In 2021, I finished my PhD in mathematics at MIT working with Clark Barwick.
During the 2021–2022 academic year, I was an NSF Postdoc and UC President's Postdoc at UC Berkeley under the direction of David Nadler.
During the 2022–2023 academic year, I'm a postdoc at the Institute for Advanced Study under Jacob Lurie.
I will return to Berkeley in Fall 2023.
My mathematical interests center around homotopy theory, algebraic geometry, and microlocal sheaf theory.
Araminta Amabel, Clark Barwick, Vladimir Chugunov, Arun Debray, Sanath Devalapurkar, Sergei Fomin, Saul Glasman, Victor Guillemin, Tim Holzschuh, Mauro Porta, Ravi Shankar, Jean-Baptiste Teyssier, Sebastian Wolf
From nonabelian basechange to basechange with coefficients (last updated September 2022)
The homotopy-invariance of constructible sheaves, with Mauro Porta and Jean-Baptiste Teyssier (last updated August 2022)
The James and Hilton–Milnor Splittings, & the metastable EHP sequence, with Sanath Devalapurkar (last updated November 2021)
- The fundamental fiber sequence in étale homotopy theory, with Tim Holzschuh and Sebastian Wolf (last updated October 2022)
Pyknotic objects, I. Basic Notions, with Clark Barwick (last updated April 2019)
On coherent topoi & coherent 1-localic ∞-topoi (last updated July 2019)
- Doc. Math. 26 (2021), 1423–1464.
- [pdf, arXiv:1912.04130, Journal version]
- Note proving the James and Hilton–Milnor Splittings in a very general context that applies to motivic spaces over an arbitrary base. We also give a new, non-computational proof of the metastable EHP sequence in an ∞-topos that essentially only makes use of the Blakers–Massey Theorem.
Exodromy for stacks, with Clark Barwick (last updated January 2019)
Extended étale homotopy groups from profinite Galois categories (last updated January 2019)
On the homotopy theory of stratified spaces (last updated April 2022)
Exodromy, with Clark Barwick & Saul Glasman (last updated August 2020)
Stability analysis of non-Newtonian rimming flow, with Vladimir Chugunov, Sergei Fomin, & Ravi Shankar (February 2016)
- [pdf, arXiv:1904.01877]
- Fills a small gap in the literature by proving that the ∞-categories of coherent ordinary topoi (in the sense of SGA4) and coherent 1-localic ∞-topoi (in Lurie's sense) are equivalent. We also collect some examples of coherent geometric morphisms coming from algebraic geometry.
- Differential Cohomology: Categories, Characteristic Classes, and Connections, jointly edited with Araminta Amabel & Arun Debray (last updated October 2021)
Differential forms, with Victor Guillemin
- [pdf, arXiv:2109.12250]
- We give an overview of differential cohomology from a modern, homotopy-theoretic perspective in terms of sheaves on manifolds. Although modern techniques are used, we base our discussion in the classical precursors to this modern approach, such as Chern–Weil theory and differential characters, and include the necessary background to increase accessibility. Special treatment is given to differential characteristic classes, including a differential lift of the first Pontryagin class. Multiple applications, including to configuration spaces, invertible field theories, and conformal immersions, are also discussed. This book is based on talks given at MIT’s Juvitop seminar run jointly with UT Austin in the Fall of 2019.
- In addition to chapters by the editors, there are also chapters by: Dexter Chua, Sanath Devalapurkar, Dan Freed, Mike Hopkins, Greg Parker, Charlie Reid, and Adela Zhang.
- On the homotopy theory of stratified spaces
Descent for sheaves on compact Hausdorff spaces (last updated September 2022)
- Profinite completions of products (last updated November 2022)
Character isomorphisms via tempered cohomology (last updated January 2022)
- [pdf, arXiv:2210.00186]
- Notes explaining why the functor sending a compact Hausdorff space K to the ∞-category of Postnikov complete sheaves on K is a sheaf on the site of compact Hausdorff spaces and finite jointly surjective families.
We use this to show that condensed cohomology and sheaf cohomology agree for locally compact Hausdorff spaces.
Viva Fukaya! (last updated January 2022)
Connectedness of cotensors (last updated March 2021)
- Extended notes from my talk at the 202One Talbot Workshop on Ambidexterity. These notes explain how to use Lurie's work on elliptic cohomology to recover character isomorphisms in chromatic homotopy theory due to Hopkins–Kuhn–Ravenel and Stapleton.
Splitting free loop spaces (last updated January 2021)
- The purpose of this note is to generalize the following observation: given an (n+1)-connected map of spaces f: X → Y, the induced morphism Lf: LX → LY on free loop spaces is n-connected. We show that in an ∞-topos, cotensoring with a finite space with cells in dimensions ≤m decreases connectedness of morphisms by m.
References for the equivalence of different approaches to ∞-categorical enhancements of derived categories (last updated October 2020)
- The purpose of this note is to prove a very general splitting result for free loop objects. For example, we show that in an ∞-category C with finite limits, given an A₂-algebra G in C that has inverses in an appropriate sense, the loop object LG splits as LG ≃ G × ΩG.
We also sketch Aguadé & Ziller's proof that the free loop space LSn splits if and only if n = 0, 1, 3, or 7.
The Bégueri Resolution (last updated July 2020)
- A point of confusion for a lot of people learning about ∞-categories is why the different approaches to ∞-categorical enhancements of derived categories (via model categories, dg categories, and ∞-categorical localizations) produce the same answer.
The explaination is scattered throughout §1.3 of Lurie's book Higher Algebra.
Since so many people have asked me about this, I've collected the relevant references in one place.
Differential cohomology theories as sheaves on manifolds
The Segal–Sugawara Construction
The lci locus of the Hilbert scheme of points & the cotangent complex (last updated April 2020)
- Note explaining Bégueri's resolution of a commutative, finite locally free group scheme by smooth affine group schemes. This resolution is a technical tool used in Česnavičius and Scholze's paper Purity for flat cohomology to transfer questions about fppf cohomology to questions about étale cohomology.
Nonabelian Poincaré Duality (last updated March 2019)
Ambidexterity §4 (last updated November 2018)
Notes on Étale Cohomology (last updated October 2017)
- Notes for a talk at the Fall 2019 Thursday Seminar introducing the basic material needed to understand Elmanto, Hoyois, Khan, Sosnilo, & Yakerson's work on motivic infinite loop spaces.
An Introduction to Goodwillie Calculus (last updated September 2017)
- Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.
An Overview of Motivic Cohomology (last updated September 2017)
- Notes on the basic setup of Goodwillie calculus in the ∞-categorical setting.
On the K-theory of Finite Fields (last updated September 2016)
Lifting Enhanced Factorization Systems to Functor 2-categories (last updated July 2016)
- Notes for a talk at the MIT graduate student lunch seminar giving a very broad overview of motivic cohomology.
- A note on a technical result about defining enhanced factorization systems on functor 2-categories.
Below are some videos from talks I've given:
Galois-theoretic reconstruction of schemes and exodromy (July 7, 2022)
Stratified étale homotopy theory (May 20, 2020)
The Homotopy Theory of Stratified Spaces (April 28, 2020)
Intro to Constructible Sheaves and Exit-Paths (April 24, 2020)
Pyknotic Spaces, Spectra, etc. (March 10, 2020)
Pyknotic Sets 2 (March 4, 2020)
- Exit-path categories in geometry and topology (September 22, 2022)
Below are some seminars that I have helped organize:
Condensed Thursday Seminar, Fall 2020
Juvitop seminar on the Cobordism Hypothesis, Fall 2020
Reading Group on Part I of An Inclusive Academy: Achieving Diversity and Excellence by Abigail Stewart and Virginia Valian, Summer 2020
Juvitop seminar on Differential Cohomology, Fall 2019
Miniature seminar on Factorization Homology, Spring 2019
Juvitop seminar on Ambidexterity, Fall 2018
MIT Student Colloquium for Undergraduates in Mathematics, 2015– 2016
- Nadler's Geometric Representation Theory Seminar, 2021–2022
I am involved in a number of service activities and am interested in problems around diversity in mathematics and higher education in general.
- During Fall 2020, Araminta Amabel, Sarah Greer, and I started the Grad-Undergrad Math Mentoring Iniative (GUMMI) at MIT
- Since Fall 2018, I have been the coordinator for PRIMES Circle at MIT
- January 2017 & January 2018, I was a mentor for the MIT Math Department's Directed Reading Program
- Fall 2017, I was the organizer for the MIT Topology Seminar
- Summer 2017, I was a mentor for MIT's √Mathroots program
- Since Spring 2016, I have been a member of the MIT Math Department's Diversity and Community Building Committee
Nadler's GRT Seminar
"Think globally, act microlocally"
Topic: Sheaf theory and microlocal sheaf theory
Meeting time: Thursdays 12:30pm–2pm
Room: #939 Evans
Spring 2022 schedule
||Singular support I
||Singular support II
||Singular support III
||Singular support IV
||More low dimensional
as an invariant
Fall 2021 schedule
Sheaves on ℝ
||Setup: Review of sheaf theory
||Setup: dg categories, derived
categories of sheaves, …
||Dennis Chen & Peter Haine
||Operations on sheaves:
Grothendieck's six functors
||Operations on sheaves II
||Operations on sheaves III
||David Nadler & John Nolan
||Operations on sheaves IV
||Operations on sheaves V
||Nearby and vanishing cycles I
||Mark Macerato & David Nadler
||Nearby and vanishing cycles II
||Examples of nearby and vanishing cycles
||Nearby and vanishing cycles III
Microlocal sheaf theory
||X. Jin, Microlocal sheaf categories and the J-homomorphism, Preprint available at arXiv:2004.14270, Sep. 2020.
||X. Jin and D. Treumann, Brane structures in microlocal sheaf theory, Preprint available at arXiv:1704.04291, April 2017.
||M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1994, vol. 292, pp. x+512, With a chapter in French by Christian Houzel, Corrected reprint of the 1990 original, ISBN: 3-540-51861-4.
||D. Nadler and V. Shende, Sheaf quantization in Weinstein symplectic manifolds, Preprint available at arXiv:2007.10154, Feb. 2021.
||P. Schapira, A short review of microlocal sheaf theory, Preprint available at webusers.imj-prg.fr/~pierre.schapira/lectnotes/MuShv.pdf, Jan. 2016.
||P. Schapira, Microlocal analysis and beyond, Preprint available at arXiv:1701.08955, Jan. 2017.
||J. Zhang, Quantitative Tamarkin category, Lecture notes available at arXiv:1807.09878, Jul. 2018.
Derived categories, dg categories, ∞-categories, etc.
||D. Clausen, Interlude on D(R), notes for a Course on algebraic de Rham cohomology. Available at sites.google.com/view/algebraicderham.
||D. Clausen, Interlude on sheaves with values in D(Z), notes for a Course on algebraic de Rham cohomology. Available at sites.google.com/view/algebraicderham.
||B. Keller, On differential graded categories, In: International Congress of Mathematicians. Vol. II. Eur. Math. Soc., Zürich, 2006, pp. 151–190. Available at arXiv:0601185.
||A. Mazel-Gee, An invitation to higher algebra, Book projection available at etale.site/teaching/w21/math-128-lecture-notes.pdf, 2021.
||J. Lurie, Higher topos theory, Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 2009, vol. 170, pp. xviii+925, ISBN: 978-0-691-14049-0; 0-691-14049-9.
||J. Lurie, Higher algebra, Preprint available at math.ias.edu/~lurie/papers/HA.pdf, Sep. 2017.
||J. Lurie, Spectral algebraic geometry, Preprint available at math.ias.edu/~lurie/papers/SAG-rootfile.pdf, Feb. 2018.
||B. Toën, Lectures on dg-categories, In: Topics in algebraic and topological K-theory. Vol. 2008. Lecture Notes in Math. Springer, Berlin, 2011, pp. 243–302. DOI: 10. 1007/978-3-642-15708-0. Available at perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf
||A. Yekutieli, Derived categories, Preprint available at arXiv:1610.09640, Apr. 2019.