We study a Kuramoto-Vicsek model of self-propelled particles with periodic boundary conditions subject to a constant angular tilt and a confining potential, and its mean-field (Fokker-Planck) behaviour. In the absence of confinement, the uniform density is stationary and we compute the critical coupling for four normalisation variants of the interaction kernel, showing that the leading instability is always spatially homogeneous. When the confining field is present, the uniform state is no longer stationary. We construct the new steady state perturbatively and apply eigenvalue perturbation theory to derive an explicit formula for the critical coupling as a function of the field strength. The threshold increases quadratically with confinement strength, and the tilt enters through the steady-state correction despite having no effect on the threshold in the absence of confinement. We verify the prediction numerically and derive self-consistency equations for stationary states with general multichromatic potentials.

We present a Palatini-type geometric formulation of classical Cosserat elasticity in which the coframe and rotational connection are treated as independent variational fields. In contrast to conventional metric-based approaches, this formulation makes the underlying geometric structure explicit and separates translational and rotational degrees of freedom at the level of the action. The governing equations are obtained directly as Euler--Lagrange equations and yield the Cosserat force and moment balance laws without imposing compatibility constraints a priori. It is further shown that these balances arise naturally from invariance of the action under spatial translations and rotations via Noether's first theorem, providing a transparent variational interpretation of micropolar mechanics. A metric-free linearization recovers the classical strain and wryness measures and establishes equivalence with standard tensorial formulations under appropriate constitutive assumptions. The proposed framework clarifies the role of the connection field, which remains implicit in classical theories, and provides a unified geometric setting for Cosserat continua. In addition, it establishes the foundation for subsequent extensions in which torsion and curvature represent evolving defect densities and lead to mesoscopic theories of defect mechanics.

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not extremals of any action integral. The existence of a parameter-invariant action principle for autoparallels is a longstanding open problem, which is equivalent to the so-called Finsler metrizability of the connection, i.e., to the fact that these autoparallels can be interpreted as Finsler geodesics. In this article, we address this problem for the class of torsion-free affine connections with vectorial nonmetricity, which includes, as notable subcases, Weyl and Schrödinger connections. For this class, we determine the necessary and sufficient conditions for the existence of a Finsler Lagrangian that metrizes the connection and depends only algebraically on it. In the cases where such a Finsler Lagrangian exists, we construct it explicitly. In particular, we show that a broad class of such connections is Finsler metrizable.

We study long-time dynamics of small even perturbations of the soliton in 1D quadratic Klein-Gordon equation. The soliton possesses both an internal mode and the unstable mode. On a codimension-one manifold of fine-tuned initial data the instability is suppressed and the internal mode decays slowly by transferring energy into the continuum. We show that this decay and the associated nonlinear frequency shift are accurately captured by a cubic resonant approximation, with the damping rate determined by a Fermi golde rule-type coefficient. This provides a quantitative description of irreversible energy transfer from the internal mode to dispersive radiation.

The Lee-Yang property of a given spin model means that its partition function has purely imaginary zeros as a function of an external magnetic field. A similar property is also used in the theory of quantum anharmonic crystals and quantum lattice fields. A number of powerful analytic methods of the mathematical theory of such models employ this property. Its suitable generalization is used in the theory of models of isotropic $D$-dimensional spins (rotors) or $D$-component quantum lattice fields. So far, the (generalized) Lee-Yang property has been established only for two-dimensional isotropic models. In this work, we prove that isotropic spin and field models living on $\mathds{Z}$ have this property for all even $D$.

We propose a non-perturbative description of the moduli spaces encoding p-form generalized Maxwell theories in any dimension, using derived differential geometry. Our approach synthesizes the Batalin--Vilkovisky formalism with differential cohomology. Within this framework we formulate Dirac charge quantization and show how such charge-quantized moduli spaces exhibit abelian duality between generalized Maxwell theories of different types. We also describe the compactification of generalized Maxwell theories along closed Riemannian manifolds by computing the pushforward of the underlying sheaves of cochain complexes that model differential cohomology.

This paper establishes the Quantum Voronovskaya--Damasclin (QVD) Theorem, providing a complete asymptotic characterization of Quantum Neural Network Operators in the approximation of arbitrary quantum channels. The result extends the classical Voronovskaya theorem from scalar approximation to the non-commutative operator framework of quantum information theory. We introduce rigorous quantum analogues of Sobolev and Hölder spaces defined through Fréchet differentiability in the Liouville representation and measured using the completely bounded (diamond) norm. Within this framework, we derive an explicit asymptotic expansion of the approximation error and identify the fundamental mechanisms governing convergence. The expansion separates integer-order differential contributions, fractional corrections associated with limited regularity, and intrinsically non-commutative effects arising from operator algebra structure. We also establish a sharp remainder estimate with explicit dependence on the regularity of the channel and the dimension of the underlying Hilbert space. Several applications demonstrate the scope of the theory. These include a quantum central limit theorem describing the fluctuation regime of quantum neural network operators, an optimal interpolation method based on operator geometric means, and a convergence acceleration procedure inspired by Richardson extrapolation. The results provide a rigorous mathematical foundation for the asymptotic analysis of quantum neural network models and establish a direct connection between classical approximation theory, operator algebras, and quantum information science, with implications for quantum algorithms and quantum machine learning.

Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.

We develop a novel numerical bootstrap for unitary, crossing-symmetric conformal field theories, focusing on moment observables defined as weighted averages over conformal data. Providing a global and coarse-grained probe of the operator spectrum, this framework yields numerically rigorous bounds on the operator distribution using standard semidefinite programming techniques. In the heavy correlator regime, these bounds remain robust and converge rapidly towards analytically-derived power laws. At finite external dimensions, low-lying moments capture corrections to analytic heavy limit results, while reproducing familiar bootstrap solutions such as Ising-model kinks on the boundary of moment space. Most importantly, the moment bootstrap reveals new features in previously unexplored regions of the bootstrap landscape. The lower bounds on moment variables exhibit two continuous families of kinks persisting across $2 < d < 6$, reflecting nontrivial spectral reorganizations connected to underlying operator decoupling phenomena. These results demonstrate that moment variables uncover bootstrap solutions and collective structures that are difficult to access within traditional numerical approaches.

Starting from a three-dimensional model based on the Ciarlet-Geymonat energy, we derive nonlinear shell models within the classical elasticity theory of compressible isotropic materials. The Neo-Hookean term involving the norm of the deformation gradient leads to an energy depending on the first, the second, and the third fundamental forms of the deformed midsurface. The coefficients appearing in the resulting shell models depend on the classical Lamé coefficients of the three-dimensional material, on the thickness of the shell, and on the mean and Gaussian curvatures of the reference configuration. This shows that the behavior of the shell is influenced not only by the elastic coefficients but also by the initial geometry of the three-dimensional thin body. The purely volumetric Ciarlet-Geymonat contribution of the three-dimensional energy leads to two-dimensional energies depending on the mean and Gaussian curvatures of both configurations, namely the undeformed and the deformed midsurfaces. Since a purely asymptotic derivation may lead to nonlinear terms for which the lower semicontinuity of the resulting functionals is not clear, we combine the asymptotic reduction through the thickness with Simpson's quadrature rule applied to the purely volumetric energy terms, ensuring that the lower semicontinuity is inherited from the three-dimensional model. After deriving the model, we establish the well-posedness of the proposed shell energies. More precisely, we prove coercivity and lower semicontinuity property of the resulting functional and show the existence of minimizers in appropriate Sobolev spaces. A key ingredient in the proofs is a polyconvexity concept in the shell theory, together with some results concerning the weak convergence of terms involving the mean curvature of the deformed midsurface.

The overarching goal of this thesis was to develop categorical methods that connect enumerative geometry, as studied in mirror symmetry, with large $N$ gauge theories. In the first part, we established a relation between graph complexes, Calabi-Yau $A_\infty$-categories, and Kontsevich's cocycle construction. The next main result is the construction of a formality $L_\infty$-morphism relating algebraic structures built from a Calabi-Yau category and one of its objects; this morphism depends on a splitting of the non-commutative Hodge this http URL generalizes the approach of categorical enumerative invariants from the closed to the open-closed setting. From a physics perspective, closed categorical enumerative invariants are encoded by the partition function of the associated closed string field theory (SFT). We explain how our open-closed morphism is an ingredient in quantizing the large N open SFT associated to an object of a Calabi-Yau category. In the final part of this thesis, based on an algebraic approach to open and closed backreacted SFT, we propose ideas towards a categorical formulation of 'Twisted Holography' at the level of partition functions, given as input a Calabi-Yau category and one of its objects.

We consider the soliton solutions of a recently proposed coupled Sasa-Satsuma-mKdV equation using the Kadomtsev-Petviashvili reduction method. The system consists of a complex-valued component coupled with a real-valued one. Under zero or nonzero boundary conditions, we derive four distinct classes of soliton solutions: bright-bright, dark-dark, bright-dark, and dark-bright. These solutions are derived from the vector Hirota equation, for which the bright, dark, and bright-dark soliton solutions are provided in the Appendix. We perform asymptotic analysis of soliton collisions for each class of solutions, in which inelastic collisions are observed between bright-bright solitons. In the dark-dark case, we identify soliton profiles similar to the Sasa-Satsuma equation, including double-hole, Mexican hat, and anti-Mexican hat solutions; this study further explores the collisions between these structures and hyperbolic tangent shaped kink solitons. Regarding the bright-dark case, beyond the expected soliton-kink interactions, we report and analyze a notable collision occurring between kink solitons.

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For a cubic Yeoh model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives $u_{xxt}$ of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analyzed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition $c_1 < \abs{v} < c_2$, while tending to the larger material wave speed $c_1$ or $c_2$ for large times.

In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL}}(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy.

Bayes' rule is widely regarded as the canonical prescription for belief updating. We show, however, that one can arbitrarily preserve pre-specified beliefs while appearing to perform Bayesian updates via "prior hacking": engineering a reference prior distribution such that, for a fixed channel and evidence, the update matches a chosen target distribution. We prove that this is generically possible in both classical and quantum settings whenever Bayesian inversions are well-defined (with the Petz recovery map as the quantum analogue to Bayes' rule), and provide constructive algorithms for doing so. We further establish a duality between prior hacking and Schrödinger bridge problems (a key object in statistical physics with applications in generative modelling), yielding in the quantum setting a unique, inference-consistent selection among candidate bridges. This formally establishes the Bayes-like updating that Schrödinger bridges are performing with respect to the process as opposed to the reference prior, both in classical and quantum settings.

We algebraically prove the asymmetry in thermal relaxation in phase space in the entire range from overdamped dynamics to underdamped dynamics. We show that for the same setup as for overdamped dynamics, even in the more general case of phase-space relaxation, i.e., underdamped dynamics, far-from-equilibrium heating is faster than cooling. Upon isolating the relevant relaxational contribution to the entropy production, we find that the asymmetry persist for underdamped dynamics that are linearly driven out of equilibrium. The coupling of positions and velocities emerging in this generalization further underscores, in a striking manner, the intricate dynamics of such thermal relaxation processes that do not pass through local equilibria. Investigating the overdamped limit, our generalized approach reveals, interestingly, that an excess free energy contribution from the velocity degrees of freedom does not trivially vanish in the overdamped limit, but is instead affected by the precise interpretation of temperature quenches in overdamped systems.

We consider damped driven Maxwell-Bloch equations which are finite-dimensional approximation of the damped driven Maxwell-Schrödinger equations. The equations describe a single-mode Maxwell field coupled to a two-level molecule. Our main result is the construction of solutions with single-frequency asymptotics of the Maxwell field in the case of quasiperiodic pumping. The asymptotics hold for solutions with harmonic initial values which are stationary states of averaged equations in the interaction picture. We calculate all harmonic states and analyse their stability. The calculations rely on the Bloch-Feynman gyroscopic representation of von Neumann equation for the density matrix. The asymptotics follow by application of the averaging theory of the Bogolyubov type.

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $\Sigma$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $\Sigma$, as the moduli space of representations of the fundamental groupoid $\Pi_1 (\Sigma)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $\Sigma$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.

In this paper, we develop a Markov decision process (MDP) formulation for the low--temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two--dimensional square lattice. We analyze how an external controller can guide the system to the all--occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action--dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency--based criterion promotes the growth from the boundary centers of the cluster, an energy--based reward function favours the growth at the corners of the cluster.

The well-posedness for the Dbar problem associated with the AKNS spectral problem is considered. In general, the relevant Dbar equation with normalization condition is quivalent to an integral equation, where the kernel involves exponents $\mathrm{e}^{\pm2ikx}$ with physical variable $x$ as a parameter. We develop a decomposition technique to control the convergence of the integral by defining a new integral operator $RT_{\mathbb{C}}(k;x)$. The small norm condition of the operator is obtained to show that there exists a unique solution for the Dbar problem. Moreover, the Dbar dressing method is extended to construct the AKNS spectral problem and the potential construction is presented via the Dbar data. Prior estimates are given to show that the map from the Dbar data to the AKNS potential is Lipschitz continuous.

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics -- the canonical model of continuous-variable decoherence in optical attenuation channels, squeezed-light interferometric sensing, and superconducting bosonic architectures -- complete positivity, the requirement that the dynamics remain physical even for systems entangled with an ancilla, creates an exact phase boundary at which the minimum reverse cost vanishes, fixes the reverse-noise budget on both sides, and makes pure nonclassical targets dynamically singular. The minimum reverse cost vanishes exactly at a critical squeezing-to-thermal ratio and is strictly positive away from it, with a sharp asymmetry: below the boundary, standard reverse prescriptions such as the fixed-diffusion Bayes reverse remain feasible at mild cost; above it, these prescriptions become infeasible, the covariance-aligned generator remains CP-feasible and uniquely optimal, and the cost can be severe. The optimal reverse noise is locked to the state's own fluctuation geometry and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact reversal of a pure quantum state is dynamically unattainable. These results establish an exact law of quantum reversibility in the canonical pure-loss setting and provide a sharp benchmark for broader theories of quantum reverse diffusion.

The paper describes a numerical method for solving acoustic multibody scattering problems in two and three dimensions. The idea is to compute a highly accurate approximation to the scattering operator for each body through a local computation, and then use these scattering matrices to form a global linear system. The resulting coefficient matrix is relatively well-conditioned, even for problems involving a very large number of scatterers. The linear system is amenable to iterative solvers, and can readily be accelerated via fast algorithms for the matrix-vector multiplication such as the fast multipole method. The key point of the work is that the local scattering matrices can be constructed using potentially ill-conditioned techniques such as the method of fundamental solutions (MFS), while still maintaining scalability and numerical stability of the global solver. The resulting algorithm is simple, as the MFS is far simpler to implement than alternative techniques based on discretizing boundary integral equations using Nyström or Galerkin.

Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the $C^1$ topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.

This paper presents a Hamiltonian reduction procedure for field theories over affine principal bundles introducing a canonical identification to describe the reduced multisymplectic space without the introduction of a connection. The main goal is to provide a Hamiltonian analogue of the Lagrangian reduction theory developed in M. Castrillón López, P. M. Chacón, and P. L. García. J. Geom. Mech., 5(4):399-414, 2013. The core of this work lies in the derivation of this canonical identification, the reduced Hamilton-Cartan equations, and a reduced covariant bracket that describes the dynamics. Finally, this theoretical framework is illustrated with a fundamental example: molecular strands.

We study the neutral scalar perturbations of asymptotically flat extremal Reissner-Nordström black holes via the quantum geometry of $\mathcal{N}=2$ $\mathrm{SU(2)}$ gauge theory with $N_f=2$ flavors. The master equation, given by a double confluent Heun equation, is mapped to the quantum Seiberg-Witten curve in the Nekrasov-Shatashvili limit. We compute the quasinormal mode frequencies non-perturbatively using the quantization condition derived from the Nekrasov-Shatashvili free energy. Our analytical results accurately reproduce the numerical benchmarks for massless fields, and capture the quasi-resonance behavior of massive probes at the strict extremal limit.

We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on $N$ strands. Allowing for arbitrary rank (dependent on $N$) and non-abelian representations, and letting $N \to \infty$, this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb-Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

Completely positive transformations play an important role in the description of state changes in quantum mechanics, including the time evolution of open quantum systems. One useful tool to describe them is the so-called Choi isomorphism, which maps completely positive transformations to positive semi-definite matrices. Accordingly, there are numerous proposals to generalize the Choi isomorphism. In the present paper, we show that the 1976 paper of Gorini, Kossakowski and Sudarshan (GKS) already holds the key for a further generalization and study the resulting GKS isomorphism. As an application, we compute the GKS matrix of the time evolution of a general open quantum system up to second order in time.

This paper presents a study of nonlinear superpositions of Riemann wave solutions admitted by quasilinear hyperbolic first-order systems of partial differential equations. In particular, we focus on the Euler system and non-elastic wave superpositions that cannot be decomposed into pairwise independent interactions of waves. A crucial tool for this analysis is the property of quasi-rectifiability of the families of vector fields determined by this system. It imposes certain conditions to be satisfied by the commutators of these vector fields. They enable us to find a parametrization of the region of superpositions of Riemann waves which leads to a simplification of the initial system of equations. In order to identify non-elastic superpositions we prove that a class of Lie modules associated with them can be uniquely transformed into a real Lie algebra through an angle-preserving transformation. We are then able to select a particular basis of vector fields associated with a given module which ensures the property of quasi-rectifiability. That, in turn, allows us to construct the reduced form of the Euler system for which a non-elastic superposition of two Riemann waves is then derived analytically. A study of the geometry of the manifold of non-elastic wave superpositions in terms of deformations of submanifolds corresponding to the Lie algebras is performed. Finally, we adapt the described approach to the general form of a hydrodynamic-type system i.e., to arbitrary Lie modules of vector fields associated with such a system, providing the criteria for their quasi-rectifiability. A geometric interpretation of non-elastic wave superpositions in this system is given.

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie algebras is necessary in order to derive irreducible representations of the Lie algebra and subsequently of the symmetry group. In this paper, we give a precise definition of the concept and prove step by step an important result $\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}$. This result is used to determine the irreducible representations of the proper Lorentz group and thus the physical objects admissible when this symmetry is present. It is shown that finite representations of the proper Lorentz group are characterized by pairs of half-integers $(j_1,j_2)$, which determine unambiguously the physical object associated to the given representation. For example, the representation $(0,0)$ of dimension $1$ is called the scalar representation, it corresponds to the Higgs field, and $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of dimension $4$ is called the Dirac spinor representation, it corresponds to matter particle called fermions. This means that the mathematical group structure determines the material content of the universe following this algebraic structure.

We present some review material relating to the topic of optimal asymptotic expansions of correlation functions and associated observables for $\beta$ ensembles in random matrix theory. We also give an introduction to a related line of study that we are presently undertaking.

Let $D$ be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in $D$ with a hard wall along $\eta = \partial D$, \[Z_{n}(D) =\frac 1{n!}\int_{D^n}\prod_{1\le k < \ell \le n}|z_k-z_\ell|^{2} \prod_{k=1}^n d^2z_k.\] We are interested in how the geometry of $\eta$ is reflected in the large $n$ behavior of $Z_n(D)$. We prove that $\eta$ is a Weil-Petersson quasicircle if and only if \[ \lim_{n \to \infty} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} = -\frac{1}{12}I^L(\eta), \] where $I^L$ is the Loewner energy, $\mathbb{D}$ is the unit disc, and $\log Z_n(\mathbb{D}) = \log \pi^n/n!$. We next consider piecewise analytic $\eta$ with $m$ corners of interior opening angles $\pi \alpha_p, p=1,\ldots, m$. Our main result is the asymptotic formula \[ \lim_{n\to\infty}\frac 1{\log n} \log \frac{Z_n(D)}{Z_n(\mathbb{D})} =-\frac 16\sum_{p=1}^m \left(\alpha_p+\frac 1{\alpha_p}-2 \right) \] which is consistent with physics predictions. The starting point of our analysis is an exact expression for $\log Z_{n}(D)$ in terms of a Fredholm determinant involving the truncated Grunsky operator for $D$. The proof of the main result is based on careful asymptotic analysis of the Grunsky coefficients. As further applications of our method we also study the Loewner energy and the related Fekete-Pommerenke energy, a quantity appearing in the analysis of Fekete points, for equipotentials approximating the boundary of a domain with corners. We formulate several conjectures and open problems.

We consider a statistical model for symmetric matrix factorization with additive Gaussian noise in the high-dimensional regime, where the rank of the signal matrix to infer $M$ scales with its size $N$ as $M=\mathrm{o}(\sqrt{\ln N})$. Allowing for an $N$-dependent rank offers new challenges and requires new methods. Working in the Bayes-optimal setting, we show that whenever the signal has i.i.d. entries, the limiting mutual information between signal and data is given by a variational formula involving a rank-one replica symmetric potential. In other words, from the information-theoretic perspective, the case of a (slowly) growing rank is the same as when $M=1$ (namely, the standard spiked Wigner model). The proof is primarily based on a novel multiscale cavity method allowing for growing rank along with some information-theoretic identities on worst noise for the vector Gaussian channel. We believe that the cavity method developed here will play a role in the analysis of a broader class of inference and spin models where the degrees of freedom are large arrays instead of vectors.

The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. This discreteness hypothesis implies that given two different points in space, there is no continuous curve (a world line) joining them. Consequently, this hypothesis is not compatible with the theory of relativity. We propose $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})^{3}$ as a model of a space-time. For simplicity, we work out our models using $\mathbb{R}\times(\mathbb{R}\times\mathcal{X})$ as the configuration space. Quantum mechanics (QM), in the sense of Dirac-von Neumann, on the Hilbert space $L^{2}(\mathbb{R}\times\mathcal{X})$ is a non-local theory: the Hamiltonians are non-local operators, and thus, spooky action at a distance is allowed. The paradigm asserting that the universe is non-locally real implies that the proposed version of QM admits realism. This version of QM can be specialized to standard QM by using Hamiltonians acting on wavefunctions supported on the region $\mathbb{R}\times\mathbb{R}$. We apply the developed formalism to the measurement problem. We propose a new mechanism for the collapse of the wavefunction. The mechanism resembles the one proposed by Ghirardi, Ramini, and Weber, but there are significant differences. The most important feature is that the Schrödinger equation describes the dynamics at all times, even at the moment of measurement. We also discuss a model for the two-slit experiment, where bright and dark states of light (proposed recently) naturally occur.

For spin and fermionic systems in any spatial dimension, we establish that the superpolynomial decay behavior of mutual information and conditional mutual information is a universal property of gapped pure- and mixed-state phases; i.e., all systems in such a phase possess this property if one system in this phase possesses this property. We further demonstrate that the (conditional) mutual information indeed decays superpolynomially in a large class of phases, including chiral phases. As a by-product, we sharpen the notion of mixed-state phases.

A basic postulate of modern compositional approaches to generalised physical theories is the generalised Born rule, in which probabilities are postulated to be computable from the composition of states and effects. In this paper we consider whether this postulate, and the strength of the identification between scalars and probabilities, can be argued from basic principles. To this end, we first consider the most naive possible process- theoretic interpretation of textbook quantum theory, in which physical processes (unitaries) along with states and effects (kets and bras) and a probability function from states and effects satisfying just some basic compatibility axioms are identified. We then show that any process theory equipped with such structure is equivalent to an alternative process theory in which the generalised Born rule holds. We proceed to consider introduction of noise into any such theory, and observe that the result of doing so is a strengthening of the identification between scalars and probabilities; from bare monoid homomorphisms to semiring isomorphisms.

We derive an intensity doubling feature of critical Brownian loop-soups on the cable-graphs of ${\mathbb Z}^d$ for $d \ge 7$ that can be described as follows: In the box $[-N, N]^d$ (and with a probability that goes to $1$ as $N$ goes to infinity), the set of all clusters of Brownian loops that do contain proper self-avoiding cycles of diameter comparable to $N$ can be decomposed into two identically distributed families: (a) The collection of clusters that do contain a large Brownian loop from the loop-soup (and therefore do automatically contain such a large cycle) (b) The collection of clusters that contain no macroscopic loop from the loop-soup (more specifically, no loop of diameter greater than $N^{\beta}$ when $\beta > 4/ (d-2)$ is fixed) but nevertheless contain a large cycle. In particular, due to the fact that these two families are asymptotically identically distributed, large cycles formed in case (b) by chains of small Brownian loops (i.e., all of diameter much smaller than $N$) will look like large Brownian loops themselves, and form a second independent "ghost" critical loop-soup in the scaling limit. Reformulated in terms of the Gaussian free field on such cable-graphs, this shows that large cycles in the collection of its sign clusters will converge in the scaling limit to a Brownian loop-soup with twice the usual critical intensity. This result had been conjectured by the first author in arXiv:2209.07901 [math.PR] ; our proof builds heavily on the second author's switching property for such loop-soups from arXiv:2502.06754 [math.PR] .

We describe the loop corrections to supercharges in supersymmetric quantum field theories using the holomorphic twist formalism. We begin by reviewing the relation between supercharge corrections and the "twice-generalized" Konishi anomaly, which corrects the semi-chiral ring. In the holomorphic twist, these corrections appear as BRST anomalies and are computed using the higher operations of an underlying $L_\infty$ conformal algebra. We then apply this formalism to obtain the complete one-loop corrections to the supercharge of four-dimensional Lagrangian supersymmetric gauge theories, including $\mathcal{N}=4$ SYM, where it admits a remarkably compact expression in terms of superfields.

Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every $t>0$, but the optimum diverges as $2/t$: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.

We determine the phase diagram of the Anderson tight-binding model on random regular graphs with Gaussian disorder and sufficiently large degree. In particular, we prove that if the degree is fixed and the number of vertices goes to infinity, the spectrum asymptotically consists of a finite delocalized interval surrounded by two unbounded localized components. Our argument uses a recent description of the spectrum of the tight-binding model on the Bethe lattice (Aggarwal--Lopatto, 2025). By viewing the Bethe lattice as the local limit of a random regular graph, and establishing suitable concentration, eigenvalue-counting, and resolvent estimates, we transfer this characterization of the spectrum of the limiting model to the finite-volume setting.