William Balderrama

Email: williamb (at) math.uni-bonn.de
Office: MZ / 4.014

I am currently an HCM postdoc at the University of Bonn. Previously, I was an RTG postdoc and lecturer in mathematics at the University of Virginia, with Nick Kuhn. Before that, I received my PhD at the University of Illinois Urbana Champaign, advised by Charles Rezk.

My work is in homotopy theory. Here's my CV. My papers can be found on the arXiv.

Papers and preprints.

17. Unstable synthetic deformations II: Infinitesimal Extensions, with Piotr Pstrągowski.
Abstract.
This paper is the second in a series devoted to the study of unstable synthetic deformations through the lens of Malcev theories: certain ∞-categorical algebraic theories P with well-behaved ∞-categories ModelP of models. In this paper, we show that Malcev theories and their models admit a well-behaved deformation theory, generalizing the classical deformation theory of rings and modules. As our main example, we prove that the Postnikov tower of a Malcev theory P is a tower of square-zero extensions, and that all of this structure is preserved by passage to ∞-categories of models. This allows us to control the difference between the ∞-categories Modelhn+rP and ModelhnP for r ≤ n, and forms the basis of a ``cofibre of τ'' formalism in our approach to unstable synthetic homotopy theory. As an application, we derive from this a variety of new Blanc–Dwyer–Goerss style decompositions of moduli spaces of lifts along the tower ModelP → ··· → ModelhP.

16. Unstable synthetic deformations I: Malcev theories, with Piotr Pstrągowski.
Abstract.
This paper is the first in a series of articles devoted to the construction and study of synthetic deformations of ∞-categories in the unstable context: that is, deformations of ∞-categories that categorify spectral sequence or obstruction-theoretic information. Our approach is based on the techniques of higher universal algebra, with deformations built from the ∞-categories of models of ∞-categorical variants of algebraic, or Lawvere, theories.

This paper sets up the foundations of our study. We introduce and study various classes of ∞-categorical and infinitary algebraic theories. We establish many basic properties of the ∞-categories of the models of different classes of theories, such as the existence of free resolutions, image factorizations, monadicity theories, and presentability, as well as recognition theorems identifying the ∞-categories that arise this way.

We give an intrinsic definition of a Malcev theory in higher universal algebra. These are characterized as those theories satisfying Quillen’s condition that all simplicial models satisfy the Kan condition, and we prove that this is equivalent to a weak grouplike condition which is easily verified in practice. We establish that the ∞-category of models of a Malcev theory may be characterized as freely adjoining geometric realizations to the theory. This leads to the notion of a derived functor between ∞-categories of models of Malcev theories, and we study some of the behavior of these derived functors with respect to connectivity and limits.

The key idea in our work is that the ∞-category of models of a Malcev theory P can be thought of as a deformation whose special fibre is the ∞-category of models of its homotopy category hP. We also recall the notion of a loop theory, a class of Malcev theories whose ∞-category of models also admits a generic fibre, given by the full subcategory of loop models. We study in detail the interaction between functors and derived functors of ∞-categories of loop models and models, establishing in particular that monads and a large class of comonads on the generic fibre lift canonically to the whole deformation.

In the last part of the paper, we show that by considering the coalgebras for these deformed comonads over ∞-categories of models, one can recover various stable deformations considered in the literature, such as filtered models or Postnikov-complete synthetic spectra. We then expand on these results by constructing ∞-categories of synthetic spaces and synthetic Ek-rings which, as will be further developed in the sequels, categorify the generalized unstable Adams spectral sequence and Goerss–Hopkins spectral sequence respectively.

15. Affineness and reconstruction in complex-periodic geometry, with Jack Davies and Sil Linskens. Submitted.
Abstract.
Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to E∞ rings equipped with the fpqc topology, these foundations give an ∞-category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of 0-affine stacks, generalizing Mathew--Meier's famous 0-affineness result. We introduce a spectral refinement of Hopkins' stack construction of an E∞ ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

14. Cpn-equivariant Mahowald invariants, with Yueshi Hou and Shangjie Zhang. Submitted.
Abstract.
We introduce the Cpn-Mahowald invariant: a relation π★SCpn-1 ⇀ π*S between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when n=1. We compute the Cpn-Mahowald invariants of all elements in the Burnside ring A(Cpn-1) = π0SCpn-1, extending Mahowald and Ravenel's computation of MCp(pk). As a consequence, we determine the image of the Cp-geometric fixed point map ΦCp : πVSCpn → π0SCpn/Cp ≅ A(Cpn-1) when V is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for n=1.

13. Equivariant v1,0⃗-self maps, with Yueshi Hou and Shangjie Zhang.
Accepted for publication in Bull. Lond. Math. Soc.
Abstract.
Let G be a cyclic p-group, X ∈ π0SG be a virtual G-set, and V be a fixed point free complex G-representation. Under conditions depending on the sizes of G, X, and V, we construct a self map v : ΣVC(X)(p) → C(X)(p) on the cofiber of X which induces an equivalence in G-equivariant K-theory. These are transchromatic v1,0⃗-self maps, in the sense that they are lifts of classical v1-self maps for which the telescope C(X)(p)[v-1] can have nonzero rational geometric fixed points.

12. Type 2 complexes constructed from Brown-Gitler spectra, with Justin Barhite, Nick Kuhn, and Donald Larson.
Math. Z. 311, 72 (2025). https://doi.org/10.1007/s00209-025-03869-6.
Abstract.
In a previous paper, one of us interpreted mod 2 Dyer-Lashof operations as explicit A-module extensions between Brown-Gitler modules, and showed these A-modules can be topologically realized by finite spectra occurring as fibers of maps between 2-local dual Brown-Gitler spectra.

Starting from these constructions, in this paper, we show that infinite families of these finite spectra are of chromatic type 2, with mod 2 cohomology that is free over A(1). Applications include classifying the dual Brown-Gitler spectra after localization with respect to K-theory.

11. A motivic analogue of the K(1)-local sphere spectrum, with Kyle Ormsby and J.D. Quigley.
J. Eur. Math. Soc. (JEMS). https://doi.org/10.4171/jems/1641.
Abstract.
We identify the motivic KGL/2-local sphere as the fiber of ψ3−1 on (2,η)-completed Hermitian K-theory, over any base scheme containing 1/2. This is a motivic analogue of the classical resolution of the K(1)-local sphere, and extends to a description of the KGL/2-localization of any motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.

10. Equivalences of the form ΣVX ≃ ΣWX in equivariant stable homotopy theory. Submitted.
Abstract.
We study G-equivariant equivalences of the form ΣVX ≃ ΣWX, where G is a compact Lie group, X is a G-spectrum, and V and W are G-representations. These equivalences encode a periodicity phenomenon in G-equivariant homotopy theory which generalizes the classical James periodicity for G = C2.

When X = C(aλm) is the cofiber of an Euler class, we construct an RO(G)-graded J-homomorphism J : πλKOG → (π★C(aλm))× which gives control over these periodicities. It also produces infinite periodic families in the G-equivariant stable stems. We illustrate this with several explicit examples.

More generally, our work gives information about RO(G)-graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the G-homotopy fixed point spectral sequence, and other equivariant Atiyah--Hirzebruch spectral sequences.

09. An elementary proof of the chromatic Smith fixed point theorem, with Nick Kuhn.
Homology Homotopy Appl. 26(1), 131-140 (2024). https://doi.org/10.4310/HHA.2024.v26.n1.a8.
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 p-group of rank r, then any finite A-space X which is acyclic in the nth Morava K-theory with n at least r will have its subspace F of fixed points acyclic in the (n-r)th Morava K-theory. 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, A-CW complexes, which suggests some open problems.

08. The Real-oriented cohomology of infinite stunted projective spaces.
Algebr. Geom. Topol. 24, 4061-4084 (2024). https://doi.org/10.2140/agt.2024.24.4061
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.

07. Total power operations in spectral sequences.
Trans. Amer. Math. Soc. 377, 4779-4823 (2024). https://doi.org/10.1090/tran/9073.
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 which may be of independent interest.

06. K-theory equivariant with respect to an elementary abelian 2-group.
New York J. Math. 28, 1531-1553 (2022). https://nyjm.albany.edu/j/2022/28-67.html.
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.

05. The motivic lambda algebra and motivic Hopf invariant one problem, with Dominic Culver and J.D. Quigley.
Geom. Topol. 29, 1489-1570 (2025). https://doi.org/10.2140/gt.2025.29.1489
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.

04. The C2-equivariant K(1)-local sphere.
Accepted for publication in Math. Z.
Abstract.
We compute the RO(C2)-graded Green functor π★LKUC2/(2)SC2.

03. Algebraic theories of power operations.
J. Topol. 16(4), 1543-1640 (2023). https://doi.org/10.1112/topo.12318.
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.

02. Deformations of homotopy theories via algebraic theories.
Adv. Math. 480 B, 110496 (2025). https://doi.org/10.1016/j.aim.2025.110496.
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.

01. Definability and decidability in expansions by generalized Cantor sets, with Philipp Hieronymi.
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.

10. A note on Fp-synthetic p-profinite spaces.
09. A proof of convergence of the chromatic tower for classifying spaces of abelian p-groups.
08. An amusing proof of the Adams differential d2(D1) = h02 h3 g2.
07. A note describing R-motivic K(1)-localization. Now absorbed into paper 11 above.
06. A note on the Curtis algorithm for ExtA(F2,H*L(k)n), with accompanying Curtis table.
05. Slides for an expository talk on filtered spectra.
04. 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. The program computes full cycle representatives; if you want these, see here (61.2 MiB, 5.2GiB uncompressed).
03. Notes on the classification of formal groups over a perfect field from the viewpoint of their Dieudonné modules. The main point was to understand the relation between exterior powers of formal groups and the determinant homomorphism Sh → S1 = Zp×.
02. Notes on a short proof of straightening / unstraightening for left fibrations over an ordinary category assuming a characterization of the covariant model structure.
01. Notes that describe some ordinary category theory using discrete (op)fibrations.