I'll discuss how a certain notion of symmetry captures the computational complexity of approximating homomorphism problems between relational structures, also known as constraint satisfaction problems. I'll present recent results on inapproximability of conflict-free and linearly-ordered hypergraph colourings and solvability of systems of equations.

Upper bounds on the number of incidences between points and lines, tubes, and other geometric objects, have many applications in combinatorics and analysis. On the other hand, much less is known about lower bounds. We prove a general lower bound for the number of incidences between points and tubes in the plane under a natural spacing condition. In particular, if you take $n$ points in the unit square and draw a line through each point, then there is a non-trivial point-line pair with distance at most $n^{-2/3+o(1)}$. This quickly implies that any $n$ points in the unit square define a triangle of area at most $n^{-7/6+o(1)}$, giving a new upper bound for the Heilbronn's triangle problem.

Joint work with Alex Cohen and Cosmin Pohoata.

Fundamental to the study of hyperbolic groups is their Gromov boundaries. The classical Cannon--Thurston map for a closed fibered hyperbolic 3-manifolds relates two such boundaries: it gives a continuous surjection from the boundary of the surface group (a circle) to the boundary of the 3-manifold group (a 2-sphere). Mj (Mitra) generalized this to all hyperbolic groups with hyperbolic normal subgroups. A generalization of the Gromov boundary to all finitely generated groups is called the Morse boundary. It collects all the "hyperbolic-like" rays in a group. In this talk we will discuss Cannon--Thurston maps for Morse boundaries. This is joint work with Ruth Charney, Antoine Goldsborough, Alessandro Sisto and Stefanie Zbinden.

to follow