# Seminars in geometry/topology

## Fall 2014

## Monday August 25, 13:15 - 15:00, room 734, Sentralbygg 2

Erik Rybakken (NTNU): *Extended Topological Field Theories*

## Monday September 1, 13:15 - 14:15, room 734, Sentralbygg 2

Moritz Groth (Radboud University Nijmegen): *Introduction to derivators*

**Abstract:** The first talk culminates in a definition of a derivator, a framework for abstract homotopy theory which was proposed independently by Heller, Grothendieck, and others. The main
claim of the theory is that a good deal of information about a homotopy theory is encoded by universal constructions like homotopy limits and homotopy colimits. These notions are recalled and illustrated by some examples. Key properties of related categorical constructions are recalled as well, hopefully making the definition of a derivator a rather natural one.

## Wednesday September 3, 13:15 - 14:15, room 734, Sentralbygg 2

Moritz Groth (Radboud University Nijmegen): *Stable derivators and canonical triangulations*

**Abstract:** In this talk we show that stable derivators give rise to canonical triangulations. The main goal of this talk is to use this theorem as a pretext to play a bit with the axioms of a derivator. Along the way we study some constructions available in pointed homotopy theories, like cofibers, fibers, suspensions, and loops. We will also show that loop objects are group objects, yielding a conceptual proof of the additivity of stable derivators.

## Thursday September 4, 12:15 - 13:15, room 734, Sentralbygg 2

Moritz Groth (Radboud University Nijmegen): *Abstract representation theory via stable homotopy theory*

**Abstract:** This third talk is on work j/w Jan Stovicek. Tilting theory is a derived version of Morita theory, say studying the existence of derived equivalences of path algebras of quivers
over fields. In this talk we will see that some aspects of this theory are formal consequences of stability, and hence valid as well for representations over rings, in quasi-coherent modules
on schemes, in the differential-graded as well as in the spectral context. If time allows then we will also construct some universal tilting modules, i.e., certain spectral bimodules, realizing these more general equivalences.

## Monday September 15, 13:15 - 14:15, room 734, Sentralbygg 2

Markus Szymik (NTNU): *Homological stability and stable homology I*

**Abstract:** This will be an opinionated introduction to the ideas in the title with focus on definitions and examples.

## Monday September 22, 13:15 - 14:15, room 734, Sentralbygg 2

Markus Szymik (NTNU): *Homological stability and stable homology II*

**Abstract:** After a brief review of the concepts introduced in the first talk, I will focus on a class of algebraic examples, and describe their stable homology in general terms. I will then discuss homological stability and, if time permits, add a remark about representation stability in this context.

## Monday September 29, 13:15 - 14:15, room 734, Sentralbygg 2

Helge Glöckner (Paderborn): *Homotopy groups of topological spaces containing a dense
directed union of manifolds*

**Abstract:** Let \(X\) be a topological space and \(M_i\) for \(i\) in \(I\) be a directed union of (possibly infinite-dimensional) topological manifolds whose union \(D\) is dense in \(M\). In the talk, a criterion will be described which ensures that the homotopy groups of \(X\) can be calculated as direct limits:

\[ \pi_n(X) = \mathrm{dirlim} \; \pi_n(X_i) \]

The result can be regarded as a non-linear analogue of a classical result by R. Palais (where \(X\) is an open subset of a locally convex space \(E\) and \(D\) is the intersection of \(X\) with a dense vector suspace \(F\) of \(E\)).

Applications in infinite-dimensioanl Lie theory are described. In particular, the result enables the calculation of the homotopy groups of Lie groups of rapidly decreasing mappings (as constructed by Boseck, Czichowski and Rudolph in 1981), which was formulated as an open problem by the latter authors.

## Monday October 6, 13:15 - 15:00, room 734, Sentralbygg 2

Espen Auseth Nielsen and Erik Rybakken (NTNU): *Factorization homology as a TQFT*

**Abstract:** How do quantum field theories arise from their algebras of observables? Physicists have knows for some time that the collection of observables of a quantum field theory has the algebraic structure of a factorization algebra. In this talk, we describe how these algebras assigned to manifolds assemble into a topological quantum field theory. To this end, we will introduce the notion of factorization homology with values in \(E_k\)-algebras.

## Monday October 13, 13:15 - 14:15, room 734, Sentralbygg 2

Alexander Schmeding (NTNU): *The Lie group of bisections of a Lie groupoid*

**Abstract:** In the talk we give a short introduction to bisections of Lie groupoids together with some elementary examples. One may think of a biscetion as a "generalised element" of a groupoid. It is well known that the bisections associated to a fixed groupoid form a group.
We will outline how the group of bisections can be endowed with a natural locally convex Lie group structure if the Lie groupoid has compact space of objects.
Moreover, we develop the connection to the algebra of sections of the associated Lie algebroid and discuss the implications of this construction on the level of Lie functors between suitable categories.

This is joint work with Christoph Wockel (Universität Hamburg), see The Lie group of bisections of a Lie groupoid", arXiv:1409.1428v2 [math.DG].

## Monday October 27, 13:15 - 15:00, room 734, Sentralbygg 2

Bjørn Dundas (University of Bergen): *Ind-spaces*

**Abstract:** The aim of algebraic topology is to solve geometric problems by assigning algebraic invariants to spaces. These invariants are often very hard to calculate, but a successful technique - both in applications like persistent homology and in theoretical work - has been to study *filtrations* of the space and try to assemble the invariants for difficult spaces from simpler building blocks.

We take this seriously, and study the category of filtrations - ind-spaces. On one level this is equivalent to study the spaces themselves, but carries the possibility of refining the classical invariants considerably - in surprising manners.

We start by treating this on a level that should be accessible with a minimum of background, and continue to refine our methods in the second 45 minutes.

In particular, we explore the phenomenon that lengths of homotopies often grow out of control. This is open to attack in our setting, since it is reflected in that colimits in Ind-spaces are quite different from colimits in spaces, making it possible to both restrict how big spaces can grow, but also how long systems of homotopies are allowed to tend. We also will want to consider the proper context of the Adams spectral sequence, where we’ll see that even finite fields have exotic modules in a filtered world.

The literature on the dual concept, pro-spaces is much better developed for the simple reason that spaces are built from cells, and this is badly mistreated by limits. However, many of the examples of pro-spaces arise naturally as mapping spaces of ind-spaces. For instance, if you map out of an infinite dimensional space, then the standard procedure is to consider the pro-space you get by considering the mapping space of ind-spaces out of the skeletal filtration. Hence, these examples are fundamentally examples that profitably could be handled by staying in the nicer category of ind-spaces.

## Monday November 17, 13:15 - 14:15, room 734, Sentralbygg 2

Geir Bogfjellmo (NTNU): *The Butcher group is a Lie group*

**Abstract:** The concept of B-series, formal expansions of numerical methods for ordinary differential equations, has been an important tool for numerical analysts over the last decades. In 1972, Butcher showed that numerical integrators which allow a B-series expansion, form an infinite-dimensional group under composition. In 1998, the Butcher group was rediscovered by Connes and Kreimer in the context of renormalization in Quantum Field Theory. Connes and Kreimer also showed that, algebraically, the Butcher group is associated with a Lie algebra.

In this talk, the motivation for using B-series and the Butcher group is revisited, before we move on to the analytic properties of the Butcher group.

We show that the Butcher group is a Lie group modeled on a Fréchet space. The Lie algebra of Connes and Kreimer reappears as a dense subalgebra of the tangent space at the identity of this group.We explore the properties of the Butcher group as an infinite dimensional Lie group, complementing the algebraic treatment by Connes and Kreimer.

Joint work with Alexander Schmeding (NTNU).

## Tuesday November 25, 13:00 - 14:00, room S24, Sentralbygg 2

Christoph Wockel (University of Hamburg): *An introduction to higher bundles and higher Lie groups*

**Abstract:** Higher versions of bundles arise in different fields of mathematics that are inspired by flavours of field theories. For instance string structures, motivated by anomaly cancellation in certain supersymmetric sigma models, can be very conveniently understood in terms of principal 2-bundles for the string 2-group.

In this talk I will give a short introduction into higher Lie groups and their associated notions of higher principal bundles. I will then explain some Lie theoretic applications of this theory (in parts work in progress with A. Schmeding). Depending on time, I will also line out some hypothetical applications in string geometry.

## Thursday December 4, 13:15 - 15:00, room F3, Gamle fysikk

Tore August Kro (Østfold University College): *K-theory and an infinity-category of matrices of manifolds*