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I'm a first-year graduate student at Harvard. I'm broadly interested in algebraic topology and arithmetic geometry, but my interests are constantly changing. Outside of math, I spend a bunch of time playing the drums, listening to instrumental prog metal, biking, and playing basketball. I was an undergraduate at MIT, where I majored in math (course 18) and minored in physics (course 8).
Here are the courses that I've taken, as well as a blog (which I have been horrible about updating). |
Writings
Some projects
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Quantization and Koszul duality.
Discusses the theory of deformation quantization from a Koszul dual perspective, roughly allowing us to view deformation quantization as the datum of an \(S^1\)-action. In characteristic \(p\), this can be refined further: restricted deformation quantization can roughly be viewed as the datum of an \(S^1\)-action along with a cyclotomic structure. Applications to the deformation theory of schemes are discussed. -
Higher chromatic Thom spectra via unstable homotopy theory (2020). Last update: June 2020. arXiv.
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: fixed some embarassing typos. -
On the James and Hilton-Milnor Splittings, and the metastable EHP sequence (2019), joint with Peter Haine. Last update: April 2020. arXiv.
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. -
Hodge theory for elliptic curves and the Hopf element \(\nu\) (2019). Last update: April 2020. arXiv.
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. Latest version fixes a small calculational error (results remain unchanged). -
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.
- 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.
Notes
- Splitting cobordism spectra. Slides for an invited talk at Tsinghua University about my work on higher chromatic Thom spectra.
- 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. Last update: May 2020.
- The Bogmolov-Tian-Todorov theorem, notes on the Bogomolov-Tian-Todorov theorem in characteristic zero and characteristic p. Proves a generalization of this theorem to the noncommutative setting (in arbitrary characteristic), as the formality of a certain dg-Lie algebra. Last update: June 2020.
- 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.
- Loop groups and their representations, notes for a talk given at Juvitop. Last update: November 2019. Has some errors that I’ll fix soon.
- 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.