My research area is algebraic topology, specifically equivariant stable homotopy theory. My current focus is on the slice filtration, a method of decomposing genuine G-spectra into certain building blocks called slices. The primary method I use to do so is to find short exact sequences of Mackey functors, which lead to cofiber sequences of spectra where I know the slice status of some of the spectra in the sequence. The goal here is to use properties of the slice filtration (which result from its status as a localizing subcategory of genuine G-spectra) to string these cofiber sequences into the slice (co)tower, and consequently determine the slices of my chosen spectrum.
Here's the birds eye view: The slice filtration is a relatively new tool in algebraic topology (circa 2009). One of its parameters is a group of equivariance, G, which must be finite. Once this group is chosen, the slice filtration takes in particular objects (genuine G-spectra) that algebraic topologists often use in their work and wish to understand. As it is so new, there are very few papers decomposing these objects into the building blocks that the slice filtration provides. I've written two of them (one with my advisor) plus my thesis which contains both papers, calculations not contained in either, and extra exposition on some of the tools used.
The slice filtration is so mysterious, that almost all of these papers focus on decomposing some of the most basic genuine G-spectra available, called positive, trivial suspensions of Eilenberg-Mac Lane spectra (for a chosen group and Mackey functor associated with the group). My Klein Four paper expands the knowledge base and considers both positive and negative trivial suspensions of a particular spectrum over the Klein Four group and finishes off a characterization of all slices over Klein Four. With the exception of (-1)-suspensions, negative suspensions were not previously considered. The quaternionic paper is one of two papers considering the slice filtration over a non-abelian group and introduces new theory relating various type of pullbacks (a useful tool relating a group to its subgroups) which align in the specific case of bottleneck subgroups.
The slices of quaternionic Eilenberg-Mac Lane spectra (available on the ArXiv)
We compute the slices and slice spectral sequence of integral suspensions of the equivariant Eilenberg-Mac Lane spectra HZ for the group of equivariance Q8. Along the way we compute the Mackey functors \pi_{k\rho} HZ.
Klein Four 2-slices and the Slices of Σ^±n HZ, Math. Z. 301 (2022), no. 4, 3895–3938. (ArXiv)
We determine a characterization of all 2-slices of equivariant spectra over the Klein four-group C2×C2. We then describe all slices of integral suspensions of the equivariant Eilenberg-MacLane spectrum HZ for the constant Mackey functor over C2×C2.
(with Kate Ponto and Kalila Sawyer) The AWM and the University of Kentucky, "Fifty Years of Women in Mathematics: Reminiscences, History, and Visions for the Future of AWM", May 2022.
2-slices over cyclic groups, Topology Seminar, Columbia University, April 2023.
2-slices over cyclic groups, Student Topology Seminar, University of Pennsylvania, February 2023.
2-slices over cyclic groups, Topology Seminar, Binghampton University, February 2023.
Slices of \Sigma^n HZ over Q8, AMS Spring Western Sectional Meeting, University of Virginia, March 2022 (canceled due to COVID-19).
Characterizing 2-slices over C2 and K4, Topology Seminar, University of Pennsylvania, December 2021.
Characterizing 2-slices over C2 and K4, Topology Seminar, Texas A&M University, December 2021.
Characterizing 2-slices over C2 and K4, Topology Seminar, University of Rochester, November 2021.
Characterizing 2-slices over C2 and K4, Topology Seminar, University of Virginia, November 2021.
Characterizing 2-slices over C2 and K4, AMS Fall Western Sectional Meeting, hosted by University of New Mexico over Zoom, October 2021.
Characterizing 2-slices over C2 and K4, Topology Seminar, University of California Los Angeles, May 2021.
2-slices over cyclic groups, Topology Seminar, University of Rochester, February 2023.
Pullbacks and the Slice Filtration, BUGCAT, Binghampton University, November 2022.
Pullbacks and the Slice Filtration, 3rd Upstate New York Topology Seminar, Syracuse University, October 2022.
Pullbacks and the Slice Filtration, AMS Fall Eastern Sectional Meeting, University of Amherst, October 2022.
Slices of Eilenberg-Mac Lane Spectra over K4 and Q8 (Thesis Defense), Topology Seminar, University of Kentucky, August 2022.
Pullbacks and the Slice Filtration, Young Topologists Meeting, University of Copenhagen, July 2022.
Slices of \Sigma^n HZ over Q8, Twinned Conference on Homotopy Theory with Applications to Arithmetic and Geometry, Fields Institute, May 2022.
Slices of \Sigma^n HZ over Q8, Topology Seminar, University of Kentucky, February 2022.
Characterizing 2-slices over C2 and K4, Graduate Student Seminar, University of Kentucky, September 2021.
Characterizing 2-slices over C2 and K4, Young Topologists Meeting, hosted by KTH Royal Institute of Technology and Stockholm University over Zoom, July 2021.
Characterizing 2-slices over C2 and K4, Graduates Reminisce Online On Topology (GROOT) Summer Seminar, June 2021.
Characterizing 2-slices over C2 and K4, 18th Annual Graduate Student Topology and Geometry Conference, Indiana University, Bloomington, April 2021.
Slices of Eilenberg-MacLane Spectra: Computational Strategies, eCHT Equivariant Algebra Reading Seminar, April 2021.
Characterizing 2-slices over C2 and K4, Topology Seminar, University of Kentucky, September 2020.
Slice Towers of Trivial Suspensions of HF2 over C2, Graduates Reminisce Online On Topology (GROOT) Summer Seminar, July 2020.
The Slice Filtration, 18th Annual Graduate Student Topology and Geometry Conference, Indiana University, Bloomington, April 2020 (canceled due to COVID-19).
Slice Towers of Trivial Suspensions of HF2 over C2, Topology Seminar, University of Kentucky, February 2020.
The Slice Filtration (Qualifying Talk), Topology Seminar, University of Kentucky, December 2019.
Smith Theory and Bredon Cohomology (Masters talk), Topology Seminar, University of Kentucky, April 2019.
Models of Nation-Building via Systems of Differential Equations (Poster), 8th Annual Research and Scholarship Symposium, Cedarville University, April 2017.
Models of Nation-Building via Systems of Differential Equations (Poster), Nebraska Conference for Undergraduate Women in Mathematics, University of Nebraska-Lincoln, February 2017.
Models of Nation-Building via Systems of Differential Equations (Talk and poster), Air Force Institute of Technology, July 2016.
Forced into Chaos: An Analysis of Chaotic Orbits in Duffing's Equation, Senior Research Presentation, Cedarville University, April 2016.