My research focus is in Equivariant Algebra: the theory of those algebraic objects which arise from equivariant (stable) homotopy theory. The algebraic objects of chief importance in this theory are Mackey Functors, Green Functors, and Tambara Functors, which are equivariant generalizations of Abelian groups and commutative rings, respectively. I am also interested in incarnations of these objects in motivic homotopy theory & beyond.

Preprints & Publications

• The Spectrum of the Burnside Tambara Functor (in preparation)
  joint work with: Maxine Calle, David Chan, David Mehrle, J.D. Quigley, and Danika Van Niel
  Following work of Nakaoka, Calle-Ginnet, and the authors, we provide a complete description of the Nakaoka spectrum of the Burnside Tambara functor for an arbitrary finite group \(G\).
• Algebraically Closed Fields in Equivariant Algebra (arXiv)
  joint work with: Jason Schuchardt and Noah Wisdom
  Using the Burklund-Schlank-Yuan abstraction of "algebraically closed" to "Nullstellensatzian", we show that a \(G\)-Tambara functor is Nullstellensatzian if and only if it is the coinduction of an algebraically closed field (for any finite group \(G\)). As a consequence we deduce an equivalence between the \(K\)-theory spectrum of any Nullstellensatzian \(G\)-Tambara functor with the \(K\)-theory of some algebraically closed field.
• On the Tambara Affine Line (arXiv)
  joint work with: David Chan, David Mehrle, J.D. Quigley, and Danika Van Niel
  Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any \(G\)-ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order \(p\); we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order \(p\) in terms of the Zariski spectra of \(\mathbb{Z}[x]\), \(\mathbb{Z}[x,y]\), and the ring of cyclic polynomials \(\mathbb{Z}[x_0, \dots, x_{p-1}]^{C_p}\). To obtain these results, we introduce a "ghost construction" which produces an integral extension of any \(C_p\)-Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors.
• Norms of Generalized Mackey and Tambara Functors (arXiv)
  Let \(G\) be a finite group. A \(G\)-Tambara functor can be defined as a product-preserving functor \(\mathcal{P}_G \to \mathsf{Set}\) (satisfying one additional condition), where \(\mathcal{P}_G\) is a category that is constructed in a straightforward way from the category of finite \(G\)-sets. By replacing the category of finite \(G\)-sets with other categories, we obtain a more general notion of "Tambara functor". This more general notion subsumes the notion of motivic Tambara functors introduced by Bachmann. In this article, we extend a result of Hoyer about \(G\)-Tambara functors to this more general context.