A number field ${\displaystyle K}$ is monogenic if the ring of integers admits a power integral basis, i.e., if ${\displaystyle {𝒪}_K=\mathbb{Z}[\alpha ]}$ for some ${\displaystyle \alpha \in {𝒪}_K}$. This talk will survey recent results on monogeneity with a focus on the speaker's recent results on radical extensions and torsion fields of abelian varieties.

In the 1980s commutator theory was generalized from groups to arbitrary algebras using the socalled term condition commutator.  Since this commutator has particularly nice properties for algebras in varieties with modular congruence lattices, the structure theory for abelian, nilpotent, solvable,.. algebras is best developed in this setting. Unfortunately a variety of semigroups is congruence modular if and only if it consists of groups.
We give an overview of properties of the binary and higher commutators for semigroups. In particular we investigate what the derived notions of nilpotence, supernilpotence, and solvability mean and how they relate to classical concepts in semigroup theory.

This is an ongoing project with Dr. Eirini Chavli (University of Stuttgart), Dr. Maud DeVischer (City University of London), Dr. Alison Parker (University of Leeds), and Dr. Ulrica Wilson (Morehouse College). The walled Brauer algebra is a subalgebra of the Brauer algebra. As a diagram algebra, the walled Brauer algebra can be represented using diagrams of the Brauer algebra that satisfy certain conditions. We will discuss previous work has been done to describe the center of this algebra in the cases where the algebra is semisimple. It was conjectured that the center in the nonsemisimple case can be described the same way. We will describe current work on this conjecture.

The Axiom of Choice is one of the most pervasive principles of foundational mathematics, with much study focusing on the consequences and equivalences of it as well as the “disasters” present in its absence. In this talk, we will introduce an alternative principle, Determinacy, and see that the absence of Choice doesn’t have to be such a disaster after all. Determinacy is a game theoretic axiom that sits in an interesting position compared with Choice, directly contradicting it but sharing some of Choice’s implications and coming with its own novel consequences. We will explore some parts of the picture of set theory under Determinacy as compared with that under Choice, consider various forms of Determinacy, and comment on the consequences of our considerations for foundations of mathematics and logic.

It is known, thanks to the work of Deligne and Mostow from 1986, that some ball quotients can be interpreted as moduli spaces of points in the line.  Such interpretation has been expanded to K3 surfaces thanks to the work of Kondo and Dolgachev.  Within this context, we will describe ongoing work with Matt Kerr and Luca Schaffler, which allows us to give geometric interpretations to the toroidal compactifications of such ball quotients.

Motivic homotopy theory over R is interesting in part because of its connection to ordinary stable homotopy theory and to C_2-equivariant homotopy theory. In this talk I will review some of these connections, and discuss work in progress with Dan Isaksen to compute R-motivic stable homotopy groups of spheres using an Adams spectral sequence. One of our main applications is to a variant of the Mahowald invariant which can be computed using knowledge of the R-motivic Adams spectral sequence.