Speaker: Rhiannon Dougall
Title: Proving a ratio limit theorem for random walks on groups using the variational principle.
Abstract: There are strong results for certain random walks on groups when we have enough structure, such as the random walk being symmetric and the group having some good structure. We can manage without symmetry if the group is small enough e.g. abelian.
We will be interested in the case of a non-degenerate (non-symmetric in general) random walk on an amenable group. We are able to obtain a ratio limit theorem, which says that the probability to return to g in n steps is asymptotically proportional to that of returning to some fixed origin; and we have an explicit description of the constant in terms of g.
We'll describe how this result follows from large deviations and the variational principle.
This is joint work with Richard Sharp.
Speaker: Susanna Terron
Title: Thompson representatives and connected sum
Abstract: In 2017 Vaughan Jones introduced a construction associating links to elements of Thompson’s group F and its generalisation F3. He then proved that all links can be obtained in such a way, opening the way to new possible connections between these objects.
In this talk I will present Jones’ construction and extend it to obtain a surjective map into the set of pointed links. I’ll then define a new algebraic structure by endowing F3 with a monoid operation, which turns our map into a surjective monoid homomorphism, with image the monoid of pointed links with operation connected sum. As a consequence, we obtain a standard form for connected sum representatives, which can then be extended to obtain representatives for a certain family of links that we will refer to as tree links.
This is based on the following paper: https://arxiv.org/abs/2511.21259
Speaker: Giovanni Forni
Title: Periodic orbits and weak chaos in polygons
Abstract: We will survey results on the dynamics in billiards in polygons, including a recent result on existence of periodic orbits and joint work with Francisco Arana-Herrera and Jon Chaika on the weak mixing property.
