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, 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, Jean-Baptiste Teyssier, Sebastian Wolf

Almost Papers

  1. Fully faithful functors and pushouts of ∞-categories, with Maxime Ramzi and Jan Steinebrunner
  2. 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

  1. Exodromy beyond conicality, with Mauro Porta and Jean-Baptiste 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.
  2. Spectral weight filtrations, with Piotr Pstrągowski (last updated September 2023)
  3. Nonabelian basechange theorems & étale homotopy theory, with Tim Holzschuh and Sebastian Wolf (last updated April 2023)
  4. The fundamental fiber sequence in étale homotopy theory, with Tim Holzschuh and Sebastian Wolf (last updated December 2022)
  5. From nonabelian basechange to basechange with coefficients (last updated September 2022)
  6. The homotopy-invariance of constructible sheaves, with Mauro Porta and Jean-Baptiste Teyssier (last updated August 2022)
  7. 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, non-computational proof of the metastable EHP sequence in an ∞-topos that essentially only makes use of the Blakers–Massey Theorem.
  8. Pyknotic objects, I. Basic Notions, with Clark Barwick (last updated April 2019)
  9. On coherent topoi & coherent 1-localic ∞-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 1-localic ∞-topoi (in Lurie's sense) are equivalent. We also collect some examples of coherent geometric morphisms coming from algebraic geometry.
  10. Exodromy for stacks, with Clark Barwick (last updated January 2019)
  11. Extended étale homotopy groups from profinite Galois categories (last updated January 2019)
  12. On the homotopy theory of stratified spaces (last updated April 2022)
  13. Exodromy, with Clark Barwick & Saul Glasman (last updated August 2020)
  14. Stability analysis of non-Newtonian rimming flow, with Vladimir Chugunov, Sergei Fomin, & Ravi Shankar (February 2016)

Books

  1. 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, 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.
  2. Differential forms, with Victor Guillemin

Reports

  1. On the homotopy theory of stratified spaces

Notes

  1. Profinite completions of products (last updated March 2024)
  2. 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.
  3. 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.
  4. Viva Fukaya! (last updated January 2022)
  5. 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 n-connected. We show that in an ∞-topos, cotensoring with a finite space with cells in dimensions ≤m decreases connectedness of morphisms by m.
  6. Overview of algebraic K-theory (last updated February 2021)
  7. 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 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.
  8. 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.
  9. 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.
  10. Differential cohomology theories as sheaves on manifolds
  11. The Segal–Sugawara Construction
  12. 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.
  13. Nonabelian Poincaré Duality (last updated March 2019)
  14. Ambidexterity §4 (last updated November 2018)
  15. Notes on Étale Cohomology (last updated October 2017)
    • Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.
  16. An Introduction to Goodwillie Calculus (last updated September 2017)
    • Notes on the basic setup of Goodwillie calculus in the ∞-categorical setting.
  17. 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.
  18. On the K-theory of Finite Fields (last updated September 2016)
  19. Lifting Enhanced Factorization Systems to Functor 2-categories (last updated July 2016)
    • arXiv:1604.06812
    • A note on a technical result about defining enhanced factorization systems on functor 2-categories.

Slides

Some slides from talks I've given:

  1. Exodromy in topology & applications (last updated March 2024)
    • For a talk at the Lloyd Roeling conference. 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.
  2. Topological quantum field theories & the cobordism hypothesis in low dimensions (last updated March 2022)

Videos

Some videos from talks I've given:

  1. 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
  2. Exit-path categories in geometry and topology (September 22, 2022)
  3. Galois-theoretic reconstruction of schemes and exodromy (July 7, 2022)
  4. Stratified étale homotopy theory (May 20, 2020)
  5. The Homotopy Theory of Stratified Spaces (April 28, 2020)
  6. Intro to Constructible Sheaves and Exit-Paths (April 24, 2020)
  7. Pyknotic Spaces, Spectra, etc. (March 10, 2020)
  8. Pyknotic Sets 2 (March 4, 2020)

Seminars

Below are some seminars that I have helped organize:

  1. Nadler's Geometric Representation Theory Seminar, 2021–2022
  2. Condensed Thursday Seminar, Fall 2020
  3. Juvitop seminar on the Cobordism Hypothesis, Fall 2020
  4. Reading Group on Part I of An Inclusive Academy: Achieving Diversity and Excellence by Abigail Stewart and Virginia Valian, Summer 2020
  5. Juvitop seminar on Differential Cohomology, Fall 2019
  6. Miniature seminar on Factorization Homology, Spring 2019
  7. Juvitop seminar on Ambidexterity, Fall 2018
  8. 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.

  1. During Fall 2020, Araminta Amabel, Sarah Greer, and I started the Grad-Undergrad Math Mentoring Iniative (GUMMI) at MIT
  2. Since Fall 2018, I have been the coordinator for PRIMES Circle at MIT
  3. January 2017 & January 2018, I was a mentor for the MIT Math Department's Directed Reading Program
  4. Fall 2017, I was the organizer for the MIT Topology Seminar
  5. Summer 2017, I was a mentor for MIT's √Mathroots program
  6. 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 J-homomorphism, 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. 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.
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.imj-prg.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. Mazel-Gee, An invitation to higher algebra, Book projection available at etale.site/teaching/w21/math-128-lecture-notes.pdf, 2021.
HTT 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.
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/SAG-rootfile.pdf, Feb. 2018.
T 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
Y A. Yekutieli, Derived categories, Preprint available at arXiv:1610.09640, Apr. 2019.