Topology Seminar Fall 2025

e Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Nikolas Schonsheck
Circles in the brain: learning local geometry and nonlinear topology via spike-timing dependent plasticity

Brains use a variety of coordinate systems to encode information. Sometimes these coordinate systems are linear and can be recovered from population activity using standard techniques. Often, however, they are not: many coordinate systems exhibit nonlinear global topology for which such tools can be less effective. Notably, circular and toroidal manifolds describe activity in neural systems across a range of species; common examples include orientation-selective simple cells in primary visual cortex, head-direction cells in thalamic circuits, and grid cells in entorhinal cortex. That such structured information appears in both sensory and deep-brain regions raises a basic question: is the propagation of nonlinear coordinate systems a generic feature of biological neural networks, or must this be learned? If learning is necessary, how does it occur? We apply methods from topological data analysis developed to quantitatively measure propagation of such nonlinear manifolds across populations to address these problems. We identify a collection of connectivity and parameter regimes for feed-forward networks in which learning is required, and demonstrate that simple Hebbian spike-timing dependent plasticity reorganizes such networks to correctly propagate circular coordinate systems. We also observe during this learning process the emergence of geometrically non-local experimentally observed receptive field types: bimodally-tuned head-direction cells and cells with spatially periodic, band-like receptive fields. 
Recent preprint: https://www.biorxiv.org/content/10.1101/2025.08.27.672728v1

Clemens Bannwart
Tagged barcodes for Morse-Smale vector fields

In this talk I will present some new invariants for Morse-Smale vector fields. In the first part of the talk we consider the gradient-like case, where we construct an invariant called ‘tagged barcode’. We start by considering the Morse complex, which is a chain complex defined in terms of singular points and flow lines of the vector field and whose homology is isomorphic to the homology of the underlying manifold. We then identify a sequence of pairs of singular points along which we can simplify the Morse complex. Recording the distances between the pairs we simplified yields the tagged barcode. 
In the second part of the talk we include closed orbits into our analysis, presenting a different method where we use a filtration of the manifold by unstable manifolds first described by Smale. We consider the spectral sequence associated with this filtration and then rearrange the algebraic information in order to obtain a chain complex. 
This is joint work with Claudia Landi. The content of the first part can be found on arXiv (identifier 2401.08466), the content of the second part is work in progress.

Aloïs Rosset
Quasitoposes in graph rewriting

Graph rewriting studies algorithms that transform graphs. Fundamental questions include whether algorithms terminate and whether different rewriting paths yield the same result. A challenge lies in the wide variety of graph definitions: multigraphs, hypergraphs, bipartite graphs, coloured graphs, labelled graphs, etc. Category theory provides a unifying framework: algorithms can be formulated abstractly and then instantiated in any suitable category of graphs. Meta-theorem can likewise be expressed in terms of categorical properties. In recent years, (elementary) quasitoposes have attracted attention, as they capture key properties for unifying algorithms, handling relabelling, and ensuring certain meta-theorems. In this talk, I will present two results from my thesis. First, I show that endowing presheaves with fuzzy structures form quasitoposes, a setting that captures relabelling in graphs. Second, I investigate LT-topologies on certain categories of graphs, including simplicial sets, and identify the quasitoposes that arise from them.

Sungyeon Hong
Complexity? Topology? Well, Cybernetics!

In this talk, I will walk you through my core interests in research, a blend of complexity science and topological data analysis – well, cybernetics in sum! After telling you a little bit of background story, I will spend some time presenting a paper recently delivered at a conference as a joint work with Ben Swift, entitled “Semantic Topologies in the Recursive Application of GenAI Models”, for a wider discussion. 

Matan Prasma
Prospects for Higher Group Theory

The characterisation of loop spaces as A-infinity spaces in the early 60’s, suggested they constitute a homotopy-coherent analogue of groups. On the other hand, Higher Category Theory, allows us to identify loop spaces with pointed, connected ∞-groupoids, or ∞-groups. It is tempting to ask how much of ordinary group theory can be carried out to the homotopy-coherent framework, perhaps to constitute ‘Higher Group Theory’?  

A decade ago, I tried to develop such a framework and obtained several partial results. In this talk, I’ll present some of these results, like normal maps and their properties, isomorphism and Sylow theorems as well as prospects for future enquiries.  

Virgile Constantin
Higher Covering Maps, Deck Transformations, and Yoneda in an ∞-Topos

A classical and elementary result in topology identifies the group of deck transformations of a normal covering map (with discrete fibers) with a quotient of the fundamental group. In this talk, I will explain how to internalize this result in a fixed ∞-topos E, obtaining an isomorphism of group objects in E. Even better, for coverings with (n-1)-truncated fibers, we obtain an analogous isomorphism of n-group objects between the deck transformation n-group and a suitable quotient of the fundamental n-group. The key input is an internal form of the Yoneda lemma (and embedding) in E; I will present and motivate its formulation, and highlight its central role in the proof. As a direct corollary, we obtain a uniqueness result for quotients of higher groups. The talk is designed to be accessible: no prior knowledge of ∞-topoi is necessary.

Ana Marija Jakšić
Topology-informed mapping of brain structure and function

Understanding how physical brain structure gives rise to cognition and individual behaviour remains one of the central unsolved problems in neuroscience. I will present how topological data analysis (TDA) allows us to extract meaningful structure–function relationships directly from raw 3D brain anatomy, without requiring prior annotations or training data. First, I will introduce TopoTome, a fully unsupervised TDA-based framework that encodes 3D biological images in topological space to segment and quantify structural organization across imaging modalities, species, and scales. Unlike deep learning, it generalizes inherently and advances brain image analysis from intensity-based volumetry to structural topology.

I will then show how we apply this framework to hundreds of individual Drosophila brains, revealing a surprisingly rich natural diversity in micro- and macro-scale brain architecture that predicts individuality in behaviour. By modelling complete brain structure across scales, we uncover interpretable topological fingerprints that forecast behaviours such as locomotion and learning. These results demonstrate that individual behaviour emerges from whole-brain structural topology, not isolated neural modules, establishing TDA as a powerful route to mechanistic brain-behaviour prediction.

Isaac Ren
Computing relative Betti diagrams of functors indexed by posets

One of the fundamental results of topological data analysis is that functors from the real line (viewed as a posetal category) to the category of finite-dimensional vector spaces split as the direct sum of simple interval modules. When replacing the real line with an arbitrary (finite) poset, this decomposition no longer holds. Instead, we turn to relative homological algebra to develop new invariants for such functors. We focus on relative projective resolutions, which are exact sequences of functors that are projective relative to a given family of “simple” modules. Under certain reasonable conditions, these resolutions consist of direct sums of simple functors, whose multiplicities we call the relative Betti diagrams. We then compute these relative Betti diagrams using local Koszul complexes, which is simpler than first computing the full resolution.

After presenting our general results, I will focus on the case of Betti diagrams relative to lower hook modules, which are closely related to signed barcodes (Botnan-Oppermann-Oudot 2024), and explore some tricks to improve their computational complexity. In particular, I will present a spectral sequence that converts kernels into cokernels, the latter being easier to handle computationally.

John Miller
Categorical fragmentation and filtered topology

I will review notions of categorical complexity, and the more recent work of Biran, Cornea and Zhang on fragmentation in triangulated persistence categories (TPCs), then go on to discuss applications of this to filtered topology. In particular, we will consider a suitable category of filtered topological spaces and detail some constructions and properties, before showing that an associated ‘filtered stable homotopy category’ is a TPC. I will then give some interesting results relating to this.

Danica Kosanović
Knotted families from graspers

I will introduce a geometric object called a grasper, and explain how it gives rise to families of embeddings of an arc or a circle into an arbitrary manifold of any dimension. These families are detected by Goodwillie-Weiss embedding calculus, so the question of their nontriviality is reduced to algebraic topology. In dimension 3 our discussion reduces to constructions of knots using gropes/claspers.

Fabien Donnet-Monay
Grothendieck topoi and Giraud’s theorem

The notion of a sheaf on a topological space can be generalised to a small category. To do so we must introduce additional structure since arbitrary categories lack a notion of coverings. This structure that we will define through the talk is called a Grothendieck topology. A Grothendieck topos is then defined as any category equivalent to the category of sheaves on a category endowed with a Grothendieck topology. We will state and give a sketch of the proof of Giraud’s theorem that gives sufficient and necessary conditions for a locally small category with all finite limits to be a Grothendieck topos. No prior knowledge of sheaf theory is needed to understand the talk.

Hugo Moeneclaey
Synthetic Stone Duality: A Synthetic Approach to Condensed Mathematics.

Synthetic Stone duality is an extension of Homotopy Type Theory (HoTT) by 5 well-chosen axioms. These axioms are validated by the interpretation of HoTT in the higher topos of light condensed anima. Therefore, any results we prove in synthetic Stone duality can be interpreted as a result about light condensed anima.
First, we will explain the general concept of synthetic mathematics, with an emphasis on geometry, HoTT, higher topoi and cohomology. Then we will present the 5 axioms of Synthetic Stone duality and give detailed proofs of some of their elementary consequences, to give a feeling of how it is to work with them.  We will then give an overview of our synthetic version of Theorem 3.2 from Peter Scholze Lecture Notes on condensed mathematics (itself adapted from Roy Dyckhoff). Our version states that the cohomology of a compact Hausdorff space with countably presented coefficients can be computed from a cover of X by a Stone space.
If time permits, we will sketch applications of this result to the shape modality, which gives a convenient interface between topological spaces and their homotopy types.

Kathryn Hess
The unreasonable effectiveness of actegories

The notion of a module over a ring can be categorified to that of an actegory, consisting essentially of a functorial action of a monoidal category on a (2-)category.  Though not always explicitly identified as such, actegories show up all over pure mathematics, mathematical physics, and theoretical computer science. I will describe several interesting examples of actegories and of morphisms between them and explain in particular how to see (homotopy) colimits and limits as morphisms between two distinct actegory structures on the 2-category of categories. I’ll conclude by explaining how this type of actegory morphism can be used to build a natural machine that produces monads or comonads, like those that are central to the construction of the discrete calculus of Bauer-Johnson-McCarthy, and present a broad range of examples of such machines.

Joint work with Kristine Bauer, Brenda Johnson, and Julie Rasmussen 

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.)