A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.  Based on joint work with Sami Douba, Francesco Fournier Facio, and Simon Machado.

Vertex operator algebras provide a succinct mathematical description of the chiral sector of two-dimensional conformal field theories. Various extensions of the framework of vertex operator algebras have been proposed in the literature which are capable of describing full two-dimensional conformal field theories, including both chiral and anti-chiral sectors. I will explain how the notion of a full vertex operator algebra can be elegantly described using the modern language of factorisation algebras developed by Costello and Gwilliam. This talk will be mainly based on [arXiv:2501.08412].

By work of Goodwillie-Weiss, given any manifold $M$ with boundary, there is a cosimplicial space whose totalization is a close approximation to the space of embedding of $[0,1]$ in $M$ with fixed behaviour at the boundary. The resulting homology spectral sequence is known to collapse rationally for $M=\mathbb{R}^n$ by work of Lambrechts-Turchin and Volic. I will explain a new proof of this result which can be generalized to a manifold of the form $M=X\times[0,1]$ with $X$ a smooth and proper complex algebraic variety. This involves constructing an action of some Galois group on the completion of the cosimplicial space. This is joint work with Pedro Boavida de Brito and Danica Kosanovic.