# Year in review

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.