William Balderrama
Email: eqr8nm (at) virginia.edu
Office: 301 Kerchof Hall
I'm an RTG postdoc and lecturer in mathematics at the University of Virginia. Prior to this, I received my PhD at the University of Illinois Urbana Champaign, advised by Charles Rezk.
My work is broadly in stable homotopy theory. Here's my CV. My papers can be found on the arXiv.
Spring 2023 teaching: Math 3350 Applied Linear Algebra. Course information will be available on Canvas.
Papers and preprints.
 An elementary proof of the chromatic Smith fixed point theorem, with Nick Kuhn (arXiv, 2023).
Abstract.
A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if A is a finite abelian pgroup of rank r, then any finite Aspace X which is acyclic in the nth Morava Ktheory with n at least r will have its subspace F of fixed points acyclic in the (nr)th Morava Ktheory. This is a chromatic homotopy version of P.A.Smith's classical theorem that if X is acyclic in mod p homology, then so is F.
The main purpose of this paper is to give an elementary proof of this new theorem that uses minimal background, and follows, as much as possible, the reasoning in standard proofs of the classical theorem. We also give a new fixed point theorem for finite dimensional, but possibly infinite, ACW complexes, which suggests some open problems.
 The Realoriented cohomology of infinite stunted projective spaces (arXiv, 2022).
Abstract.
Let ER be an evenperiodic Real Landweber exact C_{2}spectrum, and ER its spectrum of fixed points. We compute the ERcohomology of the infinite stunted projective spectra P_{j}. These cohomology groups combine to form the RO(C_{2})graded coefficient ring of the C_{2}spectrum b(ER)=F(EC_{2+},i_{∗}ER), which we show is related to ER by a cofiber sequence Σ^{σ}b(ER)→b(ER)→ER. We illustrate our description of π_{⋆}b(ER) with the computation of some ERbased Mahowald invariants.
 Total power operations in spectral sequences (arXiv, 2022).
Abstract.
We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the C_{2}equivariant Adams spectral sequence, to compute norms on π_{0} of the equivariant KUlocal sphere, and to compute power operations for the K(1)local sphere. An appendix contains material on equivariant Bousfield localizations, including an equivariant smash product theorem.
 Ktheory equivariant with respect to an elementary abelian 2group (NYJM, 2022).
Abstract.
We compute the RO(A)graded coefficients of Aequivariant complex and real topological Ktheory for A a finite elementary abelian 2group, together with all products, transfers, restrictions, power operations, and Adams operations.
 The motivic lambda algebra and motivic Hopf invariant one problem, with
Dominic Culver and J.D. Quigley (arXiv, 2021).
Abstract. We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base fields of characteristic not equal to 2. Maps of Hopf invariant one classically arise from unital products on spheres, and one consequence of our work is a classification of motivic spheres represented by smooth schemes admitting a unital product.
The classical Hopf invariant one problem was resolved by Adams, following his introduction of the Adams spectral sequence. We introduce the motivic lambda algebra as a tool to carry out systematic computations in the motivic Adams spectral sequence. Using this, we compute the E_{2}page of the Rmotivic Adams spectral sequence in filtrations f≤3. This universal case gives information over arbitrary base fields.
We then study the 1line of the motivic Adams spectral sequence. We produce differentials d_{2}(h_{a+1})=(h_{0}+ρh_{1})h_{a}^{2} over arbitrary base fields, which are motivic analogues of Adams' classical differentials. Unlike the classical case, the story does not end here, as the motivic 1line is significantly richer than the classical 1line. We determine all permanent cycles on the Rmotivic 1line, and explicitly compute differentials in the universal cases of the prime fields F_{q} and Q, as well as Q_{p} and R.
 The C_{2}equivariant K(1)local sphere (arXiv, 2021).
Abstract.
We compute the RO(C_{2})graded Green functor π_{★}L_{KUC2/(2)}S_{C2}.
 Algebraic theories of power operations (arXiv, 2021).
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.
 Deformations of homotopy theories via algebraic theories (arXiv, 2021)
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.
 Definability and decidability in expansions by generalized Cantor sets, with Philipp Hieronymi (arXiv, 2017).
Abstract.
We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number r≥3, we say a set C is a generalized Cantor set in base r if there is a nonempty K⊆{1,…,r−2} such that C is the set of those numbers in [0,1] that admit a base r expansion omitting the digits in K. While it is known that the theory of an expansion of the ordered real additive group by a single generalized Cantor set is decidable, we establish that the theory of an expansion by two generalized Cantor sets in multiplicatively independent bases is undecidable.
More fun stuff.
 An amusing proof of the Adams differential d_{2}(D_{1}) = h_{0}^{2} h_{3} g_{2}.
 A note describing Rmotivic K(1)localization.
 A note on the Curtis algorithm for Ext_{A}(F_{2},H_{*}L(k)_{n}), with accompanying Curtis table.
 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.
 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.
