# Seminars in geometry/topology

## Spring 2014

## Monday March 10, 13:15 - 14:15, room 734, Sentralbygg 2

Alexander Schmeding: * : The Lie group of real analytic diffeomorphisms is not real analytic *

**Abstract:** It is a classical result that the Lie group of real analytic diffeomorphisms of a compact real analytic manifold is a smooth Lie group. In the convenient setting of global analysis this Lie group is even a real analytic Lie group [2].
The starting point of our work is a stronger condition for real analyticity (cf. [1]): A map is called real analytic if it admits around each point an extensions to a holomorphic map on a complexification.
Using this definition, we put a natural real analytic manifold structure on the group of real analytic diffeomorphisms. However, with respect to this structure, the group operations turn out to be not real analytic.
In the talk we will sketch the construction together with an explicit counter-example to the real analyticity of the group operations.

This is joint work with R. Dahmen (Universität Darmstadt).

[1] H. Glöckner: *Infinite-dimensional Lie groups without completeness restrictions.*
In Strasburger, A., Hilgert, J., Neeb, K.–H., and Wojtyński, W. (eds.):
Geometry and Analysis on Lie Groups, vol.55 of Banach Center Publication, pp. 43–59 (2002).

[2] A. Kriegl, P.W. Michor: *The convenient setting for real analytic mappings*, pp. 105- 159 Acta Math. 165 (1990).

## Monday March 3, 13:15 - 14:15, room 734, Sentralbygg 2

Magnus Hellstrøm-Finnsen: * The Homotopy Theory of (∞,1)-Categories *

**Abstract:** The study of stable (∞,1)-categories can for example be motivated from the fact that many prominent examples of triangulated categories are given almost by definition as homotopy categories of stable (∞,1)-categories. The objective for this talk is however to discuss some of the ideas and notions in the theory of (∞,1)-categories in order to state and understand that the homotopy category of a stable (∞,1)-category is a triangulated category. Here, the study of (∞,1)-categories is formalised by quasi-categories, having the benefit that many notions for our purpose adapt intuitively and well-motivated from the theory of ordinary categories. Quasi-categories are defined to be simplicial sets with the inner horn filler property.

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

Armindo Costa (University of Warwick): * Stochastic Topology Part II *

**Abstract:** In this talk we will continue to explore the topological features of the clique complex model and introduce the Linial–Meshulam model for random 2-complexes. The fundamental groups of random complexes can be seen as a model for random groups, which may be of independent interest.

We will also discuss a probabilistic approach to the Whitehead conjecture: is it true that any subcomplex of an aspherical complex is itself aspherical?

This is joint work with Michael Farber.

## Friday January 10, 13:15 - 14:15, room 734, Sentralbygg 2

Armindo Costa (University of Warwick): *Random Simplicial Complexes Part I*

**Abstract:** Stochastic Topology is an emerging area of research which studies the topology of random objects.
The interest in random objects is two-fold:
1) Randomness models the real world: from shape recognition to data analysis and configuration spaces of mechanical systems, it is desirable to obtain statistics of relevant topological features: eg distribution of the betti numbers.
2) Probabilistic method: this non-constructive method, pioneered by Erdos, allows one to prove the existence of complex mathematical objects with prescribed properties. For example the probabilistic method may be used to disprove mathematical conjectures.
We will focus on two specific models of random simplicial complexes; the Linial-Meshulam model and the clique complex model (introduced by M. Kahle). In this talk we will survey the main results in these models. In a subsequent talk we will discuss recent results concerning the fundamental group of random complexes in both models and a probabilistic approach to the Whitehead asphericity question which remains open since 1941.