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.

13. Cpn-Mahowald invariants and v1,0⃗-self maps, with Yueshi Hou and Shangjie Zhang.
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.
Our work makes essential use of equivariant K-theory, and exhibits patterns suggesting an equivariant Adams periodicity. We refine this to produce equivariant lifts of Adams' v1-self maps on Moore spectra. Given a cyclic p-group or generalized quaternion group G, fixed point free G-representation V, and element X ∈ A(G), subject to constraints depending only on the sizes of G, X, and V, we construct a map ΣVC(X)(p) → C(X)(p) on the cofiber of X which induces an equivalence in G-equivariant K-theory.
12. Type 2 complexes constructed from Brown-Gitler spectra, with Justin Barhite, Nick Kuhn, and Donald Larson. Submitted.
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.
Accepted for publication in J. Eur. Math. Soc. (JEMS).
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.
Accepted for publication in Algebr. Geom. Topol.
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 (2024), 4779-4823.
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.
Accepted for publication in Geom. Topol.
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. Submitted.
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. Submitted.
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.