Seminars in geometry/topology

Topology Seminar 24 February, 14:00 - 15:00, Zoom

Abigail Linton (NTNU): Massey products in moment-angle complexes

Abstract: Massey products are higher cohomology operations that are generally difficult to compute. In Toric Topology, moment-angle complexes are spaces whose cohomology can be studied combinatorially. I will present systematic combinatorial constructions of non-trivial Massey products, which generalise all existing examples of Massey products in moment-angle complexes. This is joint work with Jelena Grbić (Southampton).

Topology Seminar 17 March, 15:00 - 16:00, Zoom

Markus Szymik (NTNU): Knot theory revisited

Abstract: Knot theory is a traditional topic within geometric topology. In my talk, I will explain how we can approach it with some standard tools from homotopy theory to arrive at recent results on the former and open questions for the latter.

Topology Seminar 24 March, 14:00 - 15:00, Zoom

Luca Pol (Regensburg): Representation stability and outer automorphism groups

Abstract: In this talk I will present a framework for studying families of representations of the outer automorphism groups indexed on a collection of finite groups U. One can encode this large amount of data into a convenient abelian category AU which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of Church–Ellenberg–Farb, I will discuss for which choices of U the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. As an application, I will show that some invariants coming from rational global homotopy theory exhibit such a stable behavior.

Topology Seminar 14 April, 14:00 - 15:00, Zoom

Irakli Patchkoria (Aberdeen): Classification of module spectra and Franke’s algebraicity conjecture

Abstract: This is all joint work with Piotr Pstrągowski. Given an E_1-ring R such that the graded homotopy ring pi_*R is q-sparse and the global projective dimension d of pi_*R is less than q, we show that the homotopy (q-d)-category of Mod(R) is equivalent to the homotopy (q-d)-category of differential graded modules over pi_*R. Thus for such E_1-rings the homotopy theory of their modules is algebraic up to the level (q-d). Examples include appropriate Morava K-theories, Johnson-Wilson theories, truncated Brown-Peterson theories and some variations of topological K-theory spectra. We also show that the result is optimal in the sense that (q-d) is the best possible level in general where algebraicity happens. At the end of the talk we will outline how the results for modules can be generalized to the settings where we do not have compact projective generators. This proves Franke’s algebraicity conjecture which provides a general result when certain nice homology theories provide algebraic models for homotopy theories.

Topology Seminar 21 April, 14:00 - 15:00, Zoom

Ludwig Rahm (NTNU): An operadic approach to substitution in Lie-Butcher series

Abstract: Butcher's B-series are a celebrated tool within numerical analysis. Remarkably, they are understood and studied as algebraic structures. Lie-Butcher (LB-)series are a recent generalization of B-series, with the aim to extend numerical integration techniques beyond Euclidean spaces. The program of extending results on B-series to LB-series is an active area of research. In this talk I will give an overview of algebraic structures and results related to B-series, and their LB-series generalizations. I will furthermore present how an open problem in this program was solved using operads.

Topology Seminar 5 May, 14:00 - 15:00, Zoom

Fabio Strazzeri (Institut de Robòtica i Informàtica Industrial, CSIC-UPC. Barcelona): Topological representation of cloth state for robot manipulation

Abstract: Research on robot manipulation has focused, in recent years, on grasping everyday objects, with target objects almost exclusively rigid items. Non–rigid objects, as textile ones, pose many additional challenges with respect to rigid object manipulation. In this seminar we will present how we can employ topology to study the “state” of a rectangular textile using the configuration space of n points on the plane. Using a CW-decomposition of such space, we can define for any mesh associated with a rectangular textile a vector in an euclidean space with as many dimensions as the number of regions we have defined. This allows us to study the distribution of such points on the cloth and define meaningful states for detection and manipulation planning of textiles. We will explain how such regions can be defined and computationally how we can assign to any mesh the corresponding region. If time permits, we will also explain how the CW-structure allows us to define more than just euclidean distance between such mesh-distributions.

Topology Seminar 12 May, 14:00 - 15:00, Zoom

Xin Fu (Ajou University, Republic of Korea): Cohomology of smooth toric varieties: naturality

Abstract: The notion of the moment-angle complex (MAC) Z was introduced by Buchstaber-Panov. It is defined by a union of certain product spaces of discs and circles with a natural action of a torus T. In a topological way, MAC allows us to understand a smooth toric variety as its quotient Z/K, where K is a closed subgroup of T. The cohomology ring of such quotient space or a smooth toric variety is isomorphic to a Tor product, proven by Franz recently. In this talk, we discuss the naturality of Franz’s ring isomorphism with respect to toric morphisms. In general, there are twisting terms. In some examples, we will see trivial and nontrivial twisting terms, which can also reprove the multiplicative results related to the cohomology of these spaces.

This is a joint work with Matthias Franz (Western University).

Topology Seminar 26 May, 14:00 - 15:00, Zoom

Louis Martini (NTNU): Stratified homotopy theory and the exodromy equivalence

Abstract: In this talk I give an overview of the work of Barwick, Glasman and Haine on the homotopy theory of stratified spaces and a refinement of higher Galois theory to this framework. The stratified homotopy hypothesis allows to identify a poset-stratified topological space X with its so-called exit path infinity-category, an invariant that refines the fundamental infinity-groupoid of X. Under the exodromy equivalence, space-valued representations of this infinity-category correspond to constructible sheaves of spaces on X. I will furthermore discuss how this framework can be applied in algebraic geometry to obtain a refinement of the étale homotopy type of a scheme that completely characterizes its étale infinity-topos.