Projects

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 have been working 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.

Recently, in  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.