneously encodes the unsigned Laplacian of the underlying topology and the signed Laplacian induced by antagonistic interactions within a single block-structured operator. Building on this structure, we exploit symmetry patterns of the Fiedler vector to distinguish whether the dominant mesoscale organization corresponds to assortative communities or to antagonistic factions. The resulting framework provides a Laplacian-based approach to community–faction detection in networks with signs that relies entirely on classical spectral graph theory while retaining full information about positive and negative interactions.

Speaker: Fernando Diaz-Diaz (University Carlos III of Madrid)

11:50 AM Coffee

Contributed

28 Topologically invariant coordinates for dynamic epithelia undergoing morphogenesis

Epithelia are two-dimensional tissues, sheets of tightly connected cells that form many structures in organs. Such tissues acquire their shape and function through morphogenesis, a process that involves changes in both their geometry and cellular network topology. A key question in morphogenesis is to understand how cellular processes, such as cell division or T1 transitions, contribute to tissue morphology by changing its geometry and topology. For a curved epithelium, this problem can be formulated in a continuous covariant description on the tissue surface. In this work, we propose two sets of topologically invariant coordinates, describing cellular networks, obtained by embedding a graph representation of the network into R2 using only its connectivity, without the need to take into account the underlying tissue geometry. We construct these embeddings using the spectrum of the graph Laplacian and a spring-meshwork representation. Local changes of these topologically invariant coordinates allow us to identify the cellular processes occurring during tissue development. This formalism provides a framework to investigate the coupled evolution of epithelial geometry and topology.

Speaker: Paweł Korzeb (MPI PKS)

29 Simplicial effective resistance and enumeration of spanning trees

A graph can be regarded as an electrical network in which each edge is a resistor. This point of view relates combinatorial quantities, such as the number of spanning trees, to electrical ones such as effective resistance. The second and third authors have extended the combinatorics/electricity analogy to higher dimension and expressed the simplicial analogue of effective resistance as a ratio of weighted tree enumerators. In this paper, we first use that ratio to prove a new enumeration formula for color-shifted complexes, confirming a conjecture by Aalipour and the first author, and generalizing a result of Ehrenborg and van Willigenburg on Ferrers graphs. We then use the same technique to recover an enumeration formula for shifted complexes, first proved by Klivans and the first and fourth authors. In each case, we add facets one at a time, and give explicit expressions for simplicial effective resistances of added facets by constructing high-dimensional analogues of currents and voltages (respectively homological cycles and cohomological cocycles). This is joint work with Art M. Duval, Woong Kook, and Jeremy L. Martin.

Speaker: Kang-Ju Lee (Seoul National University)

1:10 PM Lunch

Keynote

30 From tree counting to the geometry of Riemann surfaces

Since the mid-19th century, it has been known that computing electric currents in a finite network is related to the weighted count of its spanning trees. A probabilistic perspective on this question involves random spanning trees, which are a special type of determinantal point process. There are variants of this determinantal measure obtained by adding scalar and matrix-valued edge-weights on the graphs, such as those arising from networks modelling magnetic interactions or discrete gauge theories.

In this talk, I will explain how, by viewing graphs as large scale limits of Riemann surfaces, one can show in a precise sense how a large family of determinantal processes on graphs emerge as limits of canonical point processes on Riemann surfaces endowed with local systems. For instance, the uniform measure on spanning trees arises as the tropical limit of a canonical determinantal point process on Riemann surfaces, as the surface degenerates into a nodal curve whose dual graph is the original graph.

This correspondence reveals a hidden geometric origin for combinatorial objects like the uniform measure on spanning trees. It also suggests new perspectives for studying Riemann surfaces inspired by the properties of their combinatorial analogs on graphs.

Based on joint work with Omid Amini.

Speaker: Adrien Kassel (ENS Lyon)

3:40 PM Coffee

Contributed

31 Extremal conditional independence for Hüsler-Reiss distributions via modular functions

We study extremal conditional independence for Hüsler–Reiss distributions, which is a parametric subclass of multivariate Pareto distributions. As the main contribution, we introduce two set functions, i.e. functions which assign a value to the distribution and each of its marginals, and show that extremal conditional independence statements can be characterized by modularity relations for these functions. For the first function, we make use of the close connection between Hüsler–Reiss and Gaussian models to introduce a multiinformation-inspired measure for Hüsler–Reiss distributions. For the second function, we consider an invariant that is naturally associated to the Hüsler–Reiss parameterization and establish the second modularity criterion under additional positivity constraints. Together, these results provide new tools for describing extremal dependence structures in high-dimensional extreme value statistics. In addition, we study the geometry of a bounded subset of Hüsler–Reiss parameters and its relation with the Gaussian elliptope.

Speaker: Ignacio Echave-Sustaeta Rodríguez (Eindhoven University of Technology)

32 Topology of Discrete Laplacian in Non-Equilibrium Thermodynamics: From Trees to Forests

Continuous-time Markov jump processes are fundamental to modeling non-equilibrium complex systems, with their dynamics naturally governed by the discrete graph Laplacian. A central challenge in nonequilibrium thermodynamics is understanding how the structural topology of this Laplacian constrains the physical limits of the underlying network. In this talk, we present a unified combinatorial framework that first translates stationary-state behaviors into the language of spanning trees, allowing us to derive rigorous thermodynamic bounds on non-equilibrium symmetry breaking that are entirely independent of specific kinetic parameters. Building upon this foundation, we then extend our graph-theoretic approach to capture time-dependent transient dynamics using spanning forests. We reveal that global relaxation processes and spectral invariants can also be systematically encoded through combinatorial graph structures, ultimately providing a unified perspective on how the discrete Laplacian dictates both the stationary limits and the dynamic evolution of complex non-equilibrium systems.

Speaker: Shiling Liang (MPI-PKS)