About
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 was a postdoc at the Institute for Advanced Study under Jacob Lurie.
From 2023 to 2025, I'm back at Berkeley to finish my NSF postdoc.
My mathematical interests center around homotopy theory, algebraic geometry, and microlocal sheaf theory.
CV
Research/Writing
Collaborators
Araminta Amabel, Qingyuan Bai, Clark Barwick, Magnus Carlson, Vladimir Chugunov, Arun Debray, Sanath Devalapurkar, Sergei Fomin, Saul Glasman, Victor Guillemin, Tim Holzschuh, Mauro Porta, Piotr Pstrągowski, Maxime Ramzi, Ravi Shankar, Jan Steinebrunner, JeanBaptiste Teyssier, Sebastian Wolf
Papers in Progress
 Localizing invariants of constructible sheaves, with Qingyuan Bai
 Fully faithful functors and pushouts of ∞categories, with Maxime Ramzi and Jan Steinebrunner
 Reconstruction of schemes from their étale topoi, with Magnus Carlson and Sebastian Wolf
 We generalize Voevodsky's reconstruction theorem to infintie fields of positive characteristic that are finitely generated over their prime fields.
We also relax the normality assumption to the optimal one, namely absolute weak normality (seminormality in characteristic zero, and perfectness in positive characteristic).
Papers
 Exodromy beyond conicality, with Mauro Porta and JeanBaptiste Teyssier (Last updated January 2024)
 [pdf, arXiv:2401.12825]
 Generalizes the exodromy equivalence in topology beyond the setting of conically stratified spaces, with a number of applications.
 Spectral weight filtrations, with Piotr Pstrągowski (last updated September 2023)
 Nonabelian basechange theorems & étale homotopy theory, with Tim Holzschuh and Sebastian Wolf (last updated June 2024)
 Profinite completions of products (last updated March 2024)
 [pdf, arXiv:2406.00136]
 Initially posted on here in November 2022, but wasn't uploaded to the arXiv until May 2024.
 The fundamental fiber sequence in étale homotopy theory, with Tim Holzschuh and Sebastian Wolf (last updated December 2022)
 From nonabelian basechange to basechange with coefficients (last updated September 2022)
 The homotopyinvariance of constructible sheaves, with Mauro Porta and JeanBaptiste Teyssier (last updated August 2022)
 The James and Hilton–Milnor Splittings, & the metastable EHP sequence, with Sanath Devalapurkar (last updated November 2021)
 Doc. Math. 26 (2021), 1423–1464.
 [pdf, arXiv:1912.04130, Journal version]
 We prove 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, noncomputational proof of the metastable EHP sequence in an ∞topos that essentially only makes use of the Blakers–Massey Theorem.
 Pyknotic objects, I. Basic Notions, with Clark Barwick (last updated April 2019)
 On coherent topoi & coherent 1localic ∞topoi (last updated July 2019)
 [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 1localic ∞topoi (in Lurie's sense) are equivalent. We also collect some examples of coherent geometric morphisms coming from algebraic geometry.
 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 nonNewtonian rimming flow, with Vladimir Chugunov, Sergei Fomin, & Ravi Shankar (February 2016)
Books
 Differential Cohomology: Categories, Characteristic Classes, and Connections, jointly edited with Araminta Amabel & Arun Debray (last updated January 2023)
 To be published in Cambridge Studies in Advanced Mathematics.
 [pdf, arXiv:2109.12250]
 We give an overview of differential cohomology from a modern, homotopytheoretic 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.
 Differential forms, with Victor Guillemin
Reports
 On the homotopy theory of stratified spaces
Notes
 Descent for sheaves on compact Hausdorff spaces (last updated September 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.
 Character isomorphisms via tempered cohomology (last updated January 2022)
 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.
 Viva Fukaya! (last updated January 2022)
 Connectedness of cotensors (last updated March 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 nconnected. We show that in an ∞topos, cotensoring with a finite space with cells in dimensions ≤m decreases connectedness of morphisms by m.
 Overview of algebraic Ktheory (last updated February 2021)
 Splitting free loop spaces (last updated January 2021)
 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 A₂algebra 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 LS^{n} splits if and only if n = 0, 1, 3, or 7.
 References for the equivalence of different approaches to ∞categorical enhancements of derived categories (last updated October 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.
 The Bégueri Resolution (last updated July 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.
 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)
 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.
 Nonabelian Poincaré Duality (last updated March 2019)
 Ambidexterity §4 (last updated November 2018)
 Notes on Étale Cohomology (last updated October 2017)
 Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.
 An Introduction to Goodwillie Calculus (last updated September 2017)
 Notes on the basic setup of Goodwillie calculus in the ∞categorical setting.
 An Overview of Motivic Cohomology (last updated September 2017)
 Notes for a talk at the MIT graduate student lunch seminar giving a very broad overview of motivic cohomology.
 On the Ktheory of Finite Fields (last updated September 2016)
 Lifting Enhanced Factorization Systems to Functor 2categories (last updated July 2016)
 arXiv:1604.06812
 A note on a technical result about defining enhanced factorization systems on functor 2categories.
Lecture Notes & Slides
Some lecture notes and slides from talks/lecture series I've given:
 Exodromy beyond conicality (last updated July 2024)
 An introduction to "Exodromy beyond conicality" (last updated May 2024)
 For a talk that I at the Lloyd Roeling conference, UC Davis Algebraic Geometry Seminar, and Syzygies and Mirror Symmetry Virtual Seminar. To simplify the presentation, in the talk I ignored the difference between sheaves and hypersheaves. In these slides, the word
sheaf
should be interpreted as hypersheaf
.
 Topological quantum field theories & the cobordism hypothesis in low dimensions (last updated March 2022)
Videos
Some videos from talks I've given:
 Reconstruction in algebraic geometry (January 12, 2023)
 A survey/introduction to some reconstruction results in algebraic geometry aimed at an audience of graduate students across all areas of math for the IAS Spring Oppertunities Workshop
 Exitpath categories in geometry and topology (September 22, 2022)
 Galoistheoretic 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 ExitPaths (April 24, 2020)
 Pyknotic Spaces, Spectra, etc. (March 10, 2020)
 Pyknotic Sets 2 (March 4, 2020)
Seminars
Below are some seminars that I have helped organize:
 Nadler's Geometric Representation Theory Seminar, 2021–2022
 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
Service
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 GradUndergrad 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
2021–2022
"Think globally, act microlocally"
Topic: Sheaf theory and microlocal sheaf theory
Meeting time: Thursdays 12:30pm–2pm
Room: #939 Evans
Spring 2022 schedule
Microlocal perspective
1/27 
David Nadler 
Singular support I 
Notes 
2/3 
David Nadler 
Singular support II 
Notes 
2/10 
David Nadler 
Singular support III 
Notes 
2/17 
David Nadler 
Singular support IV 
Notes 
2/24 
David Nadler 
Microlocal sheaves 
Notes 
3/3 
David Nadler 
Examples of microlocal sheaves 
Notes 
3/10 
David Nadler 
More low dimensional examples 
Notes 
3/17 
David Nadler 
Microlocal sheaves as an invariant 
Notes 
Fall 2021 schedule
8/31 
David Nadler 
Organizational meeting/ Sheaves on ℝ 
Notes 
9/7 
Dennis Chen 
Setup: Review of sheaf theory 
Notes 
9/14 
Dennis Chen 
Setup: dg categories, derived categories of sheaves, … 

9/21 
Dennis Chen & Peter Haine 
Setup continued 
Notes 
9/28 
Dennis Chen 
Setup continued 

10/5 
John Nolan 
Operations on sheaves: Grothendieck's six functors 

10/12 
John Nolan 
Operations on sheaves II 

10/19 
John Nolan 
Operations on sheaves III 

10/26 
David Nadler & John Nolan 
Operations on sheaves IV 

11/2 
John Nolan 
Operations on sheaves V 

11/9 
Mark Macerato 
Nearby and vanishing cycles I 

11/16 
Mark Macerato & David Nadler 
Nearby and vanishing cycles II 

11/30 
David Nadler 
Examples of nearby and vanishing cycles 

12/7 
Mark Macerato 
Nearby and vanishing cycles III 

References
Microlocal sheaf theory
J 
X. Jin, Microlocal sheaf categories and the Jhomomorphism, Preprint available at arXiv:2004.14270, Sep. 2020. 
JT 
X. Jin and D. Treumann, Brane structures in microlocal sheaf theory, Preprint available at arXiv:1704.04291, April 2017. 
KS 
M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften. SpringerVerlag, Berlin, 1994, vol. 292, pp. x+512, With a chapter in French by Christian Houzel, Corrected reprint of the 1990 original, ISBN: 3540518614. 
NS 
D. Nadler and V. Shende, Sheaf quantization in Weinstein symplectic manifolds, Preprint available at arXiv:2007.10154, Feb. 2021. 
S1 
P. Schapira, A short review of microlocal sheaf theory, Preprint available at webusers.imjprg.fr/~pierre.schapira/lectnotes/MuShv.pdf, Jan. 2016. 
S2 
P. Schapira, Microlocal analysis and beyond, Preprint available at arXiv:1701.08955, Jan. 2017. 
Z 
J. Zhang, Quantitative Tamarkin category, Lecture notes available at arXiv:1807.09878, Jul. 2018. 
Derived categories, dg categories, ∞categories, etc.
C1 
D. Clausen, Interlude on D(R), notes for a Course on algebraic de Rham cohomology. Available at sites.google.com/view/algebraicderham. 
C2 
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. 
K 
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. 
MG 
A. MazelGee, An invitation to higher algebra, Book projection available at etale.site/teaching/w21/math128lecturenotes.pdf, 2021. 
HTT 
J. Lurie, Higher topos theory, Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 2009, vol. 170, pp. xviii+925, ISBN: 9780691140490; 0691140499. 
HA 
J. Lurie, Higher algebra, Preprint available at math.ias.edu/~lurie/papers/HA.pdf, Sep. 2017. 
SAG 
J. Lurie, Spectral algebraic geometry, Preprint available at math.ias.edu/~lurie/papers/SAGrootfile.pdf, Feb. 2018. 
T 
B. Toën, Lectures on dgcategories, In: Topics in algebraic and topological Ktheory. Vol. 2008. Lecture Notes in Math. Springer, Berlin, 2011, pp. 243–302. DOI: 10. 1007/9783642157080. Available at perso.math.univtoulouse.fr/btoen/files/2012/04/swisk.pdf 
Y 
A. Yekutieli, Derived categories, Preprint available at arXiv:1610.09640, Apr. 2019. 