EPFL Topology Seminar Fall 2026

Location: Room CM 1 517 (And sometimes Zoom)

Programme

Abstracts

Steve Oudot
Estimating the persistent homotopy type of filtered topological spaces from point samples

Given a sufficiently regular topological space X, Latschev’s theorem says that, for a dense enough sample P on X, and for suitable values of delta>0, one can recover the homotopy type of X from the Rips complex of parameter delta built on P. Our question here is: can this result be extended to recovering the persistent homotopy type of the filtration of X by the sublevel sets of a sufficiently regular function f: X->R^n? We answer the question in the affirmative, stating a persistent version  of Latschev’s theorem in which the Rips complex is filtered by the values of f at its vertices. For the sake of the exposition, I will first focus on estimating the persistent homology of the filtration from a pair of filtered Rips complexes, using techniques in topological data analysis; then, I will move on to estimating the persistent homotopy type of the filtration from a single filtered Rips complex,  using ingredients coming from homotopy theory.
The main result of this talk is joint work with Lukas Waas; the intermediate result on persistent homology estimation is joint work with Ethan André, Jingyi Li and David Loiseaux.

Shunsuke Tada
Central limit theorems for persistent Betti numbers

Various persistent homological invariants have been developed in recent years, raising the question of whether their probabilistic behavior can be understood in a unified way. In this talk, I will introduce a method based on homological algebra for deriving central limit theorems for persistent Betti numbers associated with \mathbb{R}-indexed chain complexes constructed from a homogeneous Poisson point process. The key point is that the translation invariance, weak stabilization, and bounded moment assumptions in the Penrose–Yukich framework can be verified by studying the kernels and cokernels of add-one maps between chain complexes. As applications, we recover the known central limit theorem for persistent Betti numbers arising from simplicial complex filtrations and establish central limit theorems for blurred magnitude homology and magnitude Betti numbers of random geometric graphs.

Toshitaka Aoki
Interleavings on posets via latching- and matching-type constructions

Interleaving distance is a fundamental tool in persistence theory, but its classical formulation relies on translations of the indexing poset. In this talk, I will introduce latching- and matching-type constructions associated with a height-difference function on a poset, yielding an \mathbb{R}_{\geq 0}-indexed family of adjoint endofunctors and interleavings on general posets. I will discuss basic properties and examples, including the recovery of the usual multiparameter interleaving, as well as pullback stability and stability under perturbations of the height-difference function. I hope to discuss possible connections with homotopy theory, related notions of interleaving, and potential applications of this framework.

El Mehdi Cherradi
Exploring the ∞-categorical semantics of homotopy type theory

This talk is meant to give an overview of the semantics of HoTT in (∞,1)-categories, tackling the key challenge of rigidification: the process of bridging homotopy-coherent notions phrased in models of higher categories such as quasicategories, and their type-theoretical counterparts that must account for the inherent strictness of type theory.
After quickly recalling the usual categorical semantics of extensional Martin-Löf type theory within locally cartesian closed category, the plan is to present some useful ideas to carry over some of the argument to the higher setting. Specifically, I will discuss the process of turning an elementary higher topos into a model of HoTT, and sketch important ideas used to prove the internal language conjecture for locally cartesian closed (∞,1)-categories.