$ \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} $

A lot has happened in 2021. For myself, productivity was at an all-time low for much of the year. But, mathematicians at large continued their break-neck speed and a ton of interesting papers were written and posted; see the 49 listed below. Returning to campus in the fall sparked new activity and I have gotten a lot done since then. Much of this work is bound up in a longer project which won’t see the light of day for a bit longer.

My PhD student Joel Stapleton successfully defended his PhD project which was to prove that Weibel’s conjecture on vanishing of K-theory holds for Azumaya algebras; see his paper on the arXiv, which will appear in the Annals of K-Theory.

Here are a few stats about work during the year. I finished one paper with Asher Auel on genus $1$ curves which we had been thinking about for five years. I also wrote 19 recommendation letters of various forms, wrote 2 referee reports (this is very low thanks to my sabbatical), and wrote 21 quick opinions!

I wrote 12 blog posts here and read 40 or so non-math books, mostly novels, of which the best by far were George Eliot’s Middlemarch, James Baldwin’s Go Tell it on the Mountain, Arundhati Roy’s The God of Small Things, and Fredrik Backman’s Anxious People. In terms of nonfiction, I enjoyed Ayad Akhtar’s Homeland Elegies, Tom Murphy’s Energy and Human Ambitions on a Finite Planet, and Michael Pollan’s Omnivore’s Dilemma.

I was excited to see that the Stacks Project and especially Johan de Jong won the Steele Prize for mathematical exposition from the AMS. This is a huge deal and hopefully will encourage other bold, sweeping collaborative works. Next, I would like to see the L-functions and modular forms database LMFDB win something because it is really the best.

Some notable papers, by theme

Surveys

Ravi Vakil’s spectral sequence story book.

Bhatt, Algebraic geometry in mixed characteristic, arXiv:2112.12010. Notes for his ICM 2022 address. Some material on the forthcoming Riemann–Hilbert correspondence with Lurie.

Klinger, Hodge theory, between algebraicity and transcedence, arXiv:2112.13814. Survey on o-minimality and connections to Hodge theory.

Mathew, Some recent advances in topological Hochschild homology, arXiv:2101.00668. A nice survey paper, including some work on computing syntomic complexes of singular rings.

van der Geer, Curves over finite fields and moduli spaces, arXiv:2112.08704. Great survey on rational points on moduli spaces of curves over finite fields.

Gallauer, An introduction to six-functor formalisms, arXiv:2112.10456. Another nice survey paper and a convenient reference.

Wykowski and Schedler, An investigation into Lie algebra representations obtained from regular holonomic D-modules, arXiv:2111.14774. Helpful survey focusing on the basic case of $\mathfrak{sl}_2$.

de Rham cohomology and all that

Addington and Bragg, Hodge numbers are not derived invariant in positive characteristic, arXiv:2106.09949. Proves what the title says for $3$-folds in characteristic $3$. Answers the implicit question left open by my paper with Bragg. See my post on the paper for more details.

Calaque, Campos, and Nuiten, Lie algebroids are curved Lie algebras, arXiv:2103.10728. Model category approach to the $\infty$-categories of objects from the title.

Brantner, Campos, and Nuiten, PD operads and explicit partition Lie algebras, arXiv:2104.03870. More intuition about partition Lie algebras and deformation theory in characteristic $p$. Also some interesting Koszul duality remarks using pro-coherent modules.

Moulinos, Filtered formal groups, Cartier duality, and derived algebraic geometry, arXiv:2101.10262.

Mondal, \(\bG_a^{\#}\)-perf modules de Rham cohomology, arXiv:2101.03146.

Li and Mondal, On endomorphisms of the de Rham cohomology functor, arXiv:2109.04303.

Bhatt and Scholze, Pismatic $F$-crystals and crystalline Galois representations, arXiv:2106.14735. Proves that the two theories in the title are the same.

Fargues and Scholze, Geometrization of the local Langlands correspondence, arXiv:2102.13459.

Le Bras and Vezzani, The de Rham–Fargues–Fontaines cohomology, arXiv:2105.13028.

Kelly, Kremnizer, and Mukherjee, Analytic Hochschild–Kostant–Rosenberg theorem, arXiv:2111.03502. Characteristic zero results using Raksit’s approach.

Hansen and Scholze, Relative perversity, arXiv:2109.06766.

Bhatt and Hansen, The six functors for Zariski-constructible sheaves in rigid geometry, arXiv:2101.09759.

Kubrak and Prikhodko, $p$-adic Hodge theory for Artin stacks, arXiv:2105.05319.

Bhatt and Li, Totaro’s inequality for classifying spaces, arXiv:2107.04111. Gives another proof of the result of Kubrak and Prikhodko.

Colmez and Nizioł, On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem, arXiv:2104.13448.

Colmez and Nizioł, On the cohomology of $p$-adic analytic spaces, II: the $C_{st}$-conjecture, arXiv:2108.12785.

Petrov, Universality of the Galois action on the fundamental group of \(\bP^1\setminus\{0,1,\infty\}\), arXiv:2109.09301.

Morin, Topological Hochschild homology and zeta-values, arXiv:2011.11549. Relates BMS2-style filtrations on Hochschild homology and THH to something I know little about: zeta-values.

Min and Wang, On the Hodge–Tate crystals over $\Oscr_K$, arXiv:2112.10140.

Guo, Crystalline cohomology of rigid analytic spaces, arXiv:2112.14304.

Guo, Prismatic cohomology of rigid analytic spaces over de Rham period ring, arXiv:2112.14746.

Algebraic $K$-theory

Barwick, Glasman, Mathew, and Nikolaus, $K$-theory and polynomial functors, arXiv:2102.00936. See my post for more details.

Sulyma, Floor, ceiling, slopes, and $K$-theory, arXiv:2110.04978. The $K$-theory of truncated polynomial rings, this time by computing the syntomic complexes. Great pics.

McCandless, Curves in $K$-theory and $TR$, arXiv:2102.08281. Great, modern approach to the curves in K-theory perspective on TR. See my post for more details.

Dahlhausen, $K$-theory of admissible Zariski–Riemann spaces, arXiv:2101.04131. Algebraic $K$-theory of Zariski–Riemann spaces behaves an awfully lot like they are regular.

Kerz, Saito, and Tamme, $K$-theory of non-archimedean rings II, arXiv:2103.06711.

Braunling, Hilbert reciprocity using $K$-theory localization, arXiv:2111.11580. More intertwining of class field theory and $K$-theory.

Canoncao, Neeman, and Stellari, Uniqueness of enhancements for derived and geometric categories, arXiv:2101.04404. The most general results yet in this direction.

Elmanto, Kulkarni, and Wendt, $\bA^1$-connected components of classifying spaces and purity for torsors, arXiv:2104.06273. A subject close to my heart, this paper clarifies some things about extending `unramified’ $G$-torsors off of generic points.

Lüders and Morrow, Milnor $K$-theory of $p$-adic rings, arXiv:2101.01092.

Burghardt, The dual motivic Witt cohomology Steenrod algebra, arXiv:2112.03156. Computes the algebra over, for example, quadratically closed fields of characteristic not $2$.

Konovalov, Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture, arXiv:2102.01566. Some new cases of Blanc’s lattice conjecture, largely using assembly techniques.

Burklund and Levy, On the $K$-theory of regular coconnective rings, arXiv:2112.14723. The paper gives some nice results on vanishing of $K$-theory for coconnective ring spectra, whereas most results are for connective ring spectra. (It also corrects a gap in an argument in an example of my negative $K$-theory paper with Gepner and Heller.)

Redshift

Blumberg, Mandell, and Yuan, A version of Waldhausen’s chromatic convergence for $TC$, arXiv:2106.00849.

Blumberg, Mandell, and Yuan, Chromatic convergence for the algebraic K-theory of the sphere spectrum, arXiv:2110.03733.

Moshe and Schlank, Higher semiadditive $K$-theory and redshift, arXiv:2111.10203.

Yuan, Examples of chromatic redshift in algebraic $K$-theory, arXiv:2111.10837.

Real cyclotomic spectra

Quigley and Shah, On the equivalence of two theories of real cyclotomic spectra, arXiv:2112.07462.

Dotto, Moi, and Patchkoria, On the geometric fixed-points of real topological cyclic homology, arXiv:2106.04891.

Spectral sequences

Ariotta, Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause, arXiv:2109.01017. A long-awaited paper on coherent cochain complexes. See my post for more details.

Barthel and Pstrągowski, Morava $K$-theory and filtrations by powers, arXiv:2111.06379.

Belmont and Kong, A Toda bracket convergence theorem for multiplicative spectral sequences, arXiv:2112.08689. Computing Toda brackets via spectral sequences: it works how you might hope.