Research
Current Project(s):
My thesis work focuses on heuristics for algebraic K-theory, specifically that of function fields, specifically ℓ-ranks thereof. I am very excited to share more about this project in the coming months!
I also presently am working on a project on compatibility of N_∞ operads joint with Mike Hill, Yigal Kamel, Nelson Niu, Kurt Stoekl, Danika van Niel, and Guoqi Yan. This came out of the 2024 AMS MRC in Homotopical Combinatorics and is based upon work supported by the National Science Foundation under Grant Number DMS 1916439.
Past Work:
My undergraduate senior comps [thesis] project studied reducibility in families of iterates of quadratic polynomials with integer coefficients and no linear term. The initial work on this project was joint with Moses Misplon and Michael Stoneman under the supervision of Rafe Jones at Carleton College in Northfield, MN. This is now available in published form as "Eventually stable quadratic polynomials over ℚ" joint with Hindes, Jones, Misplon, Stoll and Stoneman in New York J. Math 26(2020), 526-561. arXiv:1902.09220.
Also in undergraduate, I participated in an REU at Oregon State University. Under supervision of Clayton Petsche, Kenneth Allen and I studied the dynamics of the Hénon map over ℚ_p. This as well was published joint with Allen and Petsche and as "Non-Archimedean Hénon maps, attractors, and horseshoes" in Res. Number Theory 4 (2018), Art. 5, 30 pp. arXiv:1610.04271
RUpcoming and Previous Talks
Joint Mathematics Meetings: Special Session on MRC Homotopical Combinatorics
Location: Seattle, Washington, USA
Date: 9 January 2025
Abstract: There are a variety of related additive and multiplicative structures on G-spaces and G-spectra, all encoded by N_∞ operads. This raises the question, given a fixed additive structure, what is the maximal multiplicative structure compatible with it? Given the work of Rubin, characterising N_∞ operads as posets called G-transfer systems, this question can be translated into purely combinatorial terms. Given a fixed disk-like (additive) transfer system, can we characterize the maximal compatible transfer system? In this first of two talks, we discuss the role of saturated transfer systems, and show that a disk like transfer system is (maximally) self compatible if, and only if, it is saturated. This talk presents joint work with, Hill, Kamel, Niu, Stoeckl, Van Niel and Yan. This material is based upon work supported by the National Science Foundation under Grant Number DMS 1916439.
Slides: Right here!
AMS 2025 Spring Central Sectional Meeting: Special Session on Homotopy theory and algebraic K-theory I
Location: University of Kansas. Lawrence, Kansas, USA
Dates: 29-30 March 2025
Title: TBD
Abstract: TBD
Newton Institute: Workshop on Operads and Calculus
Location: Queen's University. Belfast, Northern Ireland, UK
Dates: 7-11 April 2025
Title: [Maximal] Compatibility of Transfer Systems and Functoriality Thereof
Abstract: In this work, we discuss some of the combinatorial aspects of N_infinity operads and bi-incomplete Tambara functors through the lens of transfer systems. Work of a number of authors has combined to show that N_infinity structures are classified up to homotopy by their associated transfer systems. Working within that framework, Chan gave a characterization of the compatibility condition linking the additive and multiplicative structures of a bi-incomplete Tambara functor in the language of transfer systems. Fixing a disk-like transfer system parametrizing additive structure, we present work towards a characterization of the maximal compatible multiplicative counterpart. We show that maximal compatiblity is functorial with homomorphism of groups and present work towards a conjectural description of the maximal compatible counterpart. This talk is based on work joint with Hill, Kamel, Niu, Stoekl, Van Niel and Yan.
Slides: Right here!
Research Statement:
Here is a current (dated 15 November, 2024) edition of my research statement:
RS_11_15_24.pdf