This brief pedagogical talk introduces key concepts of Carroll geometries, which arise as the limit of relativistic spacetimes in the vanishing speed of light regime. In this limit, light cones collapse along a timelike direction, resulting in a manifold equipped with a degenerate metric. Consequently, physics in such spacetimes exhibits peculiar properties. Despite this, the Carroll contraction is relevant to a wide range of applications, from flat-space holography to condensed matter physics. To complement this introduction, and depending on the audience’s interests, I can discuss Carroll affine connections, symmetry groups, conservation laws, and Carroll-invariant field theories.

Complex dynamics is the study of the behaviour, under iteration, of complex polynomials and rational functions. This talk is about an application of combinatorial algebraic geometry to complex dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point. Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. Two long-standing open questions are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? I will sketch an argument showing that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a special degeneration of degree-2 rational maps that tells us that Per_n has smooth point with Q-coordinates "at infinity”.