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. Here's my CV. My papers can be found on the arXiv.

Fall 2022 teaching: Math 3351 Elementary Linear Algebra. Course information will be available on Collab.

Preprints.

    1. Title. The Real-oriented cohomology of infinite stunted projective spaces (arXiv, 2022).
      Abstract. Let ER be an even-periodic Real Landweber exact C2-spectrum, and ER its spectrum of fixed points. We compute the ER-cohomology of the infinite stunted projective spectra Pj. These cohomology groups combine to form the RO(C2)-graded coefficient ring of the C2-spectrum b(ER)=F(EC2+,i∗ER), which we show is related to ER by a cofiber sequence Σσb(ER)→b(ER)→ER. We illustrate our description of π⋆b(ER) with the computation of some ER-based Mahowald invariants.
    2. Title. 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 C2-equivariant Adams spectral sequence, to compute norms on π0 of the equivariant KU-local 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.
    3. Title. K-theory equivariant with respect to an elementary abelian 2-group (arXiv, 2022).
      Abstract. We compute the RO(A)-graded coefficients of A-equivariant complex and real topological K-theory for A a finite elementary abelian 2-group, together with all products, transfers, restrictions, power operations, and Adams operations.
    4. Title. 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 E2-page of the R-motivic Adams spectral sequence in filtrations f≤3. This universal case gives information over arbitrary base fields.
      We then study the 1-line of the motivic Adams spectral sequence. We produce differentials d2(ha+1)=(h0+ρh1)ha2 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 1-line is significantly richer than the classical 1-line. We determine all permanent cycles on the R-motivic 1-line, and explicitly compute differentials in the universal cases of the prime fields Fq and Q, as well as Qp and R.
    5. 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.
    6. 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 well-behaved 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 obstruction-theoretic machinery, we obtain tools 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.
    7. 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.
    8. Title. 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 non-empty 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.

    1. A brief note identifying a form of the G-equivariant K(n)-local sphere for a finite group G. This is now absorbed into the appendix of my total power operations paper.
    2. Slides for an expository talk on filtered spectra.
    3. 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.
    4. 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).
    5. Notes on a short proof of straightening / unstraightening for left fibrations over an ordinary category assuming a characterization of the covariant model structure.
    6. Notes that describe some ordinary category theory using discrete (op)fibrations.