Jeremy Hahn

I am a postdoc at MIT working under Haynes Miller.
I study Algebraic Topology, with a particular emphasis on structured ring spectra.
Here is my CV.

Contact Information:
Email: jhahn01 at mit dot edu



Preprints and Publications Teaching



Preprints and publications

  • Real topological Hochschild homology and the Segal conjecture with Dylan Wilson (2019). We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F2. This determines the E2-page of the descent spectral sequence for the map NF2 → F2, where NF2 is the C2-equivariant Hill-Hopkins-Ravenel norm of F2. The E2-page represents a new upper bound on the RO(C2)-graded homotopy of NF2, from which the Segal conjecture is an immediate corollary.

  • On the boundaries of highly connected, almost closed manifolds with Robert Burklund and Andrew Senger (2019). 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 Krannich and Kreck, 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 HFp-Adams filtrations for all primes p. We additionally prove new vanishing lines in the HFp-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.

  • Wilson Spaces, Snaith Constructions, and Elliptic Orientations with Hood Chatham and Allen Yuan (2019). We construct a canonical family of even periodic E-ring spectra, with exactly one member of the family for every prime p and chromatic height n. At height 1 our construction is due to Snaith, who built complex K-theory from CP. At height 2 we replace CP with a p-local retract of BU<6>, producing a new theory that orients elliptic, but not generic, height 2 Morava E-theories. In general our construction exhibits a kind of redshift, whereby BP<n-1> is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K(n)-localization of our height n ring to work of Peterson and Westerland building (En)hSG from K(Z,n+1).

  • The Lubin-Tate Theory of Configuration Spaces: I with Lukas Brantner and Ben Knudsen (2019). We construct a spectral sequence converging to the Morava E-theory of unordered configuration spaces and identify its E2-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the E-theory of the weight p summands of iterated loop spaces of spheres (parametrising the weight p operations on En-algebras), as well as the E-theory of the configuration spaces of p points on a punctured surface. We read off the corresponding Morava K-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the Fp-homology of the space of unordered configurations of p particles on a punctured surface.

  • Nilpotence in normed MGL-modules with Tom Bachmann (2019). We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies E^HZ=0, then also E^MGL=0. In words, motivic homology detects vanishing of normed modules over the algebraic cobordism spectrum.

  • Exotic Multiplications on Periodic Complex Bordism with Allen Yuan (2019). Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of BU. This presents an E structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E-ring MUP. Here, we prove that these two E-rings are in fact different, though the underlying E2-rings are equivalent. Nonetheless, we prove that both rings E-orient 2-completed KU and other forms of K-theory.

  • Eilenberg-MacLane spectra as equivariant Thom spectra with Dylan Wilson (2018). We prove that the G-equivariant mod p Eilenberg-MacLane spectrum arises as an equivariant Thom spectrum for any finite, p-power cyclic group G, generalizing a result of Behrens and the second author in the case of the group C2. We also establish a construction of HZ(p), and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant HFp arises as the Thom spectrum of a more than double loop map.

  • Multiplicative Structure in the Stable Splitting of ΩSLn(C) with Allen Yuan (2017). The space of based loops in SLn(C), also known as the affine Grassmannian of SLn(C), admits an E2 or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not E2. Nonetheless, we show that the splitting becomes E2 after base-change to complex cobordism. Our proof of the A splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations. Accepted by Advances in Mathematics.

  • Real Orientations of Morava E-theories with Xiaolin Danny Shi (2017). We show that Morava E-theories at the prime 2 are Real oriented and Real Landweber exact. The proof is an application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for En with its C2-action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these C2-fixed points.

  • Nilpotence in En algebras (2017). Nilpotence in the homotopy of E-ring spectra is detected by the classical HZ-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of En-ring spectra. For all odd primes p and all chromatic heights h, we use the Cohen-Moore-Neisendorfer theorem to construct examples of K(h)-local, E2n-1-algebras with non-nilpotent pn-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound is sharp at height 1, and remark on the situation at height 2.

  • On the Bousfield classes of H-ring spectra (2016). We prove that any K(n)-acyclic, H-ring spectrum is K(n+1)-acyclic, affirming an old conjecture of Mark Hovey.

  • Appendix to: Brown Peterson cohomology from Morava E-theory by Tobias Barthel and Nathaniel Stapleton (2015). We provide the proof of a technical lemma about torsion in the Morava E theory of finite abelian groups modulo transfers. The theorems of Barthel and Stapleton then allow conclusions about BP cohomology modulo transfers. Accepted by Compositio Mathematica.

  • Quotients of even rings with Dylan Wilson (2018). A short note about quotients of even rings remaining highly structured. Applications to Brown-Peterson spectra are forthcoming.

    Work from 2013-2018 was supported by an NSF GRFP under Grant DGE-1144152. Work starting in Fall 2018 is supported by the NSF under Grant DMS-1803273.

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Teaching

Collaborators: Tom Bachmann, Tobias Barthel, Lukas Brantner, Robert Burklund, Hood Chatham, Gijs Heuts, Ben Knudsen, Andrew Senger, Xiaolin (Danny) Shi, Nathaniel Stapleton, Dylan Wilson, Allen Yuan.