2026
Andre Nies, 30 Jan, 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.
Nicola Di Vittorio, 26 Feb, 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.