I'm a fifth-year graduate student at Harvard, fortunate to be advised by Mike Hopkins and Dennis Gaitsgory. I'm broadly interested in algebraic topology, characteristic p (and v1, v2, ...) geometry, and geometric representation theory, but my interests are constantly changing. I'll be applying for postdocs during fall 2024. Outside of math, I spend a bunch of time playing the drums, listening to (mostly instrumental) prog metal, and playing basketball. I was an undergraduate at MIT, where I majored in math (course 18) and minored in physics (course 8). Email address: skdevalapurkar[at]gmail[dot]com. My office is SC 333f. CV: [pdf] I'm also on MathOverflow. Here is a blog (which I have been horrible about updating), and 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.)

• 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 \(\check{\mathfrak{g}}/\check{G}\) 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.)

• Chromatic aberrations of the geometric Satake equivalence over the regular locus (2023). Last update: Aug 2024. arXiv. (This is almost a complete rewrite of the first version.)
  Studies the question of proving an analogue of the derived geometric Satake equivalence of Ginzburg and Bezrukavnikov-Finkelberg (really, of the Arkhipov-Bezrukavnikov-Ginzburg equivalence) with coefficients in complex K-theory and elliptic cohomology. Shows, in the case of elliptic cohomology with associated elliptic curve \(E\) (for instance), that the quotient \(\widetilde{\check{\mathfrak{g}}}/\check{G}\) of the Grothendieck-Springer resolution in the spectral side of the usual ABG equivalence is to be replaced by the semistable locus in (the degree zero component of) the Kontsevich-Mori compactification \(\widetilde{\mathrm{Bun}}_{\check{G}}(E^\vee)\). (Added some new material on the interaction with power operations.)

• Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic \(p > 0\) (2023). Last update: July 2024. arXiv.
  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.
  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: Sept 2023. arXiv.
  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. To appear in 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. Published in 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. Published in 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). Published in 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.

• 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.

• 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.

---

Teaching

• In Spring 2024, I’m teaching an undergraduate course (Math 99r) on integrable systems. The website is here.
• I’m not teaching in Fall 2023, but I am organizing a seminar on relative Langlands duality with Ben Gammage.
• I’m not teaching in Spring 2023.
• In Fall 2022, I’m teaching Math 1b (Calculus, Series, and Differential Equations) at Harvard, and I’m continuing to mentor for a high school project via PRIMES USA.
• In Spring 2022, I was a TA for Math 231b (Algebraic topology II) at Harvard (fibrations, spectral sequences, abstract homotopy theory, rational homotopy theory), taught by Mike Hopkins. I am also a mentor for a high school project via PRIMES USA.
• In Fall 2021, I was a TA for Math 223a (Algebraic number theory) at Harvard (ideal class groups, local fields, basics of local class field theory), taught by Mark Shusterman.
• Every semester from Fall 2020 through Fall 2022, and Fall 2023, I’ve been a mentor for the directed reading program at Harvard; in chronological order, I’ve mentored reading projects on surfaces and Chern classes in differential geometry, function fields and elliptic curves, topological K-theory, the cobordism hypothesis, and prismatic cohomology.

---

Notes

The usual warning holds: many of these documents have not been proofread, so there’s a chance they contain mistakes. Caveat lector! Please send me an email if you have any comments.

• A String analogue of Spin-C. Miscellaneous notes I wrote a while back. Last update: August 2022.
• Exceptional isomorphisms. Notes for a talk at the Trivial Notions seminar at Harvard; currently incomplete near the end. Last update: October 2023.
• Equivariant homotopy theory and geometric Langlands. Slides for a talk at Princeton. Last update: October 2023.
• Prismatization. Notes for a talk at the Thursday seminar at Harvard. Last update: December 2022.
• Etale comparison. Notes for a talk at the Thursday seminar at Harvard. Last update: October 2022.
• Anomalies and invertible field theories. Notes for a seminar on reflection positivity at Harvard. Last update: April 2022.
• Mirror symmetry for toric varieties. Notes for a talk at Trivial Notions at Harvard. Last update: February 2022.
• Summary of six functors for spaces. Miscellaneous notes on the six functor formalism for spaces. Last update: December 2021.
• The quantum Noether theorem. Notes for a talk at Juvitop on Costello-Gwilliam. Last update: December 2021.
• Nonabelian Fourier transform/bi-Whittaker reduction. Notes for a talk at a seminar on the universal regular centralizer group scheme. Last update: October 2021.
• Morava K-theory and Poincare duality. Notes for a talk at the outdoor symplectic seminar on Abouzaid-Blumberg. Last update: June 2021.
• In Summer 2021, I ran a seminar on (deformation) quantization. The website is here.
• Divisibility of Chern numbers of PPAVs. Notes for a talk at Juvitop on Feng-Galatius-Venkatesh. Last update: April 2021.
• Periods. Notes for a talk at Trivial Notions at Harvard. Last update: April 2021.
• Chromatic analogues of the Hopkins-Mahowald theorem. Handwritten slides used for a talk at the Rochester topology seminar. Last update: March 2021.
• Discrete adic spaces. Notes for a talk on discrete adic spaces for the Thursday seminar on condensed math. Last update: November 2020.
• Whitehead products. Notes studying generalized Whitehead products following my paper with Peter Haine. Last update: September 2020.
• Splitting cobordism spectra. Slides for an invited talk at Tsinghua University about my work on higher chromatic Thom spectra. Last update: August 2020.
• The Balmer spectra of spectral stacks. Proves that the Balmer spectrum of nice spectral algebraic stacks separates into a “homotopy-theoretic” part and an “algebro-geometric” part. Writing a more detailed note, so I’ve temporarily taken this link down.
• The Bogmolov-Tian-Todorov theorem, notes on the Bogomolov-Tian-Todorov theorem in characteristic zero and characteristic p. Writing a more detailed note, so I’ve temporarily taken this link down.
• The nonabelian Hodge correspondence, notes for a talk given for CORONAGS (Corona Outbreak-Response Omnipresent (Noncommutative) Algebraic Geometry Seminar). I’ve got slides, which contain more information than the notes. Last update: March 2020.
• The Riemann-Hilbert correspondence, notes for a talk given on the D-module Day for the seminar on mixed Hodge modules. Also, I’ve got slides, which contain more information than the notes. Last update: March 2020.
• Representations of Frobenius kernels, notes for a talk given at the Langlands Support Group on modular representation theory. Last update: March 2020.
• Not mine, but a scan of Mahowald-Unell’s Bott Periodicity at the Prime 2 in the Unstable Homotopy of Spheres. Thanks to Peter May for providing the (sole surviving?) physical copy.
• Dynamical properties of quantum systems with fractional statistics, final paper for 8.06 (quantum mechanics III). Last update: April 2019.
• Some handwritten lecture notes, which I’ve actually ended up scanning. Here are my scanned lecture notes from some talks at the 2017 UIUC conference.
• Stable splittings of classifying spaces of compact Lie groups, for a talk at the Thursday seminar at Harvard. Last update: November 2018.
• The Morava K-theory of Eilenberg-MacLane spaces, for a talk at Juvitop. Last update: October 2018.
• Notes from 8.321 (graduate quantum mechanics). Last update: October 2018.
• Anderson duality for derived stacks. Last update: August 2018. Gives a non-computational argument proving that many derived stacks are Anderson self-dual.
• Lecture notes on chromatic homotopy theory. Last update: July 2018. Lecture notes from a course which I taught in January 2018.
• Equivariant versions of Wood’s theorem. Last update: July 2018. Proves using derived algebraic geometry that Wood’s theorem holds for G-equivariant K-theory (a result due to Mathew, Naumann, and Noel) and G-equivariant periodic TMF, where G is a compact abelian Lie group.
• Orientations of derived formal groups, for a talk at Juvitop. Last update: April 2018.
• Slides from my talk on “Roots of unity in \(K(n)\)-local \(\mathbf{E}_\infty\)-rings”, at JMM 2018 (in San Diego, California).
• Examples of Goodwillie Calculus, for a talk at Juvitop. Last update: October 2017.
• Milnor’s exotic spheres, for a talk at the Kan seminar. Last update: September 2017.