We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ''horizontal'' polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

We study the cohomology of a quasimap space that categorifies the constant term of the geometric Eisenstein series for the mirabolic parabolic subgroup of $GL$ over the function field $\mathbb{F}_q(C)$ of a smooth projective curve $C$. This cohomology carries a natural action of an algebra of correspondences whose commutative subalgebra is the ring of regular functions on the Coulomb branch, which here is the $A_{n}$-surface singularity. A choice of rank-one local system on $C$ induces an action of the étale fundamental group on the Coulomb branch; the scheme-theoretic fixed locus carries a natural vector bundle. Our main result identifies the cohomology of the quasimap space with the local cohomology of this vector bundle, for a generic range of parameters.

Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten-Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, cosymplectic, and contact geometries. Finally, we characterize the conditions under which fat bundles and almost cosymplectic structures of order $p$ determine a Jacobi bracket.

Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While $3$-design protocols achieve optimal worst-case performance, the minimal number of measurement bases required for such optimality has remained open. Here we prove that $Θ(d^2)$ measurement bases are both necessary and sufficient for worst-case optimal shadow estimation and construct an explicit basis family. In stark contrast, any state $2$-design already suffices for average-case optimality: the mean squared shadow norm of normalized observables is bounded by a universal constant, and we prove strong concentration for Haar-random states, yielding constant sample complexity for generic pure-state fidelity estimation. Easily implementable $2$-designs -- from mutually unbiased bases, cyclic measurements, or shallow $\mathcal{O}(\log n)$-depth circuits -- enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires $Θ(d^2)$ bases, whereas average-case performance requires only $Θ(d)$ bases, with broad implications for quantum information theory and near-term experiments.

Viscous fingering instabilities during fluid displacement in porous media can compromise the efficiency of applications such as enhanced oil recovery, CO2 sequestration, and groundwater remediation. While extensive research exists on linear stability analysis for fully immiscible and fully miscible displacements, the intermediate case of partially miscible flow with limited mass transfer between phases remains largely unexplored. This study extends linear stability analysis to a two-phase, two-component system that accounts for gravity effects, fractional flow, capillary forces, mechanical dispersion, and interphase mass transfer, focusing on the case where a partially miscible gaseous fluid displaces a liquid. We formulate an eigenvalue problem to characterize instability growth rates and cutoff wavenumbers. The resulting ordinary differential equations have discontinuous coefficients at the transition from two-phase to pure-liquid flow, resulting in discontinuous eigenfunction derivatives. We derive jump conditions for the derivatives at this transition, and solve the eigenvalue problem using the matched initial value problem method. Results demonstrate that mass transfer has a pre-dominantly stabilizing effect by reducing viscosity contrast and altering shock properties at the displacement front. This stabilizing influence is particularly pronounced for high viscosity contrasts and dampens gravity-induced instability in upward displacements. Mass transfer most significantly affects the perturbation growth rate, while its effect on the cutoff wavenumber is less pronounced. We identify a critical value for the dimensionless longitudinal dispersion coefficient where both growth rate and cutoff wavenumber are maximized, suggesting complex interactions between capillary forces and mechanical dispersion.

Efficient and provably accurate partitioned methods for fluid-poroelastic structure interaction remain challenging because explicit treatment of the Stokes-Biot interface coupling condition can compromise stability. In this work, we develop and analyze a fully discrete, second-order, explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains. The method combines BDF2 time stepping with second-order Adams-Bashforth extrapolation of interface data through a Robin reformulation, yielding a partitioned algorithm in which the Stokes and Biot subproblems are solved independently and in parallel at each time step. The main analytical contribution is a rigorous stability and error analysis for this second-order explicit coupling strategy. Using BDF2 energy identities, a sharp decomposition of the extrapolated interface terms, and discrete trace estimates, we prove a closed stability bound under a parabolic CFL condition. We then derive an a priori error estimate through a projection-based framework using a Fortin projection for the fluid variables and Ritz-type projections for the poroelastic variables. The analysis identifies consistency defects from BDF2 time discretization, Adams-Bashforth interface extrapolation, and the projected kinematic relation. It shows that the total errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are bounded by C times the sum of the kth power of the mesh size and the square of the time step, for k from 1 to 3, in bulk energy norms. Numerical experiments with manufactured solutions confirm second-order temporal convergence and optimal-order spatial convergence. We also include a moving-domain example with Navier-Stokes fluid flow, demonstrating applicability beyond the fixed-domain Stokes-Biot setting analyzed.

We reconstruct unitary quantum dynamics from a minimal axiomatic foundation built on Hilbert-space observables and isospectral evolution. The only dynamical assumption is that physical time evolution is a continuous one-parameter flow of Hermitian observables that preserves their spectra, i.e. the possible outcomes of measurement. We show that this assumption is already sufficient to force the Lax form of quantum dynamics. The Heisenberg equation, the time-dependent and time-independent Schrödinger equations, conservation laws, and good quantum numbers then follow as theorems rather than postulates. In this formulation, Lax pair theory supplies the missing dynamical bridge between the measurement structure of a Hilbert space and standard quantum evolution: the Hamiltonian is not assumed, but emerges as the generator required for an isospectral observable flow.

Projective representations arise naturally in physics and representation theory, and determining whether they can be linearized has been a fundamental problem. In this work, we study the analogous problem for quantum cellular automata (QCA) representations, which incorporate locality constraints imposed by a metric space $X$. Over an arbitrary field $\mathbb{F}$, we develop an obstruction theory for the linearization of QCA representations, using the algebraic $K$-theory spectrum of QCA constructed in previous work of the authors. The resulting obstructions are governed by the homotopy type of the QCA spaces, from which we extract universal obstruction classes to linearization. In the complex algebraic and unitary case, we also fully compute the homotopy types of the QCA spaces over a point, a line, and a plane.

We present a complete classification of concave kite central configurations in the planar 4-body problem with three equal masses. There are two different types of central configurations when the fourth mass lies inside or outside the triangle formed by the other three. Using a rigorous computer-assisted analytical method and a fixed coordinate system, we show that the central configurations in each case form a one-parameter family and obtain a complete classification of these configurations. In addition, we rigorously show the existence and types of the bifurcation points in the reduced space. We also provide two numerical global bifurcation pictures in the entire planar 4-body configuration space as the mass ratio varies from $0$ to $+\infty$, including symmetric and asymmetric concave central configurations with three equal masses.

In this work we consider an Aztec diamond model split into two unequal regions which are asymptotically fixed in size. Each region is weighted with a distinct two-periodic weighting. We refer to this model as the t-split two-periodic Aztec diamond, to signify its difference from the previous work title Split Two-Periodic Aztec Diamond, where the model was split into two equal regions. We derive an integral expression for the correlation kernel of the model and give a partial description of the scaling limit behavior, along with a conjecture for the remainder. We refer to the larger and smaller sides of the model as the dominant and non-dominant sides, and to the location of the weight change as the interface. The dominant side exhibits a limit shape that depends only on its own weighting and is identical to that of the two-periodic Aztec diamond, while the non-dominant side appears to have a novel limit shape that depends on both weightings and the location of the interface. Lastly, we consider the complete limit shape in the case where the dominant side two-periodic parameter goes to 0.

The Bures--Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three--sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton--DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function $Ψ=Φ^2$ satisfies the linear forced heat equation \begin{equation*} D_tΨ=Ψ_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures--Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature $6/(1-4t)$ and extinction time $T=1/4$. Thus the metric remains inside the monotone cone for all $t<T$ and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume--normalized flow, for which the Bures--Helstrom metric is a fixed point. Its linearization is the shifted round--sphere Laplacian $Δ_{\mathbb S^3}+3$, with spectrum \begin{equation*} σ_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.

This work explores the effects of both the standard quantum $q$-deformation and the non-standard $h$-deformation of the Hopf algebra $\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))$ on multi-qubit systems. By constructing the states of a Hilbert space of $N$ qubits through the Clebsch-Gordan coefficients associated with the deformed algebras, we show that these states naturally coincide with the eigenstates of the Hamiltonian of the $q$- and $h$-deformed Kittel-Shore models. We compare the resulting deformed states with those typically targeted in quantum information experiments, providing a bridge between algebraic constructions and experimentally relevant quantum resources. Fidelities with respect to the undeformed states are computed to establish how the quantum correlations are affected, both for few-qubit systems (including Dicke and non-Dicke states), and in the macroscopic limit ($N \to \infty$) through closed-form formulas derived for arbitrary Dicke states. The results reveal different behaviors between the two deformations. The $q$-deformation smoothly modifies the states and maintains a residual overlap with the original configurations, while the $h$-deformation rapidly makes the states orthogonal to their undeformed counterparts. Both models demand a standard $N^{-1}$ rescaling to preserve fidelity stability in the macroscopic limit.

We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, \(-V\). Specifically, we analyze the random Schrödinger operator defined by \(H = -Δ+\ve \sum_{n} ω_n χ_n - V\), where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.

We introduce a Dirac structure on the tangent bundle of a configuration manifold, called a \textit{Lagrange--Dirac structure}, which is naturally induced by the Lagrangian two-form associated with a (possibly degenerate) Lagrangian and a constraint distribution. This structure provides a unified geometric framework for Lagrange--Dirac dynamical systems, encompassing nonholonomic, degenerate Lagrangian, and symmetric systems. In the hyperregular case, the system recovers a first-order formulation of the Lagrange--d'Alembert equations. Although nonholonomic dynamics does not preserve the Lagrangian two-form, we show that the underlying Lagrange--Dirac structure is preserved up to gauge transformations, yielding a natural gauge covariance property. We also formulate an intrinsic variational principle directly on the tangent bundle, referred to as the \textit{Lagrange--d'Alembert--Dirac principle}, which recovers Hamilton's principle in the unconstrained case and the Lagrange--d'Alembert principle in the hyperregular constrained case. Furthermore, we develop a reduction theory for systems with Lie group symmetry, deriving a reduced Lagrange--Dirac structure over the Lie algebra that yields the Euler--Poincaré--Dirac equations and a corresponding reduced variational principle. Finally, we illustrate the theory through examples including charged particles, electric circuits, and systems with Lagrangians linear in velocity, and present an infinite-dimensional extension to ideal fluids that naturally incorporates the incompressibility constraint and recovers the Euler equations.

We show that a general nonlinearly perturbed Maxwell theory of electromagnetism possesses three universal fundamental properties: (i) A finite-energy electric point charge. (ii) Exclusion of finite-energy magnetic monopoles and dually charged dyons. (iii) Arbitrary smallness of the effective radius of a point electric charge and the associated local undetectedness of the electric charge and energy. In particular, this last property offers a classical explanation for the invisibility of the electron, as a point electric charge, in accordance with the smallness of its effective radius. This nonlinear theory of electromagnetism has the feature that it minimally perturbs the Maxwell theory with a nonlinearity profile that is as general as possible such that the three universal properties stated above are all maintained.

We argue that celestial OPEs must be supplemented by shadow-basis operators. Although the shadow transform does not introduce new bulk degrees of freedom, it provides a distinct primary state in the boundary celestial theory. From OPE consistency, we show that the ordinary celestial OPE does not close on Mellin-basis exchanges alone. Rather, the same exchanged bulk particle must also appear through its shadow-basis representative. This leads to a shadow-completed OPE, with the shadow OPE coefficient fixed by the ordinary collinear coefficient through the universal shadow factor. We discuss the corresponding boundary Hilbert-space interpretation, extend this argument to gluons and gravitons, and verify the shadow exchange directly in tree-level regular celestial amplitudes, including a scalar $2\rightarrow n$ analysis and an explicit five-point example.

We reconsider the "fortuity'' mechanism in the D1D5 CFT focusing on the K3 symmetric orbifold. Going beyond the counting of BPS states, we investigate perturbatively how the explicit form of the BPS cohomologies is modified by the twist-two deformations. We calculate the action of the supercharges in the sector $(h,j)=(1,0)$ for different values of the central charge and derive explicit expressions for the primary states. Equipped with this information, we compare some protected three-point couplings in the free and the gravity regime. We show that agreement between the two descriptions imposes non-trivial constraints on the identification of monotone and fortuitous states. In particular, we argue that the map relating theories with different values of the central charge must and can be defined so as to commute with the supercharges that define the cochain complex. We then study the three-point correlators between the fortuitous and monotone states identified in our analysis to assess whether the two sectors are dynamically decoupled. We find an explicit example of a non-vanishing coupling between two monotone and a fortuitous state, providing evidence that the two sectors are dynamically connected.

The quantum wormhole length in sine-dilaton gravity has been shown to equal the Krylov spread complexity in the double-scaled SYK model. In the infinite temperature limit, we compute the five-loop semiclassical expansion of DSSYK complexity by singular perturbation of the operator Liouville-type equations of motion, extending the existing one-loop results. The same method is applied to evaluate the Krylov variance and third-order cumulant, related to the connected two- and three-point functions of the length operator at coincident points. The small- and large-time behaviour of these observables is also studied. In particular, for the large-time slope of the wormhole linear growth, we provide a conjecture for the resummation of the perturbative series, and discuss non-perturbative corrections revealed by numerical data.

In this work, we investigate the canonical quantization of topological solitons appearing in braneworld scenarios. In particular, we focus on Brane-Skyrmions, topological field configurations analogous to standard Skyrmions, which emerge as solutions of the Dirac-Nambu-Goto action supplemented by an induced curvature term. By quantizing the (iso)spin collective coordinates of the Brane-Skyrmion, we obtain a Hamiltonian that we solve perturbatively via an expansion in powers of $J^2$, in contrast to the standard Skyrme model. Furthermore, we implement a Physics-Informed Neural Network (PINN) to determine the soliton profile that minimizes the energy, consistently incorporating the backreaction from the quantized spin degrees of freedom. We conclude with a discussion of the potential applications of this framework to the description of hadronic spectra. Our results highlight both the theoretical potential of brane-defect models and the growing role of neural network methods in theoretical physics.

We study the ultraviolet (UV) limit of finite-volume expectation values of vertex operators in the sine-Gordon model using the kink nonlinear integral equation (NLIE) description of the conformal limit. By analysing the integrable formulation of vacuum expectation values in the small-volume regime, we conjecture an explicit analytic expression for the leading asymptotic term in the small-volume expansion, formulated in terms of kink functions. This establishes a direct connection between the integrable finite-volume description and the expected conformal asymptotics determined by the 3-point functions of the underlying conformal field theory (CFT). The proposed formula is tested against the analytic expression known from complex Liouville conformal field theory using high-precision numerics, showing agreement to at least 19 significant digits.

In the study of duality cascades in three-dimensional gauge theories without FI parameters, an important role is played by a fundamental domain whose vertices correspond to brane configurations with vanishing relative ranks. Through the Fermi gas formalism, such brane configurations are known to be represented by factorized quantum curves. In this paper, we show that this factorized description extends naturally to quantum curves associated with del Pezzo geometries possessing exceptional Weyl-group symmetries in the presence of FI parameters. We find that the vertices of the corresponding fundamental domains, identified with Voronoi polytopes of exceptional root lattices, are realized as factorized quantum curves built from canonical operators interpreted as 5-branes dressed with FI parameters. This provides a physical realization of the vertices of the Voronoi polytopes as ``extremal'' brane configurations.

We present the construction of the generating function of single-centered black hole index in general $\mathbb{Z}_N$ CHL models. This is done by subtracting from the index of quarter BPS dyons, described by a meromorphic Siegel modular form, the generating function for the index of two-centered black holes. We use black hole bound state metamorphosis in CHL models for the construction of the generating function of two-centered black hole index. We prove the convergence of the generating function for the cases $N=2,3$.

We study Euclidean M2-brane instantons in Freund-Rubin backgrounds $\mathrm{AdS}_{4}\times \mathrm{Y}_7$. For a seven-dimensional weak $G_{2}$ manifold $\mathrm{Y}_7$, we show that the BPS condition for an M2-brane wrapping a three-cycle $Σ\subset \mathrm{Y}_7$ is equivalent to the associativity condition with respect to the nearly parallel $G_{2}$-structure. When $\mathrm{Y}_7$ is Sasaki-Einstein, we identify a special class of BPS M2-branes that preserve both real internal Killing spinors and correspond to invariant three-dimensional submanifolds inheriting a Sasakian structure. We analyse the quadratic fluctuations around BPS M2-brane instantons in these backgrounds. For the special class of M2-branes in Sasaki-Einstein manifolds, the fluctuation problem reduces to transversely elliptic complexes, and the one-loop partition function can be expressed in terms of the corresponding equivariant indices. We then apply the index formula for the one-loop partition function to invariant M2-branes in $S^{7}/\mathbb{Z}_{k}$, recovering the known result for the $S^3/\mathbb{Z}_k$ instantons and discussing more general invariant BPS cycles. As a further application, we consider M2-brane instantons with $S^3$-quotient worldvolumes in the $(p,q)$-model geometry.

We present an algorithm that constructs Calabi-Yau threefold orientifolds and their $F$-theory uplifts to elliptically-fibered Calabi-Yau fourfolds, embedded in toric varieties at codimension one and two respectively. The resulting Calabi-Yau fourfolds arise from triangulations of $6d$ reflexive polytopes -- which our method constructs from orientifold data -- and are smooth away from isolated terminal singularities. For many of our fourfolds, the construction of the mirror manifold is immediate, enabling the computation of fourfold periods, and thus the seven-brane superpotential. We present multiple examples that demonstrate these capabilities. Our algorithms work with $\mathtt{CYTools}$ and are available through a GitHub repository.

Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real-world tasks. In this article, we investigate the key features and surprises of deep learning from a physics-informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.

We revisit the Coleman--Weinberg monopole problem introduced by Kiselev, where radiative symmetry breaking makes the broken vacuum metastable. We construct the associated static monopole--critical-bubble configuration in the full coupled radial Higgs--gauge system and show that it is a saddle of the static energy functional. The metastable monopole and monopole--critical-bubble branches are characterized by their profiles, energies, and radial Hessian spectra. The monopole--bubble solution carries a negative radial mode, while the metastable monopole remains locally stable until its lowest radial Hessian eigenvalue approaches zero. The resulting branch structure gives a direct static picture of how Coleman--Weinberg monopoles lose metastability, with critical rescaled scalar mass parameter \(μ_c=0.064352(1)\).

We introduce Vistas, a tool for visualizing high-energy particle physics collisions simulated by the Pythia Monte-Carlo event generator. Vistas utilizes the browser-based event display framework Phoenix to show distinct computational stages of a high-energy collision event simulation: the hard process, parton shower, hadronization, and particle decays. Particles produced from each of these stages are represented as lines in an interactive three-dimensional graph structure, where each line is along the direction of its particle's three-momentum vector. The event can be rotated, translated and zoomed, and details for each particle can be accessed by selecting the relevant particle line. Additionally, particle lines from all stages of the simulation can be toggled on and off and can be filtered by particle-level kinematic selection requirements. This interactive environment provides an intuitive interpretation of Pythia simulation output, including detailed features such as color flow, beam remnants, and multiple parton interactions, making it a useful tool in physics education settings, from outreach activities to graduate particle-physics courses.

This work investigates the electric, thermal, and thermoelectric responses of a rotating pion gas of finite transverse radius in the presence of a background magnetic field, with the rotation axis aligned with the magnetic field. We explicitly calculate the parameter limits for $π^+$ condensation and restrict our working regime safely outside these boundaries, ensuring well-behaved transport coefficients. Notably, the system exhibits a condensation asymmetry, with $π^-$ remaining uncondensed at the parameters that induce $π^+$ condensation. Using the Boltzmann Transport Equation under the Relaxation Time Approximation, we calculate the longitudinal electrical conductivity, thermal conductivity, and the Seebeck coefficient. Our results reveal a competing interplay between the magnetic field and rotation, highlighting the substantial impact of rotation on the medium's transport properties: while the magnetic field suppresses the transport coefficients in a static medium, rotation, acting as an effective chemical potential, introduces an energy shift that favors their increase. Beyond an angular velocity, this rotational enhancement overpowers the magnetic suppression, leading to an increase in the transport coefficients with increasing magnetic field. Finally, we analyze the relative significance of charge and heat transport through the Lorenz number, providing further insight into the transport characteristics of the rotating magnetized pion medium.

This review summarizes the experimental evidence for the hypothetical X17 particle, examines the theoretical frameworks in which it can be accommodated, and discusses its potential implications for the Standard Model and couplings to known particles. Future experimental prospects are also highlighted.