### Collective behaviors of the Lohe hermitian sphere model with inertia

We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength.

### The Poincare lemma for codifferential, anticoexact forms, and applications to physics

The linear homotopy theory for codifferential operator on Riemannian manifolds is developed in analogy to the theory for exterior derivative. A new class of anticoexact forms that exist locally in a star-shaped region is defined. Their application to physics, including vacuum Dirac-K\"{a}hler equation, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation, is presented.

### Equivariant prequantization and the moment map

If {\omega} is a closed G-invariant 2-form and {\mu} a moment map, then we obtain necessary and sufficient conditions for equivariant pre-quantizability that can be computed in terms of the moment map {\mu}. We also compute the obstructions to lift the action of G to a pre-quantization bundle of {\omega}. Our results are valid for any compact and connected Lie group G.

### Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schr\"odinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system's eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication [H. Chomet et al, New J. Phys. 21, 123004 (2019)]. Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

### The forbidden region for random zeros: appearance of quadrature domains

Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zero process of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of those holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

### Lorentz meets Lipschitz

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $\alpha$-H\"older continuous Lorentzian metric admit a $\mathcal{C}^{1,\frac{\alpha}{4}}$-parametrization.

### On quasisymmetric plasma equilibria sustained by small force

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the Magnetohydrostatic (steady three-dimensional incompressible Euler) equations sustained by a small force. The solutions are also `nearly' quasisymmetric. The primary idea is that, given a desired quasisymmetry direction $\xi$, change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$--symmetric solutions of the Magnetohydrostatic equations on that background by solving a generalized Grad-Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.

### High-frequency limits and null dust shell solutions in general relativity

Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions. Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres. Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly. By the results in [J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math. 5(4), 2017], it follows that the sequence of characteristic initial value problems gives rise to a sequence of vacuum spacetimes $(\mathcal M, g_n)$ in a fixed double-null domain $\mathcal M$. Since the existence theorem requires only very low regularity, the sequence of solutions may exhibit both oscillations and concentrations, and the limit need not be vacuum. We prove nonetheless that, after passing to a subsequence, the metrics converge in $C^0$ and weakly in $W^{1,2}$ to a solution of the Einstein-null dust system with two families of (potentially measure-valued) null dust. We show moreover that all sufficiently regular solutions to the Einstein-null dust system (with potentially measure-valued null dust) adapted to a double null coordinate system arise locally as weak limits of solutions to the Einstein vacuum system in the manner described above. As a consequence, we also give the first general local existence and uniqueness result for solutions to the Einstein-null dust system for which the null dusts are only measures. This in particular includes as a special case solutions featuring propagating and interacting shells of null dust.