EPFL Topology Seminar Spring 2025

 Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Léonard Guetta
A double categorical model of (∞,1)-Categories

In this talk, I will present recent joint work with Lyne Moser (arXiv:2412.15715), in which we prove that the category of double categories can be equipped with a model category structure which is related, via a zig-zag of Quillen equivalences, to the Rezk model structure on simplicial spaces. In particular, with this model structure, the category of double categories serves as a model for the homotopy theory of (∞,1)-categories. Along with sketching the proof of this theorem, I will highlight some notable features of this model of (∞,1)-categories, such as the ability to compute homotopy colimits in a very explicit and straightforward manner. Time permitting, I will also discuss some conjectural applications of this result to future projects.

Jérôme Scherer
The rational plus construction

This is joint work with Ramon Flores and Guille Carrion. This talk will start with a historical introduction on Quillen’s plus contruction (in integral homology) and its analogues for homology with coefficients. I will focus on a functorial description as a Bousfield localization and present the universal acyclic space Berrick and Casacuberta came up with in 1999. Our contribution is the construction of a universal rationally acyclic space. This allows us to understand better the class of rationally acyclic spaces and to study the behavior of the acyclization-plus construction fiber sequence.

Bruno Gavranović
Category Theory ∩ Deep Learning: the past, the present, and the future

Despite its remarkable success, deep learning is a young field. Like the early stages of many scientific disciplines, it is permeated by ad-hoc design decisions. From the intricacies of the implementation of backpropagation, through new and poorly understood phenomena such as double descent, scaling laws or in-context learning, to a growing zoo of neural network architectures — there are few unifying principles in deep learning, and no uniform and compositional mathematical foundation. In this talk, I will give you a sense of what the necessary components of such a foundation are, the role that category theory plays in it, and the kind of models and new capabilities this systematic approach unlocks.

Nima Rasekh
From Internal ∞-Categories to the Foundation of Mathematics

Internal categories extend the concept of ordinary categories, enabling the application of categorical methods across diverse contexts, from Lie groupoids to condensed categories. A particularly elegant use of internal category theory arises in higher categorical sheaf theory, having resulted in powerful ∞-categorical techniques and results internal to Grothendieck ∞-topoi. In this talk, we seek to generalize several results to more general internal ∞-categories, only to encounter unexpected challenges that surprisingly intertwine ∞-category theory with the foundations of mathematics.

Severin Bunk
Infinitesimal higher symmetries and connections on higher bundles

Every principal bundle on a manifold has a universal symmetry group. It controls equivariant structures, and its tangent Lie algebra controls connections on the bundle.
In this talk we extend these concepts to higher, or categorified bundles. We will use a family‐version of the Lurie-Pridham Theorem from derived deformation theory to compute the associated L_∞-algebras, or rather L_∞-algebroids. That allows us to provide a unified definition of connections on higher bundles and an algebraic formulation of differential cohomology. We elaborate in particular on the case of higher U(1)‐bundles, or n‐gerbes. This is joint work with Lukas Müller (Perimeter Institute), Joost Nuiten (Toulouse) and Richard Szabo (Heriot-Watt).

Johannes Schipp von Branitz
An Informal Introduction to Homotopy Type Theory via Synthetic Homotopy Theory

Homotopy type theory is an intuitionistic type theory with the aspiration of becoming a foundation for all of mathematics. In this talk we introduce the basic type theoretic constructions and their semantic interpretation in an infinity topos, before discussing the synthetic analogues of classical homotopy theoretic constructions such as Eilenberg-MacLane spaces and Whitehead’s theorem.

Bastian Rieck
From Coarse to Fine and Back Again: Geometry, Topology, and Deep Learning

A large driver contributing to the success of deep-learning models is their ability to synthesize task-specific features from data, which typically outshine hand-crafted features. Thus, for a long time, the predominant belief was that “given enough data, all features can be learned.” However, it turns out that certain tasks require imbuing models with inductive biases such as invariance or equivariance properties that cannot be readily gleaned from the data! This is particularly true for data sets that model real-world phenomena, including those that involve relations beyond the dyadic, creating a crucial need for different approaches.
In this talk, I will present novel advances in harnessing multi-scale geometrical-topological characteristics of data, focusing in particular on how the tandem of geometry and topology can improve (un)supervised tasks in representation learning. Underscoring the generality of a hybrid geometrical-topological perspective, I will furthermore showcase applications from a diverse set of data domains, including point clouds, graphs, and higher-order combinatorial complexes.