Kim Ruane, 14 Sept 2026, CAT(0) Groups, Boundaries, and Splittings
Abstract: It is a celebrated theorem of Bowditch and others that a one-ended hyperbolic group splits over a 2-ended subgroup if and only if the Gromov boundary of the group contains a local cut point. In fact, one can “see” the JSJ decomposition of the group from the structure of the local cut point in the boundary. Papasoglu proved that any finitely presented, one-ended group that is not commensurable to a surface group, splits over a 2-ended subgroup if and only if there is a quasi-line that coarsely separates any Cayley graph of the group. For CAT(0) groups, this gives the existence of a cut pair in the boundary of the space the group is acting on. Papasoglu-Swenson were then able to “see” the JSJ decomposition of a CAT(0) group using the structure of cut pairs in the boundary. In work with Mihalik and later work with Hruska, we consider splittings of one-ended CAT(0) groups over virtually abelian groups (or more general subgroups) and explore the relationship between these splittings and topological properties of the visual boundary of the CAT(0) space. We will discuss these results here using specific examples and see some interesting tools necessary to prove an if and only if statement in the setting of CAT(0) spaces with the isolated flats property.
William Hobkirk, 16 Apr 2026, Magic Braids and Leather Seifert Surfaces
Connections between braids and links have been extensively studied in the context of the braid closure and the plat closure. The study of Seifert surfaces provides another strong connection between links and the orientable surfaces that co-bound them. In this talk I will discuss a construction that allows us to study isotopy classes of Seifert surfaces via a construction from a framed braid. This will take us on a journey through a number of related configuration spaces and corresponding (framed) braid groups as well as the strangely named Homeotopy groups. This approach was inspired by a common leatherworking technique and our results lead to some surprisingly crafty real-world applications that allow you to create seemingly impossible Magic Braids. I will tell you everything you need to know to recognise a Magic Braid in the wild.
Nicola Di Vittorio, 26 Feb 2026, Homotopy Theory Through a Categorical Lens: Derivators and Their Higher-Dimensional Analogues
Homotopy theory is a branch of mathematics that studies spaces and other mathematical objects up to deformation. Two things are considered equivalent if one can be continuously deformed into the other, in a looser sense than homeomorphism: for instance, a disk and a point are considered the same. Category theory provides a general language for talking about mathematical structures and the relationships between them. In this talk, we will see how these two areas interact in a beautiful way.
A central theme is that when doing homotopy theory, you often lose information by passing to a "homotopy category," a simplified version of your original setting where equivalent objects are identified. Remarkably, while a single homotopy category might not tell you everything you want to know, looking at homotopy categories of diagrams often recovers enough information to do useful mathematics. The notion of a derivator packages this idea into a clean set of axioms. If you have seen derived categories in algebra or geometry, those are a natural example to keep in mind.
I will give a gentle introduction to this circle of ideas and then describe work aimed at developing a "two dimensional" version of the theory, capturing some of the structure that appears in the modern theory of higher categories. No prior knowledge will be assumed, and the emphasis will be on motivation and examples.
Andre Nies, 30 Jan 2026, The trivial units property and the unique product property for torsion free groups
Abstract: A torsion free group G satisfies the unique product property if for each pair A,B of finite nonempty subsets, some product in AB can be written uniquely. G satisfies the trivial units property over a domain R if the group algebra R[G] only has the trivial units, the ones of the form rg, where r is a unit in R and g is in G. Fixing a domain, the unique product property implies the trivial units property; the converse implication is not known.
Gardam in 2021 showed that GF_2[P] (where GF_2 is the two-element field) fails the trivial unit property for the Hantzsche-Wendt group P= < a,b | b^{-1} a^2 b = a^{-2}, a{-1}b^2 a = b^{-2}> . We will discuss the computational methods involving SAT solvers that Gardam used. Extending them, we found all 18 nontrivial units supported on the ball of radius 4 (the minimum possible value) in the Cayley graph of P.
We consider the Fibonacci groups F(n,n-1) where n \ge 4 is even. We show that each such group has a solvable word problem. We use SAT solvers to show that F(4,3) fails the unique product property, and discuss work in progress that might lead to a proof that it satisfies the trivial units property over GF_2.
This is joint work with Heiko Dietrich, Melissa Lee, and Marc Vinyals.