# Robin Stoll: Homotopy automorphisms, graph complexes, and modular operads

**Time: **
Wed 2022-05-11 10.00

**Location: **
Kräftriket, House 6, Room 306 and Zoom

**Video link: **
Meeting ID: 677 9204 4713

**Doctoral student: **
Robin Stoll

**Opponent: **
Manuel Krannich (Karlsruhe Institute of Technology)

**Supervisor: **
Alexander Berglund

### Abstract

This licentiate thesis consists of two papers. In Paper I we identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of \(S^k \times S^l\), where \(3 \leq k < l \leq 2k − 2\). We express the result in terms of Lie graph complex homology. In Paper II we show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. As an application, we sketch a Koszul duality theory for modular operads.