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.


  • Title. The Borel C2-equivariant K(1)-local sphere (arXiv, 2021).
    Abstract. We compute the bigraded homotopy ring of the Borel C2-equivariant K(1)-local sphere. This captures many of the patterns seen among Im J-type elements in R-motivic and C2-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. Approximating higher algebra by derived algebra (pdf, draft, 2020).
    Abstract. We develop a form of the theory of product-preserving presheaves of ∞-groupoids as a natural setting for the construction of obstruction theories and spectral sequences approximating higher, or spectral, algebra by mere derived algebra. We account for infinitary phenomena, allowing for the inclusion of completed settings into the general theory.
  • Title. Algebraic theories of power operations (pdf, draft, 2020).
    Abstract. We develop some aspects of algebraic theories that are useful for working with some of the algebraic structures that arise in stable homotopy theory, with an emphasis on their role in certain obstruction theories. We give a general treatment of Koszul complexes which may be of independent interest. We describe in detail the tools one obtains for computing with E algebras over Fp and over Lubin-Tate spectra. As an application, we demonstrate the existence of E periodic complex orientations at heights h ≤ 2.
    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 Fpr are in natural correspondence with a quotient of the h'th Morava stabilizer group Sh, and by taking top exterior powers this gives you a map Sh → S1 = Zp×. The observation is that this differs from the standard determinant homomorphism by a twist of (-1)r(h-1).
  • 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.



  • 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.

Department of Mathematics
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Telephone: (217) 333-3350    Fax: (217) 333-9576     Email: math@illinois.edu