Transfer systems are combinatorial objects with strong connections to homotopy theory. A transfer system on a partially ordered set, P, is a "coarsening" of the partial order on P (or, in categorical language, a full subcategory of P), satsifying certain conditions. They are interesting to algebraic topologists for several reasons, but were originally introduced due to a useful correspondence with N-infinity operads. Because of this connection to equivariant homotopy theory, we are especially interested in the case where P is the subgroup lattice of some group.
At a recent Mathematical Research Community workshop, several small groups were formed to explore questions about transfer systems. The group I worked with was led by Scott Balchin and including Miguel Barrero, Steve Scheirer, Yuri Sulyma, Noah Wisdom, and Valentina Zapata Castro. We focused on generating sets for transfer systems, which in some cases can be represented by diagrams we're calling "rainbows". Work on this project is ongoing!
A hierarchical clustering is a way of selecting a collection of disjoint subsets from a data set, given some parameters. As these parameters change, the clusters of data points either break apart and disappear or appear and merge together. Questions of stability concern how a hierarchical clustering changes when the input data is perturbed. If a construction is stable, then similar data sets should give similar outputs. In particular, I have been thinking about stability for layer points, which are a condensed way of describing hierarchical clusterings. You can find some of my results in the preprint Stability for layer points.
The Steenrod algebra arises topologically from natural transformations on cohomology that commute with the suspension isomorphism. (We call these "stable cohomology operations".) In some situations, computations over the Steenrod algebra can be completed by doing an equivalent computation over a subalgebra. I spend a lot of time thinking about a particular subalgebra, A(1). (Check out Bob Bruner's beautiful drawings of A(1) resolutions to see one reason A(1) is fun to think about.)
My paper Classifying and extending Q0-local A(1)-modules gives a classification of a particular class of A(1)-modules. This has implications for determining which A(1)-modules lift to modules over the whole Steenrod algebra, which is a question I continue to find interesting.
Equivariant homotopy theory applies tools from representation theory to an appropriate category of spectra. So, for example, in the category of G-spectra for some group, G, there are multiple suspension functors, indexed by representations of G.
Through Women in Topology (WIT), I worked with Teena Gerhardt, Kathryn Hess, Inbar Klang, and Hana Kong to establish both computational and structural results for equivariant analogues of Hochschild homology and related invariants.
Our first paper, Computational tools for twisted topological Hochschild homology of equivariant spectra, gives several example computations of Hochschild homology of Green functors and twisted topological Hochschild homology, including via a novel equivariant Bokstedt spectral sequence.
In our second paper, A shadow framework for equivariant Hochschild homologies , we extend these invariants as shadows (particular types of functors on bicategories). This provides properties like Morita invariance and agreement as a direct consequence.