EPFL Topology Seminar Spring 2026

Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Kensuke Arakawa
Manifold calculus beyond space-valued functors

Manifold calculus is a homotopy-theoretic technique to study presheaves on manifolds, which decomposes them into successive approximations called polynomial approximations. First invented by Weiss to study embedding spaces, it has become an important toolset for homotopical study of manifolds.  
Like ordinary calculus, manifold calculus has two “fundamental theorems,” one which classifies polynomial presheaves, and the other that classifies homogeneous presheaves. Consistent with his goal to study embedding spaces, Weiss established these theorems for space-valued presheaves.  
From the perspective of studying manifold invariants, it is extremely natural to develop manifold calculus for presheaves with more general values, such as spectra and chain complexes. However, Weiss’s proof of the fundamental theorems relies on ad-hoc constructions on spaces, which do not seem to generalize easily. 
In this talk, I will explain that the two fundamental theorems do not depend on space-level constructions. Consequently, they extend to presheaves valued in essentially any infinity category. This talk is based on my paper “A context for manifold calculus” (arXiv:2403.03321).

Marie-Camille Delarue
Stable homology using scanning methods

Homological stability is a property that holds for various families of groups, such as the symmetric groups. We build a topological model for the monoid of the groups as a sort of category of 1-cobordisms. We can use this model to compute the group homology in a stable range by constructing a scanning map following the work of Madsen, Weiss, Galatius, Randal-Williams and others. This map allows us to express the stable homology of the groups as the homology of the infinite loop space of a certain spectrum. We will also explain how to generalize the model built for the symmetric groups to a model of the Higman–Thompson groups, which are groups of certain self-homeomorphisms of Cantor sets.

Marino Gran
Hopf formulas for cocommutative Hopf algebras

In recent years, numerous new applications of categorical Galois theory have emerged in various interesting non-abelian algebraic contexts. In particular, within semi-abelian categories, this approach has led to some new calculations of higher fundamental groups in terms of generalized commutators in categories such as that of compact groups, crossed modules, and skew braces. These categories share some structural properties with the categories of groups and of Lie algebras, and also with the category of cocommutative Hopf algebras over a field, which is also semi-abelian. This raises the natural question of whether similar homological methods can be applied to study cocommutative Hopf algebras as well.
In this talk, after reviewing some fundamental properties of semi-abelian categories and some motivating examples, I will explain that the answer to the above question is affirmative. By using the exactness properties of cocommutative Hopf algebras and the free functor universally associating a Hopf algebra with any coalgebra it is possible to establish some new Hopf-type formulae for the homology of cocommutative Hopf algebras. An important role is played by cleft extensions, namely those surjective morphisms of Hopf algebras that are split as coalgebra morphisms.
With any cleft extension, one can associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory. This new approach can also be applied to investigate the homology of cocommutative Hopf braces, which are interesting structures that naturally occur in the study of solutions to the so-called quantum Yang-Baxter equation. The category of cocommutative Hopf braces turns out to be both semi-abelian and monadic on the category of coalgebras, so that it is possible to investigate it from the perspective of non-abelian homological algebra.
This talk is based on a joint work with Andrea Sciandra.

Niall Taggart
Tested functor calculi

Functor calculus refers to a family of homotopy-theoretic frameworks that extend the ideas of differential calculus into categorical settings. Existing forms of functor calculus have proved remarkably useful, with applications ranging from algebraic $K$-theory to a variety of geometric problems. Their ubiquity suggests that it is worthwhile to search for new versions of functor calculus.
Recent work of Bandklayder, Bergner, Griffiths, Johnson, and Santhanam examines the homotopy theory encoded in the degree $n$ approximations of Goodwillie calculus and discrete calculus. In each case, they identify (or in the case of discrete calculus, construct) a model structure that captures the relevant homotopical data and is controlled by a prescribed collection of test morphisms. By exploiting these test morphisms, they show that the resulting model category is cofibrantly generated, giving an explicit description of the generating acyclic cofibrations.
In this talk, I will describe the observation that these model structures can always be realised as left Bousfield localizations with respect to the corresponding sets of test morphisms. This viewpoint naturally suggests defining new calculi directly from chosen test morphisms. I will explain that such a “tested” calculus exists provided a certain technical condition on the test morphisms is satisfied, and then show how this condition can be reformulated in far more familiar terms, namely, that a degree $n$ functor automatically satisfies the conditions of being degree $n+1$.
(The latter aspect of this talk is joint work-in-progress with Julie Bergner, Brenda Johnson, Rhiannon Griffiths and Rekha Santhanam.)

Michael Bleher
Single cells, population dynamics, and Euler characteristic profiles

The Euler characteristic profile (ECP) of a multifiltered simplicial complex records the Euler characteristic at each point in the filtration poset. While cruder than multiparameter persistent homology, ECPs are computationally much more tractable and often still sensitive enough to detect changes in the topology of the underlying data. For example, ECPs based on vector field data are able to differentiate between dynamical systems in 2 and 3 dimensions. In this talk, I present a recent project for similarly extracting dynamical information from high-dimensional point cloud data equipped with a vector field. The motivating application is single-cell RNA sequence data, where RNA velocity provides a proxy for the direction and rate of cellular state transitions. We construct multifiltered flag complexes where edge weights are derived from distances and velocities. On synthetic data generated by a stochastic model of gene expression dynamics with known ground-truth transition graphs, the resulting ECPs distinguish between competing state transition networks. Ultimately we want to use these ideas to investigate neural stem cell differentiation — both in homeostasis and when it goes wrong, as in glioblastoma. This is joint work with Marta Marszewska, Justyna Signerska-Rynkowska, Paweł Dłotko, Anna Marciniak-Czochra, and Ana Martín-Vilalba.

Truong Hoang Manh
The cotangent complex of E_1 and E_∞ operads

In recent work, we show that the cotangent complex of the E_1-operad (respectively, the E_∞-operad) can be represented as a spectrum-valued functor on the simplex (respectively, Gamma) category. In this talk, we give an explicit description of these complexes and the relation between them. Moreover, we explain how the cotangent complex of the E_1-operad is related to its Hochschild complex, in the spirit of a classical result of Quillen on the cotangent complex of associative algebras.

Nadja Egner
N-fold groupoids and n-groupoids in semi-abelian categories

The notion of semi-abelian category, introduced by G. Janelidze, L. Márki and W. Tholen in 2002, generalizes that of abelian category, and captures the homological properties that the categories of groups, associative algebras, Lie algebras and cocommutative Hopf algebras over a field have in common. Internal structures behave surprisingly well in semi-abelian categories. For example, any internal reflexive graph admits at most one internal category structure, and the categories of internal categories and internal groupoids are isomorphic. Moreover, the category of internal groupoids is equivalent to the category of internal crossed modules. The notion of internal crossed module in any semi-abelian category was introduced by G. Janelidze in 2003, and recovers in particular the classical notion of crossed module of groups. The fact that the category of internal groupoids in a semi-abelian category is itself semi-abelian implies that also the category of internal n-fold groupoids is well-behaved.
In this talk, I will prove that the full subcategory of internal n-groupoids in a semi-abelian category is a Birkhoff subcategory of the category of internal n-fold groupoids, and provide a simple description of the corresponding reflection for n=2. In the abelian context, the internal n-groupoids yield a torsion-free subcategory of the category of internal n-fold groupoids, and it is possible to characterize (higher) central extensions and compute generalized Hopf formulae for homology.
Part of this talk is based on joint work with Marino Gran.

Annika Thiele
Symplectic fillings and spinal open books

The classification of the symplectic fillings of a given contact 3-manifold poses an interesting problem. It is motivated by results that demonstrate that the symplectic fillings of a contact 3-manifold hold information on the contact structure. Most notably, the contact structure of any contact 3-manifold that admits a strong symplectic filling is tight. Although the symplectic fillings of certain contact manifolds have been classified, the general classification remains an open problem. Working towards this, one can consider the geography problem for symplectic fillings. In the paper Spine removal surgery and the geography of symplectic fillings, Sam Lisi and Chris Wendl prove the existence of a universal bound for the geography (Euler characteristic and signature) of possible minimal strong symplectic fillings of a closed contact 3-manifold with a supporting planar spinal open book decomposition. Following a brief introduction to symplectic and contact topology, the aim of my talk is to explain Lisi and Wendl’s result. For this purpose, I will provide an overview of symplectic fillings and the related geography problem, and introduce spinal open book decompositions.

Elena Botti
Topological Data Analysis for Gait Pattern Classification

The classification of gait patterns is an important challenge in movement analysis, as it supports clinical assessment and decision-making by enabling diagnosis and severity stratification. In this talk, I will discuss the potential of Topological Data Analysis (TDA) for gait pattern classification. Unlike conventional approaches that rely on explicit detection of Gait Events (GEs) to compute Spatiotemporal Gait Parameters (SGPs), TDA characterises the global structure of gait signals directly, capturing relationships and patterns in the data without requiring GEs. This is particularly relevant in real-world settings, where GE detection can be difficult due to heterogeneity in walking conditions and gait patterns, potentially biasing clinically relevant metrics and, consequently, decision-making.
Within our department, preliminary results have shown that TDA-based features can achieve classification performance comparable to that of SGPs in fall-risk assessment. These findings suggest that topology offers a competitive alternative for representing gait data, with the potential to better handle variability across subjects and pathological conditions.
Building on these results, we plan to further extend the TDA framework in two directions. First, we aim to investigate time-aware topological methods to better capture the temporal structure of gait signals. Second, we will explore Topological Deep Learning (TDL) approaches to reduce reliance on handcrafted design choices and potentially improve classification performance. By combining the robustness of topology with data-driven representation learning, this work seeks to provide robust tools for the classification of typical and pathological gait patterns.

Anna Sopena-Gilboy
A-infinity structures in complex geometry

For complex manifolds, there exists a refined notion of weak equivalence related to both Dolbeault and anti-Dolbeault cohomology. This class of weak equivalences naturally defines a stronger formality notion. In particular, satisfying the ddbar-Lemma property does not imply formality in this new sense. The goal of this talk is to introduce a novel operadic framework designed to understand this homotopical situation. I will present pluripotential A-infinity algebras as well as a homotopy transfer theorem based on this strong notion of weak equivalence.

Kelly Maggs
Persistent Minimal Models in Rational Homotopy Theory

In this talk, we will discuss a generalization of the main structure theorems of rational homotopy theory to the persistent setting. Our main motivation is the computation of an explicit finite, cellular presentation of the persistent minimal model that completely characterizes the rational homotopy type of copersistent simply-connected spaces. We achieve this via an explicit construction of the minimal model of a tame persistent CDGA as an iterated sequence of cell attachments. As an application of our results, we construct an explicit decomposition of the rational Postnikov tower of simply-connected copersistent spaces in terms of a tower of persistent Eilenberg-Maclane intervals.
Joint work with Samuel Lavenir and Kathryn Hess

Ines Garcia Redondo
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. 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 aiming at extending the notion of effective resistance from graphs to simplicial complexes. In this talk, I will present a basis-free definition of effective resistance, rooted in the original definition motivated by physics, which unifies existing approaches, allows the extension of graph theoretic results, and reveals new structural insights.
This is joint work with Claudia Landi, Sarah Percival, Anda Skeja, Bei Wang and Ling Zhou.

Kate Benjamin
Topology of spatiotemporal trajectories

In recent years, multiparameter persistent homology (MPH) has been developed to address the limitations of traditional single-parameter persistence. While much progress has been made on the theoretical aspects of MPH, practical applications have mostly been limited to the special case of two parameters. Building on recent developments in the computation of MPH in more than two parameters, we propose and implement a new algorithm for the computation of multiparameter persistence landscapes in this setting. We apply this work to compute landscapes of three-parameter persistence modules built on spatiotemporal data arising in biology.

Rémi Mollinier
Finite partial groups are genuinely finite

Partial groups are, roughly speaking, groups in which the product of a given word of elements may not always be defined. They were introduced by Chermak to study the p-local structure of finite groups and come equipped with a “domain,” which consists in the set of words for which the product is defined. At first glance, given a fixed set X, it may seem possible to define infinitely many partial group structures on X by varying the domains, even while ensuring that all these structures have coherent products. For example, if G is a finite group, one might expect that there could be infinitely many different partial group structures “contained” in G by selectively removing sets of words from the domain.
However, in a joint work with Philip Hakney, we show that finite partial groups are truly finite objects: they can be defined using only a finite set of data and, in particular, contain only finitely many “partial subgroups.” This follows from the fact that finite partial groups have finite dimension as symmetric sets.
During the talk, we will explore both the algebraic and topological approaches to partial groups and no prior knowledge of the topic will be assumed.