Research/Writing

Collaborators

Araminta Amabel, Clark Barwick, Vladimir Chugunov, Arun Debray, Sanath Devalapurkar, Sergei Fomin, Saul Glasman, Victor Guillemin, Mauro Porta, Ravi Shankar, Jean-Baptiste Teyssier

Papers

  1. From nonabelian basechange to basechange with coefficients (last updated August 2021)
  2. The homotopy-invariance of constructible sheaves (last updated August 2021)
  3. The James and Hilton–Milnor Splittings, & the metastable EHP sequence, with Sanath Devalapurkar (last updated September 2021)
    • To appear in Documenta Mathematica
    • arXiv:1912.04130
    • 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.
  4. Pyknotic objects, I. Basic Notions, with Clark Barwick (last updated April 2019)
  5. On coherent topoi & coherent 1-localic ∞-topoi (last updated July 2019)
    • 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.
  6. Exodromy for stacks, with Clark Barwick (last updated January 2019)
  7. Extended étale homotopy groups from profinite Galois categories (last updated January 2019)
  8. On the homotopy theory of stratified spaces (last updated September 2019)
  9. Exodromy, with Clark Barwick & Saul Glasman (last updated August 2020)
  10. 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 October 2021)
    • 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. Viva Fukaya! (last updated August 2021)
  2. 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: XY, 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.
  3. 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 LGG × 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.
  4. 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.
  5. 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.
  6. Differential cohomology theories as sheaves on manifolds
  7. The Segal–Sugawara Construction
  8. 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.
  9. Nonabelian Poincaré Duality (last updated March 2019)
  10. Ambidexterity §4 (last updated November 2018)
  11. Notes on Étale Cohomology (last updated October 2017)
    • Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.
  12. An Introduction to Goodwillie Calculus (last updated September 2017)
    • Notes on the basic setup of Goodwillie calculus in the ∞-categorical setting.
  13. 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.
  14. On the K-theory of Finite Fields (last updated September 2016)
  15. 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.

Videos

Below are some videos from talks I've given:

  1. Stratified étale homotopy theory (May 20, 2020)
  2. The Homotopy Theory of Stratified Spaces (April 28, 2020)
  3. Intro to Constructible Sheaves and Exit-Paths (April 24, 2020)
  4. Pyknotic Spaces, Spectra, etc. (March 10, 2020)
  5. Pyknotic Sets 2 (March 4, 2020)

Seminars

Below are some seminars that I have helped organize:

  1. Nadler's Geometric Representation Theory Seminar, Fall 2021
  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
Fall 2021

"Think globally, act microlocally"

Topic: Sheaf theory and microlocal sheaf theory
Meeting time: Tuesdays 2-3:30pm
Room: #732
Course control number: 15391(14)

Tentative schedule

Background

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
John Nolan Operations on sheaves:
Grothendieck's six functors
Mark Macerato Nearby and vanishing cycles

Microlocal perspective

Date Speaker Symplectic & contact geometry
of cotangent bundles
Singular support & involutivity
Non-characteristic propagation
Perverse sheaves

Microlocal sheaves

Date Speaker Definitions, basics
μhom
Microlocal cuttoffs
Antimicrolocalization:
from microsheaves back to sheaves

Beyond cotangent bundles

Date Speaker Weinstein manifolds
Homotopical structures
Invariance of microlocal sheaves

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.

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: Prince- ton 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.