We answer a question raised by Kerzman in 1971. More precisely, we show that the canonical solution of the ${\bar{\partial }}$-equation satisfies the $L^p$ estimate on the polydisc for $p \in [1, \infty ]$. Moreover, the $L^p$ estimates for $p \in [1, \infty ]$ of ${\bar{\partial }}$ can also be obtained on the product of bounded $C^2$ planar domains by an observation based on the method developed in [Dong et al. arXiv:2006.14484].

In this paper, we study the ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds), proposed by the third author. We show the ACC conjecture holds for singularities admitting $\epsilon $-plt blow-ups. In particular, this gives the ACC for mlds for exceptional singularities. The key ingredients in the proofs of our main results are the Birkar–Borisov–Alexeev–Borisov theorem, proved by Birkar, the boundedness of complements conjecture for arbitrary DCC coefficients, proposed by the third author and proved in this paper, and the existence of uniform $\mathbb {R}$-complementary rational polytopes.

We extend the results of [Commun. Partial. Differ. Equ. 44(12), 1431–1465 (2019)] by the third and fourth author globally in time. More precisely, we prove uniform-in-N Strichartz estimates for the solutions $\phi $, $\Lambda $ and $\Gamma $ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta }v(N^{\beta }(x-y))$ for $\beta <1$. The potential v satisfies some technical conditions, but is not small. The initial conditions have finite energy and the “pair correlation” part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda $ develop dynamically in time. The estimates are expected to improve the Fock space bounds of [Ann. Henri Poincaré 23(2), 615–673 (2021)] by the first and fifth author. This will be addressed in a subsequent paper.

We study the $\mathbb {F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $\mathbb {C}$-motivic and classical stable homotopy groups.

We prove the $C^{1,1}$-regularity for stationary $C^{1,\alpha }$ ($\alpha \in (0,1)$) solutions to the multiple membrane problem. This regularity estimate was essentially used in our recent work on Yau’s four minimal spheres conjecture [arXiv:2305.08755].

We initiate an approach to simultaneously treat numerators and denominators of Green’s functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $ (H_{\lambda ,\alpha ,\theta }u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos 2\pi (\theta +n\alpha )u(n)$ be the almost Mathieu operator on $\ell ^2({\mathbb {Z}})$, where $\lambda , \alpha , \theta \in {\mathbb {R}}$. Let $\begin{aligned} \beta (\alpha )=\limsup _{k\rightarrow \infty }-\frac{\ln \Vert k\alpha \Vert _{{\mathbb {R}}/{\mathbb {Z}}}}{|k|}. \end{aligned}$ We prove that for any $\theta $ with $2\theta \in \alpha {\mathbb {Z}}+{\mathbb {Z}}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{2\beta (\alpha )}$. This confirms a conjecture of Avila and Jitomirskaya (Ann Math (2) 170(1):303–342, 2009) and a particular case of the second spectral transition line conjecture of Jitomirskaya (XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, pp. 373–382, 1995).