I want to mention five directions where in the last years significant progress has been made in chromatic homotopy theory. This is of course not exclusive!
**Unstable chromatic homotopy theory**
Among the greatest functors in existence is the Bousfield-Kuhn functor $\Phi_n\colon \mathcal{S} \to \mathrm{Sp}$ from spaces to spectra. The composite $\Phi_n\Omega^{\infty}$ agrees with $K(n)$-localization (or $T(n)$-localization, depending on the setup). Thus the $K(n)$-localization of a spectrum just depends on its infinite loop space as a space! But its rather hard to compute with the Bousfield--Kuhn functor. Much work had been done by Mahowald, Davis, Bousfield,... at height 1, but methods for higher heights were lacking. It was a breakthrough when Behrens and Rezk found a way to express it in some cases using topological Andre-Quillen homology.
- Behrens, Rezk: [The Bousfield-Kuhn functor and Topological Andre-Quillen cohomology][1]
- Behrens, Rezk: [Spectral algebra models of unstable v_n-periodic homotopy theory][2]
A little later, Heuts brought Lie algebras into the game and actually provided a model of ``$v_n$-periodic unstable homotopy'':
- Heuts: [Lie algebras and vn-periodic spaces][3]
- Heuts: [Lie algebra models for unstable homotopy theory][4]
**The chromatic splitting conjecture**
The chromatic splitting conjecture is about how we can rebuild the sphere from its $K(n)$-localizations. A strong form of this conjecture was found to admit a counterexample at height 2, prime 2 by Beaudry in 2015. A little later, she found out with Goerss and Henn what the picture actually is at height 2, prime 2.
- Beaudry, Goerss, Henn: [Chromatic splitting for the K(2)-local sphere at p=2][5]
**Higher real K-theories**
A crucial ingredient in understanding the $K(n)$-local spheres is to understand higher real K-theories, i.e. spectra of the form $E_n^{hG}$ for $G$ a finite subgroup of the Morava stabilizer group. Classically this was mostly feasible for $n=1$ and $n=2$ and a little beyond. A breakthrough was done by Hahn and Shi, who were able to do the computation for all $n$ if $G=C_2$. Their technique was to construct a $C_2$-equivariant map $MU_{\mathbb{R}} \to E_n$.
- Hahn, Shi: [Real Orientations of Lubin-Tate Spectra][6]
This technique allowed also to construct $C_{2^n}$-equivariant maps into $E_n$ from norms of $MU_{\mathbb{R}}$ (in full generality, this is due to a recent preprint by Beaudry, Hill, Shi, Zeng) and these norms are computationally partially understood due to the Kervaire-invariant paper and subsequent work. In particular, the homotopy groups of $E_4^{hC_4}$ were computed by Hill, Shi, Wang and Xu.
**Chromatic homotopy theory at big primes**
It is known that chromatic homotopy theory drastically simplifies at "big primes". But exactly how algebraic is it then? A first answer was attempted in the nineties in a brilliant preprint by Franke, which contained a significant gap though. Recently this was very convincingly solved in two quite different ways by Barthel--Schlank--Stapleton and Pstragowski:
- Barthel, Schlank, Stapleton: [Chromatic homotopy theory is asymptotically algebraic][7]
- Pstragowski: [Chromatic homotopy is algebraic when p>n^2+n+1][8]
**The Balmer spectrum for stable equivariant homotopy**
One of the most fundamental theorems in chromatic homotopy theory is the thick subcategory theorem, which classifies all thick subcategories of finite spectra (according to prime and height). For a time analogous questions for equivariant spectra remained open, but for abelian groups the situation is now completely known and for non-abelian groups there is significant partial information. The papers are:
- Balmer, Sanders: [The spectrum of the equivariant stable homotopy category of a finite group][9]
- Barthel, Hausmann, Naumann, Nikolaus, Noel, Stapleton: [The Balmer spectrum of the equivariant homotopy category of a finite abelian group][10]
- Barthel, Greenlees, Hausmann: [On the Balmer spectrum for compact Lie groups][11]
More could be mentioned. For example, Hausmann's recent breakthrough about equivariant $MU$ and equivariant formal group laws, but his paper has less of a chromatic feel (though it is certainly related). Or the work on nilpotence, beginning with the May conjecture paper by Mathew--Naumann--Noel and extended by Hahn and others -- the May conjecture paper is already from 2014 though... And one could and should mention the series Elliptic Cohomology I - III by Lurie - but much of this was already announced in 2007.
[1]: https://arxiv.org/abs/1712.03045
[2]: https://arxiv.org/abs/1703.02186
[3]: https://arxiv.org/abs/1803.06325
[4]: https://arxiv.org/abs/1907.13055
[5]: https://arxiv.org/abs/1712.08182
[6]: https://arxiv.org/abs/1707.03413
[7]: https://arxiv.org/abs/1711.00844
[8]: https://arxiv.org/abs/1810.12250
[9]: https://arxiv.org/abs/1508.03969
[10]: https://arxiv.org/abs/1709.04828
[11]: https://arxiv.org/abs/1810.04698