William Balderrama
Email: balderr2 (at) illinois.edu
Office: B3 Coble Hall
I am a sixth year PhD student; my advisor is Charles Rezk. I'll be at UVA next year.
My work is broadly in stable homotopy. I am interested in chromatic homotopy, and in the tools that allow homotopical structure to be accessed by computationally acessible algebra.
Here's my CV.
Papers.
 Title. The Borel C_{2}equivariant K(1)local sphere (arXiv, 2021).
Abstract. We compute the bigraded homotopy ring of the Borel C_{2}equivariant K(1)local sphere. This captures many of the patterns seen among Im Jtype elements in Rmotivic and C_{2}equivariant stable stems. In addition, it provides a streamlined approach to understanding the K(1)localizations of stunted projective spaces.
More. Slides for a talk on this.
 Title. Algebraic theories of power operations (arXiv, 2021; updated from pdf, draft, 2020).
Abstract. We develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding wellbehaved theories of power operations for E_{∞} ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstructiontheoretic machinery, we obtain tools for computing with E_{∞} algebras over F_{p} and over LubinTate spectra. As an application, we demonstrate the existence of E_{∞} periodic complex orientations at heights h ≤ 2.
 Title. Deformations of homotopy theories via algebraic theories (arXiv, 2021; updated from pdf, draft, 2020)
Abstract. We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral sequences that compute homotopical data starting with purely algebraic data.
More. Slides for a talk I gave on the above two items. Slides for a variant. Slides for an older talk. Finally slides for a 10 minute video (YouTube link; file here, 16MiB) recorded for the eATEN.
 Title. Definability and decidability in expansions by generalized Cantor sets, with Philipp Hieronymi (arXiv, 2017).
More fun stuff.
 Slides for an expository talk on filtered spectra.
 A Curtis table for the mod 2 lambda algebra, complete through degree 72. The file also contains an exposition of the topic. Data generated from a little program I wrote in Common Lisp, which you can find here. It computes full cycles and full tags; if you want these, see here (70.3 MiB / 5.2 GiB uncompressed).
 Notes covering the classification of formal groups over a perfect field from the viewpoint of their Dieudonné modules. The main point was to better understand the following fact: Isomorphism classes of finite height h formal groups over a finite field F_{pr} are in natural correspondence with a quotient of the h'th Morava stabilizer group S_{h}, and by taking top exterior powers this gives you a map S_{h} → S_{1} = Z_{p}^{×}. The observation is that this differs from the standard determinant homomorphism by a twist of (1)^{r(h1)}.
 Notes on a short proof of straightening / unstraightening for left fibrations over an ordinary category assuming a characterization of the covariant model structure.
 Notes that describe some ordinary category theory using discrete (op)fibrations.
Seminars.
Teaching.
 Fall 2016: Teaching Assistant for Math 221 Section ADD.
 Fall 2017: Teaching Assistant for Math 221 sections ADL and ADM.
 Fall 2018: Teaching Assistant for Math 221 sections CDE and CDF, as well as Head TA for the course.
 Spring 2019: Teaching Assistant for Math 231 sections ADE and ADT.
 Fall 2019: Teaching Assistant for Math 221 sections CDD and CDE.
 Spring 2020: Teaching assistant for Math 231 Sections CDF and CDP.
 Fall 2020: Grader / TA for Math 125, Math 225, and Math 570.
 Spring 2021: Teaching Assistant for Math 220 sections ADG and ADH.
