EPFL Topology Seminar 2023 Spring

 Location: Real World (And sometimes Zoom)

Program

Abstracts

Jérôme Scherer,
Relative plus construction:

This is the project with Guille Carrion that we worked on during his stay in our group last Fall. A relative plus construction appears in a recent paper by Broto, Levi, and Oliver in the form of a classical Quillen cell attachment. The aim is to kill a given subgroup of the fundamental group without changing the homology with coefficients in a ring R. We provide a functorial version by working in the category of maps of spaces, which allows us to explain why the BLO version is not uniquely defined and why the kind of subgroup one can kill have different nature depending on the characteristic and the torsion in the ring R.

Samuel Lavenir,
Hilton-Milnor’s theorem in infinity-toposes:

Hilton-Milnor’s theorem is an important result in homotopy theory which provides a decomposition of the loop space on a finite wedge of suspension spaces into a certain (weak) infinite product. We have shown that a version of this theorem holds in any infinity-topos, and in hypercomplete infinity-categories with universal pushouts. This framework significantly simplifies the proof and allows to apply the theorem in a wide variety of contexts. These ideas build on recent work by Anel-Biedermann-Joyal-Finster and Devalapurkar-Haine on synthetic homotopy theory in higher toposes. After providing some context for Hilton-Milnor’s theorem, I will explain the main ideas of its higher categorical proof. If time permits, I will mention some possible applications in the theory of higher groups.

Lukas Brantner,
On deformations and lifts of Calabi-Yau varieties in characteristic p:

A smooth projective variety Z is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how Calabi-Yau varieties in characteristic p deform. More precisely, we show that if Z has degenerating Hodge–de Rham spectral sequence and torsion-free crystalline cohomology, then its mixed chracteristic deformations are unobstructed; this is an analogue of the classical BTT theorem in characteristic zero. If Z is ordinary, we show that it moreover admits a canonical lift to characteristic zero; this extends classical Serre-Tate theory. Our work generalises results of Achinger–Zdanowicz, Bogomolov-Tian-Todorov, Deligne–Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre–Tate, and Ward.

After reviewing the envisioned Tannakian viewpoint on mixed motives and giving various definitions of motivic categories I will explain how one can model a subcategory of the category of étale motives generated by a 1-motive by modules over a commutative algebra in a stable category of representations of a general linear group over the integers. This builds upon rational results by Iwanari.

Maxime Ramzi,

The multiplicativity of Euler characteristics and Becker-Gottlieb transfers: 

The Euler characteristic is a fundamental invariant of spaces and has many pleasant properties with respect to various ways of ‘decomposing’ spaces. One of these properties is its multiplicativity: if F-> E -> B is a fiber sequence of connected finite spaces, chi(E) = chi(F) chi(B).  A natural class of spaces for which the Euler characteristic still makes sense is the class of finitely dominated spaces – in this talk, I will discuss proofs of the corresponding equality when B,E and F are only assumed to be finitely dominated. I will explain the connection between this seemingly trivial (with nontrivial proof) property and deep questions in algebraic K-theory.

Parts of this talk are based on joint work with John Klein and Cary Malkiewich, and (time permitting) others on joint work with Shachar Carmeli, Bastiaan Cnossen and Lior Yanovski.

Peter Arndt,
Abstract motivic homotopy theory:

We will start with a quick introduction to motivic homotopy theory, emphasizing the parallels to the homotopy theory of topological spaces. We will then develop some basic motivic homotopy theory in an abstract setup: We replace the infinity category of motivic spaces by a presentable, cartesian closed infinity category (PCCC) and the multiplicative group scheme by a commutative group object G in this category. Starting from this, we will construct basic geometric objects like projective spaces, see a representation theorem for G-bundles, a Snaith type algebraic K-theory spectrum, Adams operations, rational splittings and a rational Eilenberg-MacLane spectrum. Examples of our abstract setup include motivic spaces built from algebraic geometry, complex and non-archimedian analytic geometry, derived geometry, log geometry, as well as the many proposals for geometry over the field with one element. Furthermore, there is an initial example of our setup, the classifying PCCC for commutative groups, which is a model for univalent homotopy type theory. Certain results and constructions, including all those from the talk, carry over from this particular PCCC with its generic commutative group to general PCCCs with commutative groups.

Hadrian Heine,

Weak omega-categories as a model for directed homotopy types

Nicolas Berkouk,
Persistence and the Sheaf-Function Correspondence:

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of k-vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exists non-trivial additive invariants of persistence modules that are continuous for the interleaving distance.