We study Kohn-Sham Density Functional Theory (DFT) models describing the electronic structure of two-dimensional materials placed in a three-dimensional environment, encapsulated between two parallel conducting electrodes. In this geometry, the Dirichlet boundary conditions at the electrodes screen the Coulomb interaction, which becomes effectively short-ranged, of Yukawa type. We prove that some nonlinear Kohn-Sham DFT models are well-posed in this setting, both for periodic materials (such as graphene) and for quasi-periodic materials (such as twisted bilayer graphene and other moiré materials for generic incommensurate twist angles).

In this paper, we give a direct proof of determinantal formulas for connected $k$-point functions for hermitian matrix models. We also give a new proof of KP integrability for them. From the viewpoint of KP hierarchy, we further give a new proof of the explicit formula for the corresponding affine coordinates. Furthermore, duality for some hermitian matrix models is proved.

Motivated in part by anomaly cancellation for integral charge spectra in chiral gauge theory, we study the symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem. A symmetric integer solution is one whose entries are paired about a common center $c\in \frac12\mathbb Z$. This symmetry reduces the problem to a sum-of-two-squares equation, $x^2+y^2=u^2+v^2$, in integer variables, subject to the appropriate parity conditions. Thus the problem is governed by representations as sums of two squares. For the full symmetric locus, let $N_{\mathrm{sym}}(H)$ denote the number of nontrivial symmetric integer solutions of height at most $H$, counted with unordered multiset conventions and summed over the admissible centers. Then \begin{align*} N_{\mathrm{sym}}(H) = \frac{4\log 2}{3\pi^2}H^3\log H+O(H^3). \end{align*} The logarithmic enhancement comes from the second moment of the sum-of-two-squares representation function. In particular, the symmetric locus is larger than one would expect from the naive $H^3$ degree-weighted box-counting scale alone. This asymptotic identifies a large arithmetically structured subfamily of the ideal degree-three solution space, and suggests that paired anomaly-free integral charge spectra reflect a fundamental number-theoretic structure.

Public-key cryptosystems eliminate the requirement for pre-shared secret keys by enabling encryption with a publicly disclosed key and decryption with a corresponding private key. In this article we generalize the public-key cryptosystems to ternary algebraic structures, with particular attention to ElGamal as a representative family. We introduce the necessary algebraic background for nonderived ternary structures, including special elements, ternary group rings, and a matrix ternarization procedure that maps binary rings and group rings to antidiagonal symbolic matrices closed under ternary multiplication. Building on these foundations, we formulate a ternary analogue of the ElGamal three-step protocol (key generation, ephemeral encryption, and decryption via querelements) and derive explicit ternary power and querelement formulas that enable correct decryption. Concrete instantiations and numerical examples over a ternary fraction field, a matrix-ternarized finite group ring, and a finite \((6,3)\)-ring (field) validate the construction and illustrate admissible word-length quantization and cycle behaviour of ternary powers. The ternary framework highlights two practical advantages: richer algebraic structure (querelements replace binary inverses) that increases algebraic complexity for attackers, and higher information density (matrix ternarization transfers paired/plaintext vectors). Formal hardness assumptions, optimized parameter choices, and comprehensive security and performance analyses remain necessary future work.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N, d, k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.

The current study presents a three-dimensional linear stability analysis of particle-laden Couette-Poiseuille flow suspended in a Newtonian fluid between two parallel plates, with the lower plate coated by a porous medium. The influence of suspended particles is examined using a two-domain formulation in which particles are confined to the fluid layer and do not penetrate the porous substrate. The particle-laden suspension is modeled using the dusty-gas framework, while the flow within the porous layer is described by the volume-averaged Navier-Stokes (VANS) equations. In particle-laden flows over impermeable walls, particle inertia may either stabilize or destabilize the flow depending on the governing parameters. In contrast, the presence of a porous layer introduces an additional permeability-dependent destabilizing mechanism that fundamentally modifies these classical trends. Consequently, particle loading can reduce the critical Reynolds number at sufficiently high permeability, even in parameter regimes where particles stabilize the corresponding rigid-wall flow. The coupled formulation also introduces additional disturbance branches associated with fluid-particle coupling near the permeable interface. Although these modes remain stable across the parameter space investigated, they modify the eigenspectrum and influence the dominant instability by altering coupling pathways. Furthermore, unlike impermeable-wall Couette-Poiseuille flow, where increasing the Couette component generally stabilizes the flow, the porous-wall configuration exhibits a monotonic decrease in the critical Reynolds number over the range examined. These results demonstrate that porous boundaries can fundamentally alter established stability behavior in particle-laden shear flows through permeability-dependent coupling between the suspension and the porous substrate.

We show that isotropic random walks on the complex projective space provide a canonical and analytically tractable stochastic-geometric framework for the exploration of quantum-state space. The approach combines harmonic analysis on compact rank-one symmetric spaces with stochastic pure-state evolution and yields explicit analytical expressions for transition kernels, fidelity statistics, and geometric observables associated with the Fubini--Study metric. In particular, the framework provides a solvable reference model for isotropic depolarization and Haar equilibration, reproducing Haar-random fidelity statistics and the invariant measure on projective Hilbert space without specifying a microscopic Lindblad generator. In the short-time regime, the stochastic evolution converges to Brownian diffusion generated by the Fubini--Study Laplace--Beltrami operator, while the long-time limit exhibits concentration-of-measure behaviour characteristic of high-dimensional random quantum states. We further derive analytical and asymptotic results for the first-passage-time problem, including closed-form expressions in the Brownian limit for the mean first passage time and the long-time tail of the first-passage-time distribution. For high-fidelity target states, the mean first passage time exhibits a strong dimension-dependent divergence originating from the concentration properties of the Fubini--Study geometry.

Originally motivated by creating first-person computer visualizations within Riemannian manifolds -- the author was led to study deformable-body mechanics, as rigid-body mechanics is not available in a generic Riemannian manifold due to its lack of nontrivial isometry group. Hyperelasticity is a particularly nice sub-category of continuum mechanics in which a deformable, elastic body's behavior is determined by a stored energy density function. This allows problems to be posed variationally, and powerful tools brought to bear on studying and solving them. This article presents numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics in 2-dimensional Riemannian manifolds in which a flat hyperelastic body $B$ is embedded into a region $\Omega$ in a nowhere-flat surface $S$ of revolution $z=z\left(r\right)$ such that $\left|K\left(r\right)\right|$ decreases as $r\to\infty$, where $K$ denotes the Gaussian curvature of $S$. For example, the funnel $z=-r^{-1}$ or the paraboloid $z=\frac{1}{2}r^{2}$. Because $B$ is flat, the body can't achieve a zero-stored-energy configuration, and restorative forces arise in the body to move it toward a region of lower stored energy -- meaning, toward a flatter configuration. With the addition of a gravitational potential $U\left(r\right)=z\left(r\right)$ on $S$, forces act on the body to pull it toward $r=0$. If the body has sufficient stiffness and remains within the region $\Omega$, then the body has an equilibrium configuration in which the body's deformation-response forces perfectly cancel the gravitational forces. Such a configuration represents a kind of "levitation" phenomenon within this surface. The numerical implementation of this problem will be detailed and the resulting numerical solutions and various consequences discussed.

We prove a singularity theorem in which the classical focusing hypothesis of Hawking--Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition $\mathsf{TCD}^e_p(0,N)$, yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.

A model for predicting the residual stress gradient in a thin film segment is developed on the basis of the theory of dislocation pile-ups. The initial shear stress within the film is relaxed via the formation of a pile-up of screw dislocations against the impenetrable film-substrate interface. Plastic strain is related to the dislocation density, leading to a fundamental equation, which links the residual stress to this density. The distribution of dislocations within the pile-up for an arbitrary, non-uniform residual stress profile is derived analytically by applying the force balance condition. This results in a singular integro-differential equation for the residual stress profile. The equation is solved numerically by a collocation method for various initial stress distributions: constant, linear, parabolic, and exponential functions. The solutions demonstrate that the established residual stress profile strongly depends on the film segment's thickness-to-width ratio and the initial stress distribution. As this ratio increases, stress relaxation becomes more effective away from the film-substrate interface. In all cases, equilibrium requires a pile-up containing dislocations with both positive and negative Burgers vectors. The total number of dislocations and their density distribution vary significantly with the initial stress profile. This model provides a critical step towards more complex models of residual stress formation in constrained material systems, specifically, thin films.

A fundamental concept in linear algebra and its applications to network analysis is the Perron--Frobenius (PF) theorem, which underpins eigenvector-based centrality measures such as eigenvector centrality, PageRank, and hubs and authorities. By invoking the PF theorem, we know for strongly connected networks with positive edge weights that the eigenvector corresponding to the largest eigenvalue of the weight matrix yields a well-defined centrality measure (namely, eigenvector centrality). Traditional formulations of the PF theorem and associated centrality measures assume that networks have real-valued weights. However, many networks in areas such as quantum information, quantum chemistry, electrodynamics, and machine learning have complex-valued edge weights. In this paper, we study generalizations of the PF theorem to complex-valued matrices, establish connections between these generalizations, and propose generalized eigenvector-based centrality measures to analyzing node importances in networks with complex edge weights. We also prove results about the existence of complex-weighted networks that satisfy generalized PF properties and calculate associated centrality measures for several examples, which we draw from application areas such as electron transport, circuit analysis, mathematical chemistry, and communication networks.

We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

In this chapter we review some results describing the high-frequency eigenmodes of the Laplacian on compact manifolds, or Euclidean domains, for which the geodesic flow is chaotic. We focus on the macroscopic distribution of these eigenmodes, which is described by the concept of semiclassical measure. The main result on the question is the Quantum Ergodicity theorem, originally due to Schnirelman. We provide the detailed proof of this theorem, including the adjustments necessary to treat the case of manifolds with boundary. We also discuss the Quantum Unique Ergodicity conjecture, and some progress towards this conjecture for strongly chaotic (Anosov) systems. In particular, we describe the constraints on admissible semiclassical measures, in terms of their Kolmogorov-Sinai entropy, as well as more recent delocalization results.

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schrödinger operator is a Cantor set and has zero weight with respect to all spectral measures.

Origami-inspired metamaterials exploit the interplay between geometry and elasticity to achieve programmable mechanical responses. Yet the origin and tunability of snap-through instabilities in non-rigidly foldable patterns remain poorly understood. Here we show that the Mars tessellation, a degree-4 vertex origami pattern composed of alternating square and rhombic faces, is not rigidly foldable because the folding-speed ratios required for vertex compatibility cannot be propagated consistently across neighboring units. This geometric incompatibility forces the facets to bend during folding, giving rise to a reproducible snap-through discontinuity in the force-displacement curve with a mean force drop of about 92.6 +/- 5.5 %, marking a transition between metastable states. Laser scoring of additional diagonal creases, guided by strain-field simulations, enables continuous tuning of the snap magnitude. These results reveal a general mechanism by which geometric frustration can be harnessed to program multistability in thin-sheet metamaterials.

Given a finite collection of Lagrangian submanifolds $\mathscr L$ in a compact symplectic manifold $X$, we construct a cyclic, filtered, strictly unital curved $A_{\infty}$ category $\mathcal L$ and develop Floer theory of closed-open maps and open-closed maps. Using them, we prove that, whenever the map from the quantum cohomology of $X$ to the Hochschild cohomology of the Fukaya category $\mathcal L$ with objects $\mathscr L$ is injective, the following consequences follow: (1) any other Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by $\mathscr L$, and (2) the Hochschild homology and cohomology of the Fukaya category are isomorphic to quantum cohomology. In the exact case a similar result was obtained in [Ab]. We also provide some applications.

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $\rho$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $\rho$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

We propose a quantum bosonic qubit model on a fcc lattice that realizes the canonical source structure of mimetic dark matter as a defect of a rank-two lattice Gauss law. The standard contribution from general relativity is implemented similarly to previous works in the literature, while the mimetic sector modifies the constraint equations through additional source terms. Different theories such as mimetic dark matter, vector mimetic dark matter, and tensor-vector-scalar models are implemented on the lattice. In all these cases, a generalized Gauss law incorporates an additional Gauss-law (topological) defect depending on the type of generalization, but always fitting into the structure of the defects from the general relativity contribution. The mimetic constraint is treated in its full ADM form, retaining the normal derivative of the scalar field, and the known ghost and gradient instabilities of the minimal continuum mimetic theory are summarized. The lattice construction is therefore presented as a formal realization of the canonical source structure rather than as a complete cosmological model.

Random-matrix statistics are usually imposed at the level of matrix entries or spectral correlations. Here we formulate a complementary inverse problem: can a matrix with prescribed random-matrix moments be generated from a structured set of latent vectors? We introduce a pair-resolved vector ansatz consisting of two vector families, P and Q, construct a complex-symmetric non-Hermitian matrix M = a1P P T + a2QQT. The transpose is intentionally not a conjugate transpose; hence the reconstructed bilinear overlap matrices are not Hermitian Gram matrices once the algebraic parameters become complex. The free parameters of the vectors are fixed by complex algebraic constraints matching diagonal and off-diagonal random-matrix moments, together with a mixed-overlap condition suppressing systematic correlations between the two bilinear sectors. A fast machine-precision solve for N = 8 returns six complex branches. We therefore supplement moment matching with reproducible branch diagnostics: residual error, approximate vector orthogonality, non-Hermiticity, imaginary spectral weight, inverse participation ratio, maximum component weight, and eigenvector conditioning. Optional entanglement and low-weight Pauli-moment diagnostics can be added when N = 2n. This protocol constitutes a finite-dimensional inverse reconstruction of hidden vectorspace representations behind apparent random-matrix behavior. It is static and algebraic: it probes moment-induced delocalization, non-Hermitian branch structure, and complex spectral statistics, but it does not by itself establish dynamical chaos in the sense of sensitive dependence on nearby initial conditions.

Recently, there has been growing interest in simulating time-dependent Hamiltonians using quantum algorithms, driven by diverse applications, such as quantum adiabatic computing. While techniques for simulating time-independent Hamiltonian dynamics are well-established, time-dependent Hamiltonian dynamics is less explored and it is unclear how to systematically organize existing methods and to find new methods. Sambe-Howland's continuous clock elegantly transforms time-dependent Hamiltonian dynamics into time-independent Hamiltonian dynamics, which means that by taking different discretizations, existing methods for time-independent Hamiltonian dynamics can be exploited for time-dependent dynamics. In this work, we systemically investigate how Sambe-Howland's clock can serve as a unifying framework for simulating time-dependent Hamiltonian dynamics. Firstly, we demonstrate the versatility of this approach by showcasing its compatibility with analog quantum computing and digital quantum computing. Secondly, for digital quantum computers, we illustrate how this framework, combined with time-independent methods (e.g., product formulas, multi-product formulas, qDrift, and LCU-Taylor), can facilitate the development of efficient algorithms for simulating time-dependent dynamics. This framework allows us to (a) resolve the problem of finding minimum-gate time-dependent product formulas; (b) establish a unified picture of both Suzuki's and Huyghebaert and De Raedt's approaches; (c) generalize Huyghebaert and De Raedt's first and second-order formula to arbitrary orders; (d) answer an unsolved question in establishing time-dependent multi-product formulas; (e) and recover continuous qDrift on the same footing as time-independent qDrift. Thirdly, we demonstrate the efficacy of our newly developed higher-order Huyghebaert and De Raedt's algorithm through digital adiabatic simulation.

The Coherent Ising Machine (CIM) is a quantum network of optical parametric oscillators (OPOs) intended to find ground states of the Ising model. This is an NP-hard problem, related to several important minimization problems, including the max-cut graph problem. In order to enhance its potential performance, we analyze the coherent coupling strategy for the CIM in a highly quantum regime. To explore this limit, without assuming gaussianity, we employ accurate numerical simulations. Due to the inherent complexity of the system, the maximum network size is limited. While master equation methods can be used, their scalability diminishes rapidly for larger systems. Instead, we use Monte Carlo wave-function methods, which scale as the wave-function dimension, and use large numbers of samples. These simulations involve Hilbert spaces exceeding $10^{7}$ dimensions. To evaluate success probabilities, we use quadrature probabilities. We demonstrate the potential for quantum computational advantage by reducing the time required to reach maximum success probability in a low-dissipation regime enabled by initial quantum superpositions and entanglement. Furthermore, we demonstrate that tailored time-dependent couplings can amplify these quantum effects. Comparisons with classical CIM models give evidence that quantum tunneling effects in this strong coupling limit can overcome trapping in false minima. This can greatly increase success rates, indicating a potential for quantum advantage. Finally, we perform a coherence analysis based on the state purity to examine the role of quantum coherence in CIM performance and to determine how state purity correlates with improved optimization outcomes.

The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of [Devetak, Junge, King, Ruskai, CMP 2006] to multi-index Schatten norms. As an application, we strengthen the additivity statement of [Van Himbeeck and Brown, 2025] thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].

The topological string/spectral theory correspondence establishes a precise, non-perturbative duality between topological strings on local Calabi-Yau threefolds and the spectral theory of quantized mirror curves. While this duality has been rigorously formulated for the closed topological string sector, the open string sector remains less understood. Building on the results of [1-3], we make further progress in this direction by constructing entire, off-shell eigenfunctions for the quantized mirror curve from open topological string partition functions. We focus on local $\mathbb{F}_0$, whose mirror curve corresponds to the Baxter equation of the two-particle, relativistic Toda lattice. We then study the standard and dual four-dimensional limits, where the quantum mirror curve for local $\mathbb{F}_0$ degenerates into the modified Mathieu and McCoy-Tracy-Wu operators, respectively. In these limits, our framework provides a way to construct entire, off-shell eigenfunctions for the difference equations associated with these operators. Furthermore, we find a simple relation between the on-shell eigenfunctions of the modified Mathieu and McCoy-Tracy-Wu operators, leading to a functional relation between the operators themselves.

We propose an approach to Carrollian geometry using principal $\mathbb{R}^\times$-bundles ($\mathbb{R}^\times:= \matthbb{R} \setminus \{0\}$) equipped with a degenerate metric whose kernel is the module of vertical vector fields. The constructions allow for non-trivial bundles, and a large class of Carrollian manifolds can be analysed in this formalism. A key result in this is that once a principal connection has been selected, there is a canonical non-degenerate metric that can be leveraged to circumvent the difficulties associated with a degenerate metric. Within this framework, we examine the Levi-Civita connection and null geodesics.

Global control provides a promising route to implementing multi-qubit gates without individual qubit addressing. This is especially appealing for permutation-invariant (PI) gates, whose symmetry is often broken when they are compiled into individually addressed one- and two-qubit gates. Important examples include SWAP, $\sqrt{i\text{SWAP}}$, and the n-qubit controlled-Z gate, which is equivalent, up to two single-qubit Hadamard gates, to the multi-qubit Toffoli gate. Motivated by this global-control perspective, we show that all PI unitaries on an arbitrary number of qubits can be realized using the Tavis-Cummings (TC) interaction, the multi-qubit version of the Jaynes-Cummings interaction, together with global uniform z and x fields. Here, the $n$ qubits are identically coupled to a single bosonic mode (oscillator), which is initialized in and returned to its vacuum state. A corollary is that all PI states, including GHZ and Dicke states, can be prepared using the same global control. For the case n=2 qubits, which is particularly important in quantum computing, we also find explicit pulse sequences for implementing all PI qubit unitaries that conserve angular momentum in the z direction, using only the TC interaction and global z fields. This includes controlled-Z, SWAP, and $\sqrt{i\text{SWAP}}$.

Developments in Carrollian gravity and holography necessitate the use of singular Carroll vector fields, a feature that cannot be accommodated within standard Carrollian geometry. We introduce Carrollian Lie algebroids as a framework to study such singular Carrollian geometries. In this approach, we define the Carroll distribution as the image of the kernel of the degenerate metric under the anchor map. The Carroll distribution is, in general, a singular Stefan--Sussmann distribution that will fluctuate between rank-1 and rank-0, and so captures the notion of a singular Carroll vector field. As an example, we show that an invariant Carrollian structure on a principal bundle leads to a Carrollian structure on the associated Atiyah algebroid that will, in general, have a singular Carroll distribution. Mixed null-spacelike hypersurfaces, under some simplifying assumptions, also lead to examples of Carrollian Lie algebroids. Furthermore, we establish the existence of compatible connections on Carrollian Lie algebroids, and as a direct consequence, we conclude that Carrollian manifolds can always be equipped with compatible affine connections.

We study quantum and classical systems associated with the quantum corner symmetry group $\mathrm{QCS}=\widetilde{\mathrm{SL}}(2,\mathbb{R})\ltimes \mathrm{H}_3,$ which arises in the context of quantum gravity. We relate quantum observables -- specified by representation-theoretic data -- to their classical counterparts using generalized Perelomov coherent states and the framework of Berezin quantization. This procedure links abstract representation-theoretic input to geometric classical observables, such as area. We conclude by applying the formalism to static, spherically symmetric spacetimes admitting a horizon.

In this paper, we give a unified treatment of the local well-posedness for the wave kinetic equation in almost critical weighted $L^r$ spaces with $2 \leq r \leq \infty.$ The proof builds on ideas from our earlier works \cite{AmLe24, AmLemain25}. Our approach is based solely on kinetic tools, with no appeal to Fourier theory.