Office: Simonyi 203

Some papers of mine (some original, others expository). Most of these papers were based on work supported by the National Science Foundation under Grant No. 0906194. For selected older versions, go here
Elliptic Cohomology III: Tempered Cohomology.
This paper constructs a "tempered" cohomology theory associated to any oriented Pdivisible group over a commutative ring spectrum A, and shows that there is an accompanying theory of local systems which satisfies ambidexterity for pifinite spaces. As an application, we reproduce the character theory of HopkinsKuhnRavenel, Stapleton, and the classical character theory of finite groups. 

Ultracategories.
This paper gives a new proof of Makkai's "strong conceptual completeness" theorem, which asserts that a small pretopos C can be recovered from the category of models Mod(C) together with some additional structure given by the ultraproduct construction. 

Level Structures on Elliptic Curves.
This paper proves an integral analogue of a result of Scholze, which asserts that the moduli stack of elliptic curves at infinite level is "integral perfectoid" (after completing at some prime number p). Updated December 2019 (substantially simplified based on suggestion of O. Gabber). 

Broken Lines and Associative Algebras.
Introduces a moduli stack of "broken lines" and shows that associative algebras can be identified with factorizable sheaves on the moduli stack of broken lines. Joint work with Hiro Tanaka. 

Revisiting the de RhamWitt Complex.
An elementary approach to the construction of the de RhamWitt complex and the proof of its basic properties. Joint work with Bhargav Bhatt and Akhil Mathew. (Updated December 2019; various revisions.) 

Elliptic Cohomology II: Orientations.
Develops a theory of formal groups over commutative ring spectra and discusses various applications, like the construction of elliptic cohomology. Last update: April 2018 (added a pointwise criterion for pdivisibility). 

Spectral Algebraic Geometry (Under Construction).
An unfinished copy of my third book, which is an expanded compilation of several of the papers below (as well as some new material), in a form that I hope is much more userfriendly. Roughly 67% done (so many references are broken). Last update: February 2018. 

Elliptic Cohomology I.
A discussion of abelian varieties in the setting of spectral algebraic geometry. Last update: February 2018 (minor revisions). 

A RiemannHilbert Correspondence in Characteristic p.
Establishes a version of the RiemannHilbert correspondence for ptorsion etale sheaves on an arbitrary F_pscheme. Joint work with Bhargav Bhatt. Last update: November 2017. 

Higher Algebra.
A version of my second book. Last update: September 2017; rewrote section on the associative operad and added material on A_n algebras. 

Weil's Conjecture for Function Fields I.
The first volume of an expanded account of joint work with Dennis Gaitsgory, applying homotopytheoretic ideas to the problem of computing Tamagawa numbers of algebraic groups over function fields. Hopefully much more readable than the earlier account linked below (but defers some important steps to the sequel). Last update: May 2017. (Copyright Notice.) 

Higher Topos Theory.
The latest version of my book on higher category theory. The book has now gone to press, but I will continue to keep an updated copy here (big thanks to Bruce Williams for showing me how to fix the formatting). Last update: April 2017 (reworked discussion of retracts and idempotents, fixing some errors, and added hyperlinks). (Copyright Notice.) 

On Brauer Groups of LubinTate Spectra I.
Joint work with Mike Hopkins, computing the K(n)local Brauer group of Morava Etheory (rough draft). Last update: March 2017. 

Weil's Conjecture for Function Fields.
A second draft of my joint work with Dennis Gaitsgory on the proof of Weil's Tamagawa number conjecture for function fields. The proofs in section 7 have been somewhat simplified and there is a new section 10 which verifies the GrothendieckLefschetz trace formula for Bun_G(X) (so that the paper now contains a complete proof of Weil's conjecture). Last update: December 2014. 

Rotation Invariance in Algebraic KTheory.
This paper discusses the paracyclic Waldhausen construction for stable inftycategories and shows that its failure to descend to a cyclic construction is "measured" by a certain map constructed geometrically from Bott periodicity and the complex Jhomomorphism. Last update: September, 2015 (removed an erroneous remark about the Browder operation). 

Ambidexterity in K(n)Local Stable Homotopy Theory.
Joint with Mike Hopkins. Investigates some surprising duality phenomena in the world of K(n)local homotopy theory. Mostly finished, though it is a bit rough in places. Last update: December 2013. 

Representability Theorems.
A proof of Artin's representability theorem for spectral DeligneMumford stacks. (Essentially the main result of my thesis, but in a somewhat different setting.) This will be the last of the DAG papers for a while. First Draft: March 14, 2012. 
DAGXIV 
Rational and padic Homotopy Theory.
Last update: December 15, 2011. 
DAGXIII 
Proper Morphisms, Completions, and the Grothendieck Existence
Theorem.
An exposition of some foundational material (described in the title) in the setting of spectral algebraic geometry. Last update: November 8, 2011. 
DAGXII 
Descent Theorems.
Some descent theorems for quasicoherent sheaves and quasicoherent stacks in the setting of spectral algebraic geometry. Last update: September 28, 2011. 
DAGXI 
Formal Moduli Problems.
A study of formal moduli problems in the setting of commutative and noncommutative derived algebraic geometry. Contains detailed proofs of the results claimed in my ICM address. Rough draft. Last update: September 1, 2011. 
DAGX 
Closed Immersions.
A study of closed immersions in spectral algebraic geometry, and the operation of gluing along closed immersions. As an application we develop the rudiments of a theory of derived complex analytic spaces. Last update: June 2011. 
DAGIX 
QuasiCoherent Sheaves and Tannaka Duality Theorems.
An exposition of the theory of quasicoherent sheaves in the setting of spectral algebraic geometry. Last update: May 2011. 
DAGVIII 
Spectral Schemes.
Introduces the definition of scheme and DeligneMumford stack in the setting of algebraic geometry over structured ring spectra. Last update: June 2011 (updated version has a new section on fpqc descent). 
DAGVII 
Structured Spaces.
The fifth bit of my PhD thesis. It described a general theory of "spaces" (i.e., inftytopoi) with structure sheaves, like sheaves of commutative rings. Last update: February 2011. 
DAGV 
Survey article on elliptic cohomology.
This is a survey of the theory of elliptic cohomology, with emphasis
on the the insight offered by derived algebraic geometry.
(Updated April 2007). More details will be available later.


Expository article on topological field theories.
This paper gives an informal account of a proof of the BaezDolan
cobordism hypothesis and related matters. (Last update: May 2009)
A more detailed account will appear here eventually.


Moduli Problems for Ring Spectra.
An informal account of the role of E_n algebras in deformation theory. 
Paper. 
(infty,2)Categories and the Goodwillie Calculus I.
This is mostly devoted to some of the technical details of particular models for the theory of (infty,2)categories. The ultimate goal is to give some applications to the Goodwillie calculus, but there's very little of that in this paper. Last update: October 2009. 
GI 
Tannaka Duality for Geometric Stacks.
A preprint. 

On SimplyLaced Lie Algebras and their Minuscule Representations.
My undergraduate thesis. 

Some notes that I once prepared on the theory of Hadamard spaces
(metric spaces of nonpositive curvature).


An exposition of the BorelWeilBott theorem on the cohomology of
holomorphic line bundles over flag varieties.
