EPFL Topology Seminar Fall 2024

 Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Lyne Moser,
New methods to construct model categories:

Model categories provide a good environment to do homotopy theory. While weak equivalences are the main players in a model category and encode how two objects should be thought of as being “the same”, the additional data of cofibrations and fibrations typically facilitates computations of homotopy (co)limits and derived functors. However, because of their robust structure, model categories are usually hard to construct. In joint work with Guetta, Sarazola, and Verdugo, we develop new techniques for constructing model structures from given classes of cofibrations, fibrant objects, and weak equivalences between them. The requirement that one only needs to provide a class of weak equivalences between fibrant objects seems more natural in practice as the fibrant objects are often the “well-behaved” objects and so weak equivalences should only be expected to exhibit a good behavior between these objects. As a straightforward consequence of our result, we obtain a more general version of the usual right-induction theorem along an adjunction, where fibrations and weak equivalences are now only right-induced between fibrant objects. If time permits, I will mention some applications of these new methods.

Jan-Paul Lerch
Decomposition of generalised block modules

Interval decomposability of multiparameter persistence modules is a strong property, which can be characterised locally in some cases. A known example are 2-parameter (pointwise finite dimensional) block modules, originating in the context of levelset persistence. They satisfy an exactness property, which completely determines this class of (pointwise finite-dimensional) modules.
In this talk we discuss a generalised class of block modules in n parameters and how they are completely determined by a generalised local property.
This is joint work with V. Lebovici and S. Oudot (https://arxiv.org/abs/2402.16624).
(If time permits, I will also talk about the most recent project.)

Dan Isaksen
HF2-synthetic tools for computing stable homotopy groups

HF2-synthetic stable homotopy theory is a deformation of classical stable homotopy theory. I will describe what this means in concrete terms, and I will demonstrate in practical terms how it helps us compute stable homotopy groups.

Jérôme Scherer
Module spaces over homology theories and completion towers

This is joint work with W. Chacholski and W. Pitsch. To any coaugmented functor X=>EX one can associate its “modules”, i.e. spaces X which are retracts of EX. When E is ordinary homology, this functor is the infinite symmetric product and modules are generalized Eilenberg-Mac Lane spaces (GEMs). Given two such functors E and F we introduce a new one, we denote by a bracket [E, F], as a pullback of two natural transformations E=>EF and F=>EF. We study modules over such brackets and construct two towers by bracketing systematically either on the left or the right. For ordinary homology we obtain two well-known towers: the Bousfield-Kan completion tower and the modified one as introduced by Dror Farjoun.

Niall Taggart
Power operations and free spectral Lie algebras

Power operations are essential in making cohomology an effective tool for studying spaces, with examples including Steenrod operations on ordinary cohomology, Dyer-Lashof operations on iterated loop spaces, and Adam’s operations on complex K-theory. In this talk I will describe an approach to study certain power operations on free spectra Lie algebras. This approach hinges on the relationship between calculus of functors and splitting results of Cohen, Moore and Neisendorfer. This is joint work with Blans, Boyde, Heuts and Nervo.

Ruben Mud
Betweenness in Enriched Categories

The geometrically intuitive notion of betweenness was first formally introduced by Pasch in 1882. Since then, it has found applications in graph theory, combinatorics, artificial intelligence, and circuit design. In this talk, we will argue that enriched categories form a natural framework to reason about betweenness geometry. It turns out that there is a functorial correspondence between the category of betweenness spaces and the category of enriched categories. Using this correspondence, we obtain new interpretations of betweenness with potential applications to causal discovery.

Samuel Lavenir
Fiberwise extensions of localizations and colocalizations

In this talk, I will discuss the problem of extending functors to fiberwise constructions in the setting of higher topoi. I shall explain the relevance of such extensions both from a practical point of view, and through the lens of the internal logic of the topos. After discussing some known results about fiberwise extensions of localization functors, we will focus on the dual situation of cellularizations. We will see that extending such colocalizations is almost never possible, and relate our result to classical statements by Chachólski and Hernández. Along the way, emphasis will be put on the pertinence of higher categorical language to formalize the problem of fiberwise constructions.

Emily Roff
Homotopy by degrees, and the magnitude-path spectral sequence

The past decade has seen a proliferation of homology theories for graphs. In particular, large literatures have grown up around magnitude homology (due to Hepworth and Willerton) and path homology (Grigor’yan, Lin, Muranov and Yau). Though their origins are quite separate, Asao proved in 2022 that in fact these homology theories are intimately related. To every directed graph one can associate a certain spectral sequence—the magnitude-path spectral sequence, or MPSS—whose page E^1 is exactly magnitude homology, while path homology lies along a single axis of page E^2. In this talk, based on joint work with Richard Hepworth, I will explain the construction of the sequence and argue that each one of its pages deserves to be regarded as a homology theory for directed graphs, satisfying a Künneth theorem and an excision theorem, and with a homotopy-invariance property that grows stronger as we turn the pages of the sequence. The ‘nested’ family of homotopy categories associated to the pages is not yet well understood. But I will describe a new cofibration category structure on the category of directed graphs, associated to page E^2 of the MPSS.

Varun Sharma
Center for Molecular Medicine of the Austrian Academy of Sciences

Biological systems exhibit emergent properties that arise from complex interactions across multiple organizational scales, from the molecular level to whole tissues. Understanding these emergent phenomena requires an exploration of the mapping between these different scales, and how lower-scale interactions can give rise to higher-order phenotypic structures. In this talk, I propose equipping Neural Cellular Automata (NCA) with prior knowledge of the single cell transcriptomic manifold state (by means of manifold learning) as an ideal framework for connecting these biological scales to model a specific self-organizing tissue phenotype. NCAs offer a mechanism for examining the dynamic evolution of state spaces across biological hierarchies, providing insight into how discrete components interact to yield organized, functional structures. This work aims to explore how NCAs can act as a bridge for multi-scale encodings, enabling us to formally study attractor manifolds across scales that characterize cell transcriptomic state, cell morphology state, and tissue morphology phenotypes. By drawing connections between state transition dynamics and the broader context of multi-scale topological spaces, we begin to understand how cellular interactions encode complex tissue-level phenotypes, providing a pathway towards unifying mechanistic and functional perspectives of biological systems. I invite the audience to discuss how we might further leverage the lens of topology and category theory to help formalize and takle these multi-scale systems biological challenges.

Qi Zhu
Condensed Anima as a Fractured Topos

Clausen-Scholze and Barwick-Haine being the main players in the game, it is no surprise that condensed mathematics has been popular in recent years. They define condensed objects to be sheaves on the category of compact Hausdorff spaces CHaus for quite a simple Grothendieck topology and so it is not far-reaching that this category should inherit various nice structures and properties. In joint work with Nima Rasekh, we realize one such structure, namely we realize the ∞-topos of condensed anima Cond(An) as a fractured ∞-topos in the sense of Carchedi and Lurie.
In this talk, I will give a crash course on condensed mathematics and explain the notion of fractured topoi with the goal of putting a fractured structure on Cond(An). Time permitting, I will give an application by proving that the ∞-topos Cond(An) has enough points.