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.

Inspired by interaction of gravitational waves and dark matters, we study the Bondi–Sachs formalism for Einstein massless scalar field with zero cosmological constant. We provide asymptotic expansions for the Bondi–Sachs metrics as well as the scalar fields and prove the peeling property. We also prove the positivity of the Bondi energy–momentum under condition $c=d=0$ at some retarded time $u_0$. This condition ensures that asymptotically null hypersurfaces near $u=u_0$ are asymptotically null initial data sets of order 2 and the positive energy theorem for null infinity can be applied.

In any 5-dimensional closed Sasakian manifold, we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed Riemann surface S into $N^5$ equipped with an integer multiplicity bounded in $L^\infty $. Moreover this map, equipped with this multiplicity, satisfies a weak version of the Hamiltonian Minimal Equation. We conjecture that any solution to this equation is a smooth branched Legendrian immersion away from isolated Schoen–Wolfson conical singularities with non-zero Maslov class.

Fixing two positive integers d and k, a positive number v, and a positive integer I, we prove that the K-semistable domain of the log pair $(X, \sum _{j=1}^kD_j)$ is a rational polytope lying in the k-dimensional simplex $\overline{\Delta ^k}$, where X is a Fano variety of dimension d, $D_j\sim _{\mathbb {Q}} -K_X$, $(-K_X)^d=v$, $I(K_X+D_j)\sim 0$, and $(X, \sum _{j=1}^kc_jD_j)$ is a K-semistable log Fano pair for some $c_j\in [0,1)\cap {\mathbb {Q}}$. Moreover, we show that there are only finitely many polytopes that may appear as the K-semistable domains for such log pairs. Based on this, we establish a wall crossing theory for K-moduli with multiple boundaries.