In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multinatural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main application. We settle the question on the height of higher associative structures on the mod $p^k$ Moore spectrum $\mathrm{M}_p(k)$ at odd primes. More precisely, for any odd prime $p$, we show that $\mathrm{M}_p(k)$ admits a Thomified $\mathbb{A}_n$-structure if and only if $n \lt p^k$. We also prove a weaker result for $p=2$.

We show that $C^\infty$-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho\mathfrak{w} \in H^s \times H^s \times H^{s'}$, $2\lt s'\lt s$. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., $\mathfrak{w}=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for $\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$. We therefore, conjecture that the optimal result should be $(v,\varrho,\mathfrak{w} \in H^s\times H^s\times H^{s'}$, $s>2$, $s' > \frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustical spacetime.

We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollár's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.

We define the Chow $t$-structure on the $\infty$-category of motivic spectra $\mathcal{SH}(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $\mathcal{SH}(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $\mathcal{SH}(k)^{\mathrm{cell},c\heartsuit}$ as the category of even graded $\mathrm{MU}_{2*}\mathrm{MU}$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum $\mathbbm{1}_{c=0}$ is algebraic.

Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: to integral results; to all base fields other than just $\mathbb{C}$ to the entire $\infty$-category of motivic spectra $\mathcal{SH}(k)$, rather than a subcategory containing only certain cellular objects.

We also discuss a strategy for computing motivic stable homotopy groups of ($p$-completed) spheres over an arbitrary base field $k$ using the Postnikov--Whitehead tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.