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I'm a member at the IAS in fall 2026; I will then head to Johns Hopkins University as a (tenure-track) assistant professor in Jan 2027. I'm broadly interested in algebraic topology, characteristic p (and v1, v2, ...) geometry, and geometric representation theory, but my interests are constantly changing.
Here is a table of examples of relative Langlands duality in the context of generalized cohomology theories that I studied in some of my writings below. |
Writings
Some projects
Comments or suggestions for improving any of these documents are greatly appreciated! (By request, I’ve updated some of the files below to have internal hyperlinks.)
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Ramification in \(\mathbf{E}_n\)-algebras (2026), joint with Ishan Levy. Last update: Oct 2026. arXiv.
Studies \(\mathbf{E}_n\)-algebras with ramification in their \(\pi_0\) and in particular show that the existence of ramification places strong limitations on their chromatic behavior. -
Relative Langlands duality for linear periods (2026), joint with Tsao-Hsien Chen, David Nadler, and Mark Macerato. Last update: Oct 2026. arXiv.
Studies local relative Langlands duality for \(\mathrm{GL}_{p+q}/(\mathrm{GL}_p \times \mathrm{GL}_q)\). -
Chromatic variants of geometric Satake equivalence (2023). Last update: June 2026. arXiv. (This is a complete rewrite of the previous version.)
Studies the question of proving an analogue of the derived geometric Satake equivalence of Ginzburg and Bezrukavnikov-Finkelberg with coefficients in general \(\mathbf{E}_\infty\)-rings. -
Calculus and cohomology (or, nonlinear numbers) (2026). Last update: June 2026.
Survey/announcement of some recent work (partly from my thesis and from some of the documents below) on calculus built from 1-dimensional formal groups.
Here are some slides for a talk I gave at Berkeley in Oct 2026. -
Appendix to Perverse Microsheaves (2025) by Laurent Côté, Christopher Kuo, David Nadler, and Vivek Shende. arXiv.
Appendix discussing orientation data for symplectic manifolds. -
Spherochromatism in representation theory and arithmetic geometry (2025). Last update: Jan 2026.
This is my PhD thesis (it’s a bit rough in some places!). The word “spherochromatism” in the title is a term from optics, which refers to when a lens (in this case, interpreted as the notion of “linearization” through the homotopy theorist’s notion of spectra) to focus all the wavelengths of color (in this case, interpreted as the various chromatic strata in homotopy theory) to the same point. (See here for a vivid example in photography.) Some, but not all, of this work appears in the documents below; but some of the parts that have already appeared in the documents below have also been edited and refined. (Fixed small errors in Jan 2026) -
\(\mathrm{THH}(\mathbf{Z})\) and the image of J (2025), joint with Arpon Raksit. Last update: May 2025. arXiv. JAMS
Building on work of Bokstedt-Madsen, identifies the \(p\)-complete topological Hochschild homology of the integers with the Frobenius twist of the connective image of J spectrum (for odd \(p\)). Various applications are presented, like a new take on the work of Bokstedt-Madsen, height 1 analogues of the Beilinson fiber square, calculations of the \(K(1)\)-local algebraic K-theory of various ring spectra, etc. -
ku-theoretic spectral decompositions for spheres and projective spaces (2024). Last update: April 2024. arXiv.
Studies the local geometric relative Langlands conjecture of Ben-Zvi, Sakellaridis, and Venkatesh in the case of affine homogeneous spherical varieties of relative rank one (as listed by Akhiezer), as well as an analogue for coefficients in connective complex K-theory \(\mathrm{ku}\). (For instance, working with \(\mathrm{ku}\)-coefficients changes the spectral side \(\check{\mathfrak{g}}^\ast/\check{G}\) of derived geometric Satake to a stack encoding the Hochschild-Kostant-Rosenberg filtration on the free loop space of \(B\check{G}\).) Most of the article is in fact about the geometric relative Langlands conjectures in the general case and its relationships to homotopy theory; when specialized to the rank \(1\) case, our results are “explained” by EHP sequences and Hopf fibrations. (Some simplifications added to the arXiv version in the link above; also lightly updated Conjecture 3.5.11.) -
Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic \(p > 0\) (2023). Last update: July 2024. arXiv. Forum of Math, Sigma.
Shows that if a smooth proper variety \(X\) over \(\mathbf{F}_p\) of dimension \(< p^n\) has degenerating HKR spectral sequence, and \(\mathrm{QCoh}(X)\) lifts to the truncated Brown-Peterson spectrum \(\mathrm{BP}\langle n-1\rangle\), then the Hodge-de Rham spectral sequence degenerates at the \(E_1\)-page. This is contrast to recent work of Petrov, which proves a negative result in this direction. -
The Smith fiber sequence and invertible field theories (2023), and A Long Exact Sequence in Symmetry Breaking (2024); joint with Arun Debray, Cameron Krulewski, Leon Liu, Natalia Pacheco-Tallaj, and Ryan Thorngren. Last update: May 2024. arXiv and arXiv. JHEP.
Studies defects in symmetry breaking phases via the anomaly of their defects, and obstructions to the existence of symmetry breaking phases with a local defect using the Anderson dual of a cofiber sequence of Thom spectra. The second of these separates out the physics from the math. -
Derived geometric Satake for \(\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}\) (2024). Last update: March 2024. arXiv.
Uses the methods of “\(\mathrm{ku}\)-theoretic…” and hypermatrices following Cayley, Gelfand-Kapranov-Zelevinsky, and Bhargava to study the local geometric relative Langlands conjecture of Ben-Zvi, Sakellaridis, and Venkatesh for the spherical variety \(\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}\), whose dual is \(\mathrm{SL}_2^{\times 3}\) acting on \((\mathbf{A}^2)^{\otimes 3}\). Also studies the case of the spherical variety \(\mathrm{PSO}_8/\mathrm{G}_2\), and suggests an analogous story might exist if \(\mathrm{PGL}_2\) and \(\mathrm{G}_2\) are replaced by the rank \(3\) Dwyer-Wilkerson exotic \(2\)-compact group. (The link above has minor additions to the arXiv version.) -
Topological Hochschild homology, truncated Brown-Peterson spectra, and a topological Sen operator (2023). Last update: October 2023. arXiv.
Proves a form of Bokstedt periodicity for \(\mathbf{E}_3\)-forms of a truncated Brown-Peterson spectrum, by studying \(\mathrm{THH}\) relative to Ravenel’s Thom spectra \(X(p^n)\), which played a crucial role in the Devinatz-Hopkins-Smith proof of the nilpotence theorem. This is used to describe a higher chromatic analogue of the “Sen operator” of Bhatt-Drinfeld-Lurie; their behavior is controlled by Cohen-Moore-Neisendorfer fibrations and is Koszul dual to the nilpotence/Ravenel filtration of \(\mathrm{MU}\). (Fixed the construction of \(\Theta_n\).) -
\(p\)-typical curves on \(p\)-adic Tate twists and de Rham-Witt forms (2023), joint with Shubhodip Mondal. Last update: March 2025. arXiv. Advances.
Continuing early work of Bloch, Kato, Artin, and Mazur, shows that de Rham-Witt forms can be constructed via curves on \(p\)-adic syntomic cohomology (as constructed in Bhatt-Morrow-Scholze). The argument relies on refining a result of Hesselholt’s, as well as an evenness result of Darrell and Riggenbach (which we reprove using different methods). -
Generalized \(n\)-series and de Rham complexes (2023), joint with my PRIMES student Max Misterka. Last update: March 2023. arXiv.
Studies some basic algebraic and combinatorial properties of “generalized \(n\)-series” over a commutative ring. In particular, proves several generalizations of \(q\)-analogues of basic combinatorial results, and also studies basic properties of a “\(F\)-de Rham complex” built from the data of a formal group law. This latter construction appears in unpublished work of Arpon Raksit, and also arises naturally in my “Chromatic aberrations…” article (via the \(T \rtimes \mathbf{G}_m^\mathrm{rot}\)-equivariant homology of the affine Grassmannian \(\mathrm{Gr}_T\) of a torus). -
Examples of disk algebras (2023), joint with Jeremy Hahn, Tyler Lawson, Andrew Senger, and Dylan Wilson. Last update: Feb 2023. arXiv.
Shows that many familiar ring spectra, like \(\mathrm{BP}\), \(\mathrm{BP}\langle n\rangle\), \(X(n)\), and spherical polynomial algebras on even degree classes admit framed \(\mathbf{E}_2\)-algebra structures; this allows one to define \(\mathrm{THH}\) relative to these ring spectra. -
Higher chromatic Thom spectra via unstable homotopy theory (2020). Last update: August 2022. arXiv. Algebr. Geom. Topol.
Proves that a conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem, coupled with a conjecture about the \(\mathbf{E}_3\)-centrality of a certain element in the homotopy of Ravenel’s \(X(n)\) spectra, leads to a construction of truncated Brown-Peterson spectra, \(\mathrm{bo}\), and \(\mathrm{tmf}\) as Thom spectra (albeit not over the sphere); this is a higher chromatic analogue of the Hopkins-Mahowald theorem constructing \(\mathrm{H}\mathbf{F}_p\) as a Thom spectrum. The construction of tmf via this method is used to show that these conjectures imply that the string orientation \(\mathrm{MString} \to \mathrm{tmf}\) splits.
Last update: moved (and greatly expanded) certain results to some forthcoming papers. -
Hodge theory for elliptic curves and the Hopf element \(\nu\) (2019). Last update: August 2022. arXiv, but the arXiv is having issues with the spectralsequences TeX package that I’ve used in the newer version; so the version on the arXiv is currently out-of-date. Bull. Lond. Math. Soc.
Shows that the sheaf on the moduli stack of elliptic curves associated to \(\nu\) is isomorphic to the de Rham cohomology of the universal elliptic curve. Uses this to study the \(\mathbf{E}_1\)-quotient of \(\mathrm{tmf}\) by \(\nu\) by relating it to the moduli stack of elliptic curves with a chosen splitting of the Hodge filtration; this results in a calculation of the resulting descent/Adams-Novikov spectral sequence. This \(\mathbf{E}_1\)-quotient is a homotopical lift of quasimodular forms; the article contains a calculation of the integral ring of quasimodular forms. Latest version fixes a small calculational error (results remain unchanged). -
Loop groups and intertwining of positive-energy representations (2021). Last update: September 2021. Chapter 22 of this book.
A brief introduction to representations of loop groups; in particular, focusing on a theorem of Pressley-Segal stating that any positive-energy representation of \(LG\) extends to a projective intertwining action of \(\mathrm{Diff}^+(S^1)\). Edited by the indicated authors of the book on arXiv. -
On the James and Hilton-Milnor Splittings, and the metastable EHP sequence (2019), joint with Peter Haine. Last update: November 2020. arXiv. Doc. Math.
Studies the James and Hilton-Milnor splittings and the metastable EHP sequences in higher category theory; this allows a generalization of the James and Hilton-Milnor splittings to motivic spaces over any base scheme. Two proofs of the metastable EHP sequence are provided; one is new and non-calculational, and (essentially) only utilizes the Blakers-Massey theorem. -
The Ando-Hopkins-Rezk orientation is surjective (2019). Last update: January 2020. arXiv.
Shows that the Ando-Hopkins-Rezk orientation \(\mathrm{MString} \to \mathrm{tmf}\) is surjective on homotopy. This is done by constructing an \(\mathbf{E}_1\)-ring \(B\) with an \(\mathbf{E}_1\)-map \(B\to \mathrm{MString}\), and showing that the composite \(B \to \mathrm{MString} \to \mathrm{tmf}\) is surjective on homotopy. File for generating the Adams chart via Hood Chatham’s Ext calculator. -
Roots of unity in \(K(n)\)-local rings (2017). Proc. Amer. Math. Soc. Last update: November 2019. arXiv.
Shows that for \(n>0\), any \(K(n)\)-local \(H_\infty\)-ring \(R\) with a primitive \(p^k\)-th root of unity in \(\pi_0 R\) is trivial. This implies that the Lubin-Tate tower does not lift to a tower of derived stacks over Morava E-theory. - Two books on algebraic topology, notes from course taught by Haynes Miller.
Last update: April 2018.
These notes (especially the stuff from 18.906) have been heavily modified by Haynes, and published as a book; see here.
- Algebraic Topology I (2018). Published online with the AMS; also available on MIT OpenCourseWare.
- Haynes’ edits of the notes Algebraic Topology II (2018). Those notes are a cleaned up version of the somewhat-cleaned-up version of my edited notes.
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The Dieudonn’e modules and Ekedahl-Oort types of Jacobians of hyperelliptic curves in odd characteristic (2017), with John Halliday. Last update: December 2017. arXiv. Provides explicit formulae for the Frobenius and Verschiebung acting on the mod p Dieudonn’e module of the Jacobian of a hyperelliptic curve, when p is an odd prime. These formulae are used to settle some questions posed by Glass and Pries from 2004. The code used is available at this github repository.
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The Lubin-Tate stack and Gross-Hopkins duality (2017). Last update: July 2018. arXiv.
Uses derived algebraic geometry to provide proofs and generalizations of some duality phenomena in K(n)-local stable homotopy theory. (Fixed some typos and technical errors, and some results on K(n)-local Spanier-Whitehead self-duality of higher real K-theories.) -
Talbot proceedings: Obstruction theory for structured ring spectra (2017), joint with Eva Belmont et. al. Last update: September 2017.
The proceedings from the Talbot workshop on obstruction theory for structured ring spectra, which took place from May 21-27, 2017. - The importance of theoretical research (2018). Article published in The Tech.