Below are two summaries of my research area, a short summary for mathematicians and a summary for a broad audience.
Here is my Research Statement (last updated: September 2022).
Here is a short (~10 minutes) video summary of my research. This video is aimed at folks with some background in algebraic topology.
My research is in equivariant homotopy theory, which is a subfield of algebraic topology. I study algebraic invariants for spaces with chosen symmetries, that is, spaces endowed with a continuous action by some fixed finite group. Much of my work has focused on spaces with order 2 symmetries, and the equivariant analog to singular cohomology, which is known as \(RO(C_2)\)-graded Bredon cohomology.
My research is in a field of mathematics known as algebraic topology. Algebraic topology studies the different ways we can assign meaningful algebraic data to geometric objects. This allows us to then use algebraic tools to answer geometric questions that were otherwise inaccessible. Assigning numerical data to geometric shapes is a recurrent theme in mathematics: we can measure the slope of a line, the angles in a triangle, or the volume of a sphere. As the geometric objects we analyze increase in complexity, so do the data we need to describe them. Algebraic topology provides techniques to study properties of complex shapes, especially those in higher dimensions. While this may seem aimlessly abstract, higher dimensional objects appear in fields from data analysis to string theory, and one needs concrete ways to describe and measure their properties. (They are also just fun to think about!)
I study ways we can use algebraic data to analyze properties of geometric objects with a chosen symmetry (for example a circle has mirror symmetry). The algebraic data needed to capture properties of the geometric object and its symmetries is quite complicated, which is why mathematicians are just recently beginning to understand how these tools work. My research explores how we can concretely compute and understand this algebraic data.
Abstract: Let G be a finite group and V be a G-representation. We investigate the RO(G)-graded Bredon cohomology with constant integral coefficients of the space of ordered configurations in V. In the case that V contains a trivial subrepresentation, we show the cohomology is free as a module over the cohomology of a point, and we give a generators-and-relations description of the ring structure. In the case that V does not contain a trivial representation, we give a computation of the module structure that works as long as a certain vanishing condition holds in the Bredon cohomology of a point. We verify this vanishing condition holds in the case that dim(V)≥3 and G is any of Cp, Cp2 (p a prime), or the symmetric group on three letters.
Abstract: We prove that Hill's characteristic function \(\chi\) for transfer systems on a lattice \(P\) surjects onto interior operators for P. Moreover, the fibers of \(\chi\) have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a "matchstick game" that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups.
Abstract: For the cyclic group C2 we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring \(\mathbb{Z}/\ell\), for \(\ell\) a prime. This is fairly simple for \(\ell\) odd, but for \(\ell=2\) depends on a new splitting theorem. As corollaries of the splitting theorem we compute the associated Picard group and the Balmer spectrum for compact objects in the derived category, and we obtain a complete classification of finite modules over the \(C_2\)-equivariant Eilenberg-MacLane spectrum \(H\underline{\mathbb{Z}/2}\). We also use the splitting theorem to give new and illuminating proofs of some facts about \(RO(C_2)\)-graded Bredon cohomology, namely Kronholm's freeness theorem and the structure theorem of C. May.
Abstract: Let \(C_2\) denote the cyclic group of order 2. We compute the \(RO(C_2)\)-graded cohomology of all \(C_2\)-surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether the action is orientation preserving versus reversing in the orientable case, and one other invariant.
Abstract: In this paper, the authors build on their previous work to show that periodic rational G-equivariant topological K-theory has a unique genuine-commutative ring structure for G a finite abelian group. This means that every genuine-commutative ring spectrum whose homotopy groups are those of \(KU_{\mathbb{Q},G}\) is weakly equivalent, as a genuine-commutative ring spectrum, to \(KU_{\mathbb{Q},G}\). In contrast, the connective rational equivariant K-theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.
Abstract: In this paper, we calculate the image of the connective and periodic rational equivariant complex K-theory spectrum in the algebraic model for naive-commutative ring G-spectra given by Barnes, Greenlees and Kędziorek for finite abelian G. Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. We further deduce a structure theorem for module spectra over rational equivariant complex K-theory.
Abstract: Let \(C_2\) denote the cyclic group of order 2. Given a manifold with a \(C_2\)-action, we can consider its equivariant Bredon \(RO(C_2)\)-graded cohomology. We develop a theory of fundamental classes for equivariant submanifolds in \(RO(C_2)\)-graded cohomology with constant \(\mathbb{Z}/2\)-coefficients. We show the cohomology of any \(RO(C_2)\)-surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general, the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.
Abstract: A surface with an involution can be viewed as a \(C_2\)-space where \(C_2\) is the cyclic group of order two. Up to equivariant isomorphism, all involutions on surfaces were classified in Bujalance et al. (Math Z 211:461-478, 1992) and recently classified using equivariant surgery in Dugger (J Homotopy Relat Struct 14(4):919-992, 2019). We use the classification given in Dugger (2019) to compute the \(RO(C_2)\)-graded Bredon cohomology of all \(C_2\)-surfaces in constant \(\mathbb{Z}/2\) coefficients as modules over the cohomology of a point. We show the cohomology depends only on three numerical invariants in the nonfree case, and only on two numerical invariants in the free case.
Given in the electronic Computational Homotopy Theory Seminar (eCHT) in February 2021.
Link to video recording (45 minutes)
Abstract: When working rationally, we can often translate topological questions to algebraic questions through the use of algebraic models. Recent work of Wimmer gives an algebraic model for rational genuine-commutative equivariant ring G-spectra for G a finite group. In this talk, we describe how we use this model to see there is a unique genuine commutative structure on rational equivariant K-theory for G an abelian group. This is joint work with Anna Marie Bohmann, Jocelyne Ishak, Magdalena Kędziorek, and Clover May.
Given for the Junior Mathematician Research Archive (JMRA) in August 2020.
Link to video recording (26 minutes)
Abstract: Let \(C_2\) denote the cyclic group of order two. Given a manifold with a \(C_2\)-action, we can consider its equivariant Bredon \(RO(C_2)\)-graded cohomology. In this talk, we explain how a version of the Thom isomorphism theorem in \(RO(C_2)\)-graded cohomology in constant \(\mathbb{Z}/2\) coefficients can be used to develop a theory of fundamental classes for equivariant submanifolds. We then show these classes can be used to understand the cohomology of \(C_2\)-surfaces, including the ring structure.
Slides from a talk I gave in a special session at the JMM in January 2020. My talk was about about equivariant fundamental classes in constant Z/2 coefficients.
Slides from a talk I gave in a special session at the AMS Spring Sectional Meeting in March 2019. My talk was about the RO(C2)-graded cohomology of C2-surfaces and equivariant fundamental classes in constant Z/2 coefficients.
Slides from a talk I gave at the 2018 Young Topologists Meeting. My talk was about the RO(C2)-graded cohomology in both constant Z/2 coefficients and constant integer coefficients.