I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we can model them as topological spaces with an addition and multiplication which satisfy the usual ring axioms up to homotopy (but these homotopies have to satisfy higher axioms, resulting in higher homotopies etc.). As for usual commutative rings, we can take the algebraic K-theory of an $E_{\infty}$-ring $R$ -- actually, the output can be viewed as an $E_{\infty}$-ring $K(R)$ again of which the usual algebraic K-groups $K_i(R)$ are just the homotopy groups.

It is a central tenet of chromatic homotopy theory to classify spaces (or $E_{\infty}$-rings) by their height. Any discrete space (like $\mathbb{Z}$) has height at most $0$, topological K-theory (or the space $BU\times \mathbb{Z}$ representing it) has height $1$; higher heights are bit more subtle to define: Technically speaking, the height of an $E_{\infty}$-ring $R$ is the maximum $n$ such that $L_{K(n)}R$ is nonzero (or, if we view $R$ indeed as a space: the maximum $n$ such that the $n$-th Bousfield Kuhn functor of $R$ is nonzero).

It has long been known that $K(\mathbb{Z})$ has height $1$ and actually it almost agrees with its part of pure height $1$ (i.e. the map $K(\mathbb{Z})_p \to L_{K(1)}K(\mathbb{Z})$ is an equivalence above degree $0$ if I recall correctly; this is closely related to the Quillen-Lichtenbaum conjecture, proven by Voevodsky et al.). Motivated by this and (very few) other examples, (one version of) Rognes's redshift conjecture states:

For any $E_{\infty}$-ring, the height of $K(R)$ is always one higher than that of $R$.

This has been proven this year in the monumental paper The Chromatic Nullstellensatz by Burklund, Schlank and Yuan. I want to make a few more remarks:

• The work of Burklund-Schlank-Yuan is actually at its core about the role which Morava E-theory plays in chromatic stable homotopy theory and proves wonderful results about it. But combined with earlier work of Yuan about the algebraic K-theory of Morava E-theory, it proves the red-shift conjecture in the version above so-to-speak as a by-product.
• There has been other very important work on the redshift-conjecture and its variants recently. For example, Hahn and Wilson had shown before that the spectra $BP\langle n\rangle$ satisfy redshift (which was the first known example valid at all heights). Moreover, they show also a statement which roughly says that $K(BP\langle n\rangle)$ is well-approximated by its $K(n+1)$-localization. Even more recently, Ausoni, Bayindir and Moulinos have shown redshift for the Morava K-theories $K(n)$ themselves, which are only associative; this implies redshift for many more ring spectra which are not $E_{\infty}$. Note also the recent concrete calculations of Angelini-Knoll, Ausoni, Culver, Höning and Rognes in this context, and a new way to do similar calculations discovered in recent work of Hahn, Raksit and Wilson.
• There has been already other very interesting work in homotopy theory this year, among which I want to mention two papers by Burklund: one gives the first sublinear bound on the $p$-exponent of the stable homotopy groups of spheres; the other shows that quotients in the stable homotopy theory have much better multiplicative properties than anybody expected before.

I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we can model them as topological spaces with an addition and multiplication which satisfy the usual ring axioms up to homotopy (but these homotopies have to satisfy higher axioms, resulting in higher homotopies etc.). As for usual commutative rings, we can take the algebraic K-theory of an $E_{\infty}$-ring $R$ -- actually, the output can be viewed as an $E_{\infty}$-ring $K(R)$ again of which the usual algebraic K-groups $K_i(R)$ are just the homotopy groups.

It is a central tenet of chromatic homotopy theory to classify spaces (or $E_{\infty}$-rings) by their height. Any discrete space (like $\mathbb{Z}$) has height at most $0$, topological K-theory (or the space $BU\times \mathbb{Z}$ representing it) has height $1$; higher heights are bit more subtle to define: Technically speaking, the height of an $E_{\infty}$-ring $R$ is the maximum $n$ such that $L_{K(n)}R$ is nonzero (or, if we view $R$ indeed as a space: the maximum $n$ such that the $n$-th Bousfield Kuhn functor of $R$ is nonzero).

It has long been known that $K(\mathbb{Z})$ has height $1$ and actually it almost agrees with its part of pure height $1$ (i.e. the map $K(\mathbb{Z})_p \to L_{K(1)}K(\mathbb{Z})$ is an equivalence above degree $0$ if I recall correctly; this is closely related to the Quillen-Lichtenbaum conjecture, proven by Voevodsky et al.). Motivated by this and (very few) other examples, (one version of) Rognes's redshift conjecture states:

For any $E_{\infty}$-ring, the height of $K(R)$ is always one higher than that of $R$.

This has been proven this year in the monumental paper The Chromatic Nullstellensatz by Burklund, Schlank and Yuan. I want to make a few more remarks:

• The work of Burklund-Schlank-Yuan is actually at its core about the role which Morava E-theory plays in chromatic stable homotopy theory and proves wonderful results about it. But combined with earlier work of Yuan about the algebraic K-theory of Morava E-theory, it proves the red-shift conjecture in the version above so-to-speak as a by-product.
• There has been other very important work on the redshift-conjecture and its variants recently. For example, Hahn and Wilson had shown before that the spectra $BP\langle n\rangle$ satisfy redshift (which was the first known example valid at all heights). Moreover, they show also a statement which roughly says that $K(BP\langle n\rangle)$ is well-approximated by its $K(n+1)$-localization. Note also the recent concrete calculations of Angelini-Knoll, Ausoni, Culver, Höning and Rognes in this context, and a new way to do similar calculations discovered in recent work of Hahn, Raksit and Wilson.
• There has been already other very interesting work in homotopy theory this year, among which I want to mention two papers by Burklund: one gives the first sublinear bound on the $p$-exponent of the stable homotopy groups of spheres; the other shows that quotients in the stable homotopy theory have much better multiplicative properties than anybody expected before.

I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we can model them as topological spaces with an addition and multiplication which satisfy the usual ring axioms up to homotopy (but these homotopies have to satisfy higher axioms, resulting in higher homotopies etc.). As for usual commutative rings, we can take the algebraic K-theory of an $E_{\infty}$-ring $R$ -- actually, the output can be viewed as an $E_{\infty}$-ring $K(R)$ again of which the usual algebraic K-groups $K_i(R)$ are just the homotopy groups.

It is a central tenet of chromatic homotopy theory to classify spaces (or $E_{\infty}$-rings) by their height. Any discrete space (like $\mathbb{Z}$) has height at most $0$, topological K-theory (or the space $BU\times \mathbb{Z}$ representing it) has height $1$; higher heights are bit more subtle to define: Technically speaking, the height of an $E_{\infty}$-ring $R$ is the maximum $n$ such that $L_{K(n)}R$ is nonzero (or, if we view $R$ indeed as a space: the maximum $n$ such that the $n$-th Bousfield Kuhn functor of $R$ is nonzero).

It has long been known that $K(\mathbb{Z})$ has height $1$ and actually it almost agrees with its part of pure height $1$ (i.e. the map $K(\mathbb{Z})_p \to L_{K(1)}K(\mathbb{Z})$ is an equivalence above degree $0$ if I recall correctly; this is closely related to the Quillen-Lichtenbaum conjecture, proven by Voevodsky et al.). Motivated by this and (very few) other examples, (one version of) Rognes's redshift conjecture states:

For any $E_{\infty}$-ring, the height of $K(R)$ is always one higher than that of $R$.

This has been proven this year in the monumental paper The Chromatic Nullstellensatz by Burklund, Schlank and Yuan. I want to make a few more remarks:

• The work of Burklund-Schlank-Yuan is actually at its core about the role which Morava E-theory plays in chromatic stable homotopy theory and proves wonderful results about it. But combined with earlier work of Yuan about the algebraic K-theory of Morava E-theory, it proves the red-shift conjecture in the version above so-to-speak as a by-product.
• There has been other very important work on the redshift-conjecture and its variants recently. For example, Hahn and Wilson had shown before that the spectra $BP\langle n\rangle$ satisfy redshift (which was the first known example valid at all heights). Moreover, they show also a statement which roughly says that $K(BP\langle n\rangle)$ is well-approximated by its $K(n+1)$-localization. Even more recently, Ausoni, Bayindir and Moulinos have shown redshift for the Morava K-theories $K(n)$ themselves, which are only associative; this implies redshift for many more ring spectra which are not $E_{\infty}$. Note also the recent concrete calculations of Angelini-Knoll, Culver, Höning and Rognes in this context, and a new way to do similar calculations discovered in recent work of Hahn, Raksit and Wilson.
• There has been already other very interesting work in homotopy theory this year, among which I want to mention two papers by Burklund: one gives the first sublinear bound on the $p$-exponent of the stable homotopy groups of spheres; the other shows that quotients in the stable homotopy theory have much better multiplicative properties than anybody expected before.

I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we can model them as topological spaces with an addition and multiplication which satisfy the usual ring axioms up to homotopy (but these homotopies have to satisfy higher axioms, resulting in higher homotopies etc.). As for usual commutative rings, we can take the algebraic K-theory of an $E_{\infty}$-ring $R$ -- actually, the output can be viewed as an $E_{\infty}$-ring $K(R)$ again of which the usual algebraic K-groups $K_i(R)$ are just the homotopy groups.

It is a central tenet of chromatic homotopy theory to classify spaces (or $E_{\infty}$-rings) by their height. Any discrete space (like $\mathbb{Z}$) has height at most $0$, topological K-theory (or the space $BU\times \mathbb{Z}$ representing it) has height $1$; higher heights are bit more subtle to define: Technically speaking, the height of an $E_{\infty}$-ring $R$ is the maximum $n$ such that $L_{K(n)}R$ is nonzero (or, if we view $R$ indeed as a space: the maximum $n$ such that the $n$-th Bousfield Kuhn functor of $R$ is nonzero).

It has long been known that $K(\mathbb{Z})$ has height $1$ and actually it almost agrees with its part of pure height $1$ (i.e. the map $K(\mathbb{Z})_p \to L_{K(1)}K(\mathbb{Z})$ is an equivalence above degree $0$ if I recall correctly; this is closely related to the Quillen-Lichtenbaum conjecture, proven by Voevodsky et al.). Motivated by this and (very few) other examples, (one version of) Rognes's redshift conjecture states:

For any $E_{\infty}$-ring, the height of $K(R)$ is always one higher than that of $R$.

This has been proven this year in the monumental paper The Chromatic Nullstellensatz by Burklund, Schlank and Yuan. I want to make a few more remarks:

• The work of Burklund-Schlank-Yuan is actually at its core about the role which Morava E-theory plays in chromatic stable homotopy theory and proves wonderful results about it. But combined with earlier work of Yuan about the algebraic K-theory of Morava E-theory, it proves the red-shift conjecture in the version above so-to-speak as a by-product.
• There has been other very important work on the redshift-conjecture and its variants recently. For example, Hahn and Wilson had shown before that the spectra $BP\langle n\rangle$ satisfy redshift (which was the first known example valid at all heights). Moreover, they show also a statement which roughly says that $K(BP\langle n\rangle)$ is well-approximated by its $K(n+1)$-localization. Even more recently, Ausoni, Bayindir and Moulinos have shown redshift for the Morava K-theories $K(n)$ themselves, which are only associative; this implies redshift for many more ring spectra which are not $E_{\infty}$. Note also the recent concrete calculations of Angelini-Knoll, Ausoni, Culver, Höning and Rognes in this context, and a new way to do similar calculations discovered in recent work of Hahn, Raksit and Wilson.
• There has been already other very interesting work in homotopy theory this year, among which I want to mention two papers by Burklund: one gives the first sublinear bound on the $p$-exponent of the stable homotopy groups of spheres; the other shows that quotients in the stable homotopy theory have much better multiplicative properties than anybody expected before.