About
I was an undergraduate and at MIT and just finished my PhD in mathematics at MIT working with Clark Barwick.
This year I'm an NSF Postdoc and UC President's Postdoc at UC Berkeley under the direction of David Nadler. I'll be at the Institute for Advanced Study during the 2022–2023 academic year, and will return to Berkeley in Fall 2023.
My mathematical interests center around homotopy theory, algebraic Ktheory, and algebraic geometry.
CV
Research/Writing
Collaborators
Araminta Amabel, Clark Barwick, Vladimir Chugunov, Arun Debray, Sanath Devalapurkar, Sergei Fomin, Saul Glasman, Victor Guillemin, Mauro Porta, Ravi Shankar, JeanBaptiste Teyssier
Papers
 From nonabelian basechange to basechange with coefficients (last updated August 2021)
 The homotopyinvariance of constructible sheaves (last updated August 2021)
 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]
 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, 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 September 2019)
 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 October 2021)
 [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
 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.
 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.
Videos
Below are some videos from talks I've given:
 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, Fall 2021
 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
Fall 2021
"Think globally, act microlocally"
Topic: Sheaf theory and microlocal sheaf theory
Meeting time: Tuesdays 23:30pm
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 



Noncharacteristic 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 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.
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: Prince ton 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. 