Robert Burklund

I am an associate professor at the University of Copenhagen.
My preprints can be found on the arXiv and my blog can be found here.
email: rb at math dot ku dot dk

Here is my CV.


  • The chromatic nullstellensatz with Tomer Schlank and Allen Yuan [Ann. of Math. (2)] [arxiv] We show that Lubin--Tate theories attached to algebraically closed fields are characterized among T(n)-local E_infty-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every T(n)-local E_infty-ring R, the collection of E_infty-ring maps from R to such Lubin--Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero T(n)-local E_infty-ring R admits an E_infty-ring map to such a Lubin--Tate theory. As consequences, we construct E_infty complex orientations of algebraically closed Lubin--Tate theories, compute the strict Picard spectra of such Lubin--Tate theories, and prove redshift for the algebraic K-theory of arbitrary E_infty-rings.

  • On the high-dimensional geography problem with Andrew Senger [Geom. Topol.] [arxiv] In 1962, Wall showed that smooth, closed, oriented, (n-1)-connected 2n-manifolds of dimension at least 6 are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form which he called an n-space. In this paper, we complete the determination of which n-spaces are realizable by smooth, closed, oriented, (n-1)-connected 2n-manifolds for all n≠63. In dimension 126 the Kervaire invariant one problem remains open. Along the way, we completely resolve conjectures of Galatius-Randal-Williams and Bowden-Crowley-Stipsicz, showing that they are true outside of the exceptional dimension 23, where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of Einfty-ring spectra by Ando-Hopkins-Rezk. By previous work of many authors, including Wall, Schultz, Stolz and Hill-Hopkins-Ravenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.

  • On the K-theory of regular coconnective rings with Ishan Levy [Selecta Math. (N.S.)] [arxiv] We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its pi_0. We prove this as a consequence of a more general devissage result for stable infinity categories. Applications of our result include giving general conditions under which K-theory preserves pushouts, generalizations of An-invariance of K-theory, and an understanding of the K-theory of categories of unipotent local systems.

  • Adams-type maps are not stable under composition with Ishan Levy and Piotr Pstragowski [Proc. Amer. Math. Soc.] [arxiv] We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of E_infty-algebras is a transfinite composition of Adams-type maps.

  • On the boundaries of highly connected, almost closed manifolds with Jeremy Hahn and Andrew Senger [Acta Math.] [arxiv] Building on work of Stolz, we prove for integers 0 ≤ d ≤ 3 and k > 232 that the boundaries of (k-1)-connected, almost closed (2k+d)-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups.
    Our technique is to recast the Galatius and Randal-Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its F_p-Adams filtrations for all primes p. We additionally prove new vanishing lines in the F_p-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in BP<n>-based Adams spectral sequences.

  • An extension in the Adams spectral sequence in dimension 54 [Bull. Lond. Math. Soc.] [arxiv] We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2-extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration of the computational leverage provided by F_2-synthetic spectra.

  • The trace of the local A1-degree with Thomas Brazelton, Stephen McKean, Michael Montoro and Morgan Opie [Homology Homotopy Appl.] [arxiv] We prove that the local A1-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A1-degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja-Storch form and the local A1-degree.


  • A note on the Segal conjecture for large objects with Vignesh Subramanian [arxiv] The Segal conjecture for C_p (as proved by Lin and Gunawardena) asserts that the canonical map from the p-complete sphere spectrum to the Tate construction for the trivial action of C_p on the p-complete sphere spectrum is an isomorphism. In this article we extend the collection of spectra for which the canonical map from X to X^tCp is known to be an isomorphism to include any p-complete, bounded below spectrum whose mod p homology, viewed a module over the Steenrod algebra, is complete with respect to the maximal ideal I in A.

  • E_infty-coalgebras and p-adic homotopy theory with Tom Bachmann [arxiv] We show that for any separably closed field k of characteristic p>0, the canonical functor from nilpotent p-adic spaces to E_infty-coalgebras over k (given by singular chains with coefficients in k) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.

  • K-theoretic counterexamples to Ravenel's telescope conjecture with Ishan Levy, Jeremy Hahn and Tomer Schlank [arxiv] At each prime p and height n+1≥2, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for Z acting by Adams operations on BP<n>, we prove that the T(n+1)-localized algebraic K-theory of BP<n>^hZ is not K(n+1)-local. We also show that Galois hyperdescent, A1-invariance, and nil-invariance fail for the K(n+1)-localized algebraic K-theory of K(n)-local E_infy-rings. In the case n=1 and p≥7 we make complete computations of T(2)_*K(R), for R certain finite Galois extensions of the K(1)-local sphere. We show for p≥5 that the algebraic K-theory of the K(1)-local sphere is asymptotically L2^f-local.

  • Quivers and the Adams spectral sequence with Piotr Pstragowski [arxiv] In this paper, we describe a novel way of identifying Adams spectral sequence E_2-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of p-local integral homology, we are able to give a decomposition of the E_2-term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of infinity-categories of the second author and Patchkoria.

  • The Adams differentials on the classes h_j^3 with Zhouli Xu [arxiv] In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes h_j, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill--Hopkins--Ravenel proved that the classes h_j^2 support non-trivial differentials for j ≥ 7, resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes h_j^2 for j ≥ 7 and the fate of h_6^2 remains unknown. In this paper, in Adams filtration 3, we prove an infinite family of non-trivial d_4-differentials on the classes h_j^3 for j ≥ 6, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory---C-motivic stable homotopy theory and F_2-synthetic homotopy theory---both in an essential way. Along the way, we also show that h_j^2 survives to the Adams E_5-page and that h_6^2 survives to the Adams E_9-page.

  • Multiplicative structures on Moore spectra [arxiv] In this article we show that S/8 is an E_1-algebra, S/32 is an E_2-algebra, S/p^(n+1) is an E_n-algebra at odd primes and, more generally, for every h and n there exist generalized Moore spectra of type h which admit an E_n-algebra structure.

  • How big are the stable homotopy groups of spheres? including an appendix joint with Andrew Senger [arxiv] In this article we show that the p-torsion exponent of the stable stems grows sublinearly in n and the p-rank of the E2-page of the Adams spectral sequence grows as exp(theta(log(n)^3)). Together these bounds provide the first subexponential bound on the size of the stable stems. Conversely, we prove that a certain, precise, version of the failure of the telescope conjecture would imply that the upper bound provided by the Adams E2-page is essentially sharp -- answering the titular question: As big as the fate of the telescope conjecture demands.
    In an appendix joint with Andrew Senger we consider the unstable analog of this question. Bootstrapping from the stable bounds we prove that the size of the p-local homotopy groups of spheres grows like exp(O(log(n)^3)), providing the first subexponential bound on the size of the unstable stems.

  • Galois reconstruction of Artin-Tate R-motivic spectra with Jeremy Hahn and Andrew Senger [arxiv] We explain how to reconstruct the category of Artin-Tate R-motivic spectra as a deformation of the purely topological C2-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of C2-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of tau philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of R-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the tau map, which is a feature conspicuously absent from the cellular category.

  • Inertia groups in the metastable range with Jeremy Hahn and Andrew Senger [arxiv] We prove that the inertia groups of all sufficiently-connected, high-dimensional (2n)-manifolds are trivial. Specifically, for m≫0 and k>5/12, suppose M is a (km)-connected, smooth, closed, oriented m-manifold and Σ is an exotic m-sphere. We prove that, if M#Σ is diffeomorphic to M, then Σ bounds a parallelizable manifold. Our proof is an application of higher algebra in Pstragowski's category of synthetic spectra, and builds on previous work of the authors.

Synthetic Cookware

I am writing a book on synthetic spectra and their applications. The intention is that this should eventually become a self-contained introductory account of the synthetic viewpoint on the Adams spectral sequence. An emphasis is placed on tools, techniques and examples demonstrating their use. A draft is available here.