In this paper we give a geometric description of the Jacobi equations associated to a first-order Lagrangian field theory using a prolongation of the Lagrangian $L$ on a $k$-cosymplectic formulation. Moreover, using an appropriate modification of the prolonged Lagrangian, we obtain a variational formulation of field theories with dissipation.

In a seminal paper published in 2000 Kenyon developed a method to study the height function of the planar dimer model via discrete complex analysis tools. The core of this method is a set of identities representing height correlations through the inverse Kasteleyn operator. Scaling limits of these identities (if exist) produce a set of correlation functions written in terms of a Dirac Green's kernel with unknown boundary conditions. It was proven in [Chelkak, Laslier, Russkikh, 23] that, under natural assumptions, these correlations always define a Gaussian free field in a simply connected domain. This was generalized to doubly connected domains in the recent work [Chelkak, Deiman, 25], where the field is shown to be a sum of Gaussian free field and a discrete Gaussian component. We generalize this result further to arbitrary bordered Riemann surfaces.

Existence of the eigenvalues of the discrete-time quantum walks is deeply related to localization. Also, for the study of open quantum systems, non-Hermitian systems have attracted much attention. As mathematical models for such systems, non-unitary quantum walks with the chiral symmetry are essential for the study of the topological insulator. In this paper, we give the whole picture of the eigenvalues of a non-unitary one-dimensional two-state quantum walks with one defect and the chiral symmetry.

We recast quantum entanglement as a cohomological obstruction to reconstructing a global quantum state from locally compatible information. We address this by considering presheaf cohomologies of states and entanglement witnesses. Sheafification erases the global-from-local signature while leaving within-patch multipartite structure, captured by local entanglement groups introduced here. For smooth parameter families, the obstruction admits a differential-geometric representative obtained by pairing an appropriate witness field with the curvature of a natural unitary connection on the associated bundle of amplitudes. We also introduce a Quantum Entanglement Index (QEI) as an index-theoretic invariant of entangled states and explain its behavior. Finally, we outline a theoretical physics approach to probe these ideas in quantum many-body systems and suggest a possible entanglement-induced correction as an experimental target.

We have constructed the mean-field trivial solution of the $\varphi^4$ theory $O(N)$ model in four dimensions in two previous papers using the flow equations of the renormalization group. Here we establish a relation between the trivial solutions we constructed and perturbation theory. We show that if an UV-cutoff is maintained, we can define a renormalized coupling constant $g$ and obtain the perturbative solutions of the mean-field flow equations at each order in perturbation theory. We prove the local Borel-summability of the renormalized mean-field perturbation theory in the presence of an UV cutoff and show that it is asymptotic to the non-perturbative solution.

We implement numerical techniques to simulate D-random feuilletages, candidates for higher-dimensional random geometries introduced in L. Lionni and J.-F. Marckert, Math. Phys. Anal. Geom. 24 (2021) 39. Using finite-size scaling techniques, our approach allows to give a numerical estimation of the Hausdorff dimension $d_H$ of these feuilletages. The results obtained are compatible with the formal result known for the Brownian map, which corresponds to the D=2 random feuilletage. For the D=3 case, our numerical study finds a good agreement with the conjectured value $d_H=8$.

A consistent implementation of the gravitational constraints in de Sitter space requires gauging the full $SO(1,d)$ isometry group. In this paper, we develop a framework that enables the gauging of the full de Sitter isometry while consistently incorporating multiple observers on arbitrary geodesics. We achieve this by introducing the concept of \textit{covariant observer}, whose geodesic is a dynamical entity that transforms under the isometry group. Upon quantization, the geodesic becomes a fluctuating degree of freedom, providing a quantum reference frame for $SO(1,d)$. Inspired by the timelike tube theorem, we propose that the algebra of observables is generated by all degrees of freedom within the fluctuating static patch, including the quantum fields modes and other observers. The gauge-invariant subalgebra of observables is an averaged version of the modular crossed product algebra, and we establish its type II character by constructing a trace. This yields a well-defined von Neumann entropy. For semiclassical states, by imposing a UV cutoff in QFT and proposing a quantum generalization of the first law, we demonstrate that the algebraic and generalized entropies are in match. Our work generalizes the notion of a local algebra to that of a \textit{fluctuating region}, representing an average of algebras over all possible static patches and configurations of other geodesics. This provides a complete, covariant, and multi-observer extension of the CLPW construction and lays the foundation for a fully relational quantum gravitational description of de Sitter space.

In this paper we study two-dimensional discrete operators whose eigenfunctions at zero energy level are given by rational functions on spectral curves. We extend discrete operators to difference operators and show that two-dimensional finite-gap Schrodinger operators at fixed energy level can be obtained from difference operators by passage to the limit.

We study interacting particle systems of Kuramoto-type. Our focus is on the dynamical relation between the partial differential equation (PDE) arising in the continuum limit (CL) and the one obtained in the mean-field limit (MFL). Both equations arise when we are considering the limit of infinitely many interacting particles but the classes of PDEs are structurally different. The CL tracks particles effectively pointwise, while the MFL is an evolution for a typical particle. First, we briefly discuss the relation between solutions of the CL and the MFL showing how to generate solutions of the CL starting from solutions of the MFL. Our main result concerns a dynamical relation between important invariant manifolds of the CFL and the MFL. In particular, we give an explicit proof that the unstable manifold of the homogeneous steady state of the CL is the direct dynamical analogue of the famous Ott-Antonsen manifold for the MFL.

The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|\sigma_i|} f_j / \partial (x^1)^{\sigma_i^1}\cdots \partial (x^d)^{\sigma_i^d}$ (of orders $0 \leqslant |\sigma_i| \leqslant k$), where the multi-indices $\sigma_i$ mark (all or part of) fibre variables $u_{\sigma_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras.

We consider random Young diagrams with respect to the measure induced by the decomposition of the $p$-th exterior power of $\mathbb{C}^{n}\otimes \mathbb{C}^{k}$ into irreducible representations of $GL_{n}\times GL_{k}$. We demonstrate that transition probabilities for these diagrams in the limit $n,k,p\to\infty$ with $p\sim nk$ converge to the large $N$ limiting law for the eigenvalues of random matrices in Jacobi Unitary Ensemble. We compute the characters of Young--Jucys--Murphy elements in $\bigwedge^{p}(\mathbb{C}^{n}\otimes\mathbb{C}^{k})$ and discuss their relation to surface counting. We formulate several conjectures on the connection between the correlators in both random ensembles.

Gauss's principle of least constraint transforms a dynamics problem into a pure minimization problem, where the total magnitude of the constraint force is the cost function, minimized at each instant. Newton's equation is the first-order necessary condition for minimizing the Gaussian cost, subject to the given kinematic constraints. The principle of minimum pressure gradient (PMPG) is to incompressible fluid mechanics what Gauss's principle is to particle mechanics. The PMPG asserts that an incompressible flow evolves from one instant to another by minimizing the L2-norm of the pressure gradient force. A candidate flow field whose evolution minimizes the pressure gradient cost at each instant is guaranteed to satisfy the Navier-Stokes equation. Consequently, the PMPG transforms the incompressible fluid mechanics problem into a pure minimization framework, allowing one to determine the evolution of the flow field by solely focusing on minimizing the cost. In this paper, we show that the resulting minimization problem is a convex Quadratic Programming (QP) problem-one of the most computationally tractable classes in nonlinear optimization. Moreover, leveraging tools from analytical mechanics and the Moore-Penrose theory of generalized inverses, we derive an analytical solution for this QP problem. As a result, we present an explicit formula for the projected dynamics of the spatially discretized Navier-Stokes equation on the space of divergence-free fields. The resulting ODE is ready for direct time integration, eliminating the need for solving the Poisson equation in pressure at each time step. It is typically an explicit nonlinear ODE with constant coefficients. This compact form is expected to be highly valuable for both simulation and theoretical studies, including stability analysis and flow control design. We demonstrate the framework on the lid-driven cavity problem.

Topological quantum error-correcting codes (QECC) encode a variety of topological invariants in their code space. A classic structure that has not been encoded directly is that of obstruction classes of a fiber bundle, such as the Chern or Euler class. Here, we construct and analyze extensions of toric codes. We then analyze the topological structure of their errors and finally construct a novel code using these errors to encode the obstruction class to a fiber bundle. In so doing, we construct an encoding of characteristic classes such as the Chern and Pontryagin class in topological QECC. An example of the Euler class of $S^2$ is constructed explicitly.

Over 125 years ago, Henry Selby Hele-Shaw realized that the depth-averaged flow in thin gap geometries can be closely approximated by two-dimensional (2D) potential flow, in a surprising marriage between the theories of viscous-dominated and inviscid flows. Hele-Shaw flows allow visualization of potential flows over 2D airfoils and also undergird important discoveries in the dynamics of interfacial instabilities and convection, yet they have found little use in modeling flows in microfluidic devices, although these devices often have thin gap geometries. Here, we derive a Hele-Shaw approximation for the flow in the kinds of thin gap geometries created within microfluidic devices. Although these equations have been reported before, prior work used a less direct derivation. Here, we obtain them via a modified Method of Weighted Residuals (MWR), interpreting the Hele-Shaw approximation as the leading term of an orthogonal polynomial expansion that can be systematically extended to higher-order corrections. We provide substantial numerical evidence showing that approximate equations can successfully model real microfluidic and inertial-microfluidic device geometries. By reducing three-dimensional (3D) flows to 2D models, our validated model will allow for accelerated device modeling and design.

The requirement that a trapped spacetime domain forms in finite time for distant observers is logically possible and sometimes unavoidable, but its consequences are not yet fully understood. In spherical symmetry, the characterization of the near-horizon geometry of these physical black holes is complete and shows marked differences from their eternal counterparts. Whether these differences lead to observable signatures remains unclear. We construct an approximate near-horizon metric that encapsulates them and is suitable for modeling. The timelike apparent horizon of physical black holes provides a natural surface for a consistent membrane description: we obtain closed-form expressions for the redshift, proper acceleration, and extrinsic curvature, and assign a two-dimensional viscous-fluid stress tensor via junction conditions. These results also provide an additional perspective on the relation between Rindler and near-horizon geometries. Among dynamical generalizations of surface gravity, only a subset applies to these models. We complete their analysis and recover the intuitive definition of surface gravity -- the acceleration in the frame of a near-horizon observer, redshifted to infinity -- directly from the membrane acceleration.

For an open-closed homotopy algebra (OCHA), the previous work indicates that there is an open-closed version of Hochschild cohomology with a canonical Gerstenhaber algebra structure. If this OCHA is further cyclic and unital in the sense of Kajiura and Stasheff, we produce a BV algebra structure on this cohomology via a cochain-level identity formulated with cyclic brace operations.

Riemann vanishing theorem is a main ingredient of the conventional technique related to the Jacobi inversion problem. In the case of curves with a holomorphic involution, it has been exhaustively expounded in wellknown Fay's Lectures on theta functions. The case of real algebraic curves with involution is presented with less completeness in the literature. We give a detailed presentation of that case, including real curves of non-separating type (with involution) not considered before with this relation. We obtain the Novikov--Veselov realness conditions in a different set-up.

We study the long-time behavior of radially symmetric solutions to the Vlasov-Poisson equation consisting of an attractive point mass and a small, suitably localized and absolutely continuous distribution of particles: if the latter is initially localized on hyperbolic trajectories for the associated Kepler problem, we obtain global in time, unique Lagrangian solutions that asymptotically undergo a modified scattering dynamics (in the sense of distributions). A key feature of this result is its low regularity regime, which does not make use of derivative control, but can be upgraded to strong solutions and strong convergence by propagation of regularity.

The first part of this review tries to provide a self-contained view of supersymmetry breaking from the bottom-up perspective. We thus describe N=1 supersymmetry in four dimensions, the Standard Model and the MSSM, with emphasis on the ``soft terms'' that can link it to supergravity. The second part deals with the top-down perspective. It addresses, insofar as possible in a self-contained way, the basic setup provided by ten-dimensional strings and their links with supergravity, toroidal orbifolds, Scherk-Schwarz deformations and Calabi-Yau reductions, before focusing on a line of developments that is closely linked to our own research. Its key input is drawn from ten-dimensional non-tachyonic string models where supersymmetry is absent or non-linearly realized, and runaway ``tadpole potentials'' deform the ten-dimensional Minkowski vacua. We illustrate the perturbative stability of the resulting most symmetrical setups, which are the counterparts of circle reduction but involve internal intervals. We then turn to a discussion of fluxes in Calabi-Yau vacua and the KKLT setup, and conclude with some aspects of Cosmology, emphasizing some intriguing clues that the tadpole potentials can provide for the onset of inflation. The appendices collect some useful material on global and local N=1 supersymmetry, in components and in superspace, on string vacuum amplitudes, and on convenient tools used to examine the fluctuations of non-supersymmetric string vacua.

In joint work with Adam Black and Reuben Drogin, we develop a new approach to understanding the diffusive limit of the random Schrodinger equation based on ideas taken from random matrix theory. These lecture notes present the main ideas from this work in a self-contained and simplified presentation. The lectures were given at the summer school "PDE and Probability" at Sorbonne Université from June 16-20, 2025.

We prove the existence of the bundle predual to the tangent bundle (called precotangent bundle) for Grassmannians of reflexive Banach spaces and $p$-restricted Grassmannians of the polarized Hilbert space.

We study a broad class of qudit stabilizer codes, termed $\mathbb{Z}_N$ bivariate-bicycle (BB) codes, arising either as two-dimensional realizations of modulated gauge theories or as $\mathbb{Z}_N$ generalizations of binary BB codes. Our central finding, derived from the polynomial representation, is that the essential topological properties of these $\mathbb{Z}_N$ codes can be determined by the properties of their $\mathbb{Z}_p$ counterparts, where $p$ are the prime factors of $N$, even when $N$ contains prime powers ($N = \prod_i p_i^{k_i}$). This result yields a significant simplification by leveraging the well-studied framework of codes with prime qudit dimensions. In particular, this insight directly enables the generalization of the algebraic-geometric methods (e.g., the Bernstein-Khovanskii-Kushnirenko theorem) to determine anyon fusion rules in the general qudit situation. Moreover, we analyze the model's symmetry-enriched topological order (SET) to reveal a quasi-fractonic behavior, resolving the anyon mobility puzzle in this class of models. We also present a computational algebraic method using Gröbner bases over the ring of integers to efficiently calculate the topological order and its SET properties.

The occurrence of strong coupling or nonlinear scaling behavior for kinetically rough interfaces whose dynamics are conserved, but not necessarily variational, remains to be fully understood. Here we formulate and study a family of conserved stochastic evolution equations for one-dimensional interfaces, whose nonlinearity depends on a parameter n, thus generalizing that of the stochastic Burgers equation, whose behavior is retrieved for n=0. This family of equations includes as particular instances a stochastic porous medium equation and other continuum models relevant to various hard and soft condensed matter systems. We perform a one-loop dynamical renormalization group analysis of the equations, which contemplates strong coupling scaling exponents that depend on the value of $n$ and may or may not imply vertex renormalization. These analytical expectations are contrasted with explicit numerical simulations of the equations with n=1,2, and 3. For odd n, numerical stability issues have required us to generalize the scheme originally proposed for n=0 by T. Sasamoto and H. Spohn. Precisely for n=1 and 3, and at variance with the n=0 and 2 cases (whose numerical exponents are consistent with non-renormalization of the vertex), numerical strong coupling exponent values are obtained which suggest vertex renormalization, akin to that reported for the celebrated conserved KPZ equation. We also study numerically the statistics of height fluctuations, whose probability distribution function turns out (at variance with cKPZ) to have zero skewness for long times and at saturation, irrespective of the value of n. However, the kurtosis is non-Gaussian, further supporting the conclusion on strong coupling asymptotic behavior. The zero skewness seems related with space symmetries of the n=0 and 2 equations, and with an emergent symmetry at the strong coupling fixed point for odd values of n.

Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter's subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.

We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(\frac12-)$-Hölder continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).

Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal--Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal--Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct and study a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences.

Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity (LQG) due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\sqrt{4/3}$-LQG surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\sqrt{4/3}$-LQG surface decorated by both an SLE$_{12}$ curve and the ``dual'' SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015). Our paper is the starting point of recent works connecting LQG and random permutons such as the Baxter permuton.

The n-qubit Pauli group and its normalizer the n-qubit Clifford group have applications in quantum error correction and device characterization. Recent applications have made use of the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.

This paper establishes the global well-posedness and long-time dynamics of the general Ericksen--Leslie system for isotropic nematic liquid crystals under a constant magnetic field. On the two-dimensional torus $\mathbb{T}^2$, a liquid crystal molecule coincides with itself under rotations by integer multiples of $\pi$, which results in special boundary conditions. We prove the existence of global-in-time strong solutions by developing novel high-order energy estimates and employing compactness techniques. A key challenge lies in controlling the orientation of the liquid crystal molecules. After achieving a uniform bound for the molecular orientation angle in $\mathbb{S}^1$, we further characterize the long-time behavior of the solutions. This is accomplished by applying the Lojasiewicz--Simon inequality, which reveals the convergence of the solutions as time approaches infinity.

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_{\mu}(f)= F_{\mu}(f)-\displaystyle\int\sum\nolimits_{\lambda_i(x)<0} \lambda_i(x)d\mu(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+\alpha}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $\mu$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+\alpha}$ maps.

In this paper, we revisit Fujikawa's path integral formulation of the chiral anomaly and develop a generalised framework for systematically defining a regularised functional measure. This construction extends the $\eta$ regularisation scheme to operator language, making the connection between spectral asymmetry and measure transformation fully explicit. Before recovering Fujikawa's expression for the chiral anomaly from the regularised measure, we explore the deeper number-theoretic structure underlying the ill-defined spectral sum associated with the anomaly, interpreting it through the lens of smoothed asymptotics. Our approach unifies two complementary perspectives: the analytic regularisation of Fujikawa and the topological characterisation given by the Atiyah-Singer index theorem. We further investigate how the measure transforms under changes to the regularisation scale and derive a function $\iota_E(\Lambda)$ that encodes this dependence, showing how its Mellin moments govern the appearance of divergences. Finally, we comment on the conceptual relationship between the regularised measure, $\eta$ regularisation, and the generalised Schwinger proper-time formalism, with a particular focus on the two-dimensional Schwinger model.

In two- and higher-dimensional non-Hermitian lattices, systems can exhibit geometry-dependent bands, where the spectrum and eigenstates under open boundary conditions depend on the bulk geometry even in the thermodynamic limit. Although geometry-dependent bands are widely observed, the underlying mechanism for this phenomenon remains unclear. In this work, we address this problem by establishing a higher-dimensional non-Bloch band theory based on the concept of "strip generalized Brillouin zones" (SGBZs), which describe the asymptotic behavior of non-Hermitian bands when a lattice is extended sequentially along its linearly independent axes. Within this framework, we demonstrate that geometry-dependent bands arise from the incompatibility of SGBZs and, for the first time, derive a general criterion for the geometry dependence of non-Hermitian bands: non-zero area of the complex energy spectrum or the imaginary momentum spectrum. Our work opens an avenue for future studies on the interplay between geometric effects and non-Hermitian physics, such as non-Hermitian band topology.

The Hubbard model in the $U\to\infty$ limit has recently been shown to have resonant valence bond (RVB) ground states on the corner-sharing sawtooth and pyrochlore lattices in the dilute doping limit of a single vacancy. The two results were obtained by different approaches which do not apply to one another. We make the first step towards unifying them by studying the quasi-1D lattice of a pyrochlore stripe, where all corners are not shared between two tetrahedra, and the valence bond configurations are not fixed by the location of the vacancy. The energy level ordering of irreducible representations of each tetrahedron shows that a chain of them has exponentially degenerate partial RVB or dimer-monomer ground states where each tetrahedron hosts one spin-$1/2$ monomer and one spin-$0$ dimer. The exact ground states in the infinitely long chain limit are analytically solved by introducing basis transformations between local Hilbert spaces of neighboring tetrahedra, and its energy agrees with the extrapolation of numerical exact diagonalization results of finite sized systems.

Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.

The $GW$ approximation has become a method of choice for predicting quasiparticle properties in solids and large molecular systems, owing to its favorable accuracy-cost balance. However, its accuracy is the result of a fortuitous cancellation of vertex corrections in the polarizability and self-energy. Hence, when attempting to go beyond $GW$ through inclusion of vertex corrections, the accuracy can deteriorate if this delicate balance is disrupted. In this work, we explore an alternative route that theoretically goes beyond $GW$: the parquet formalism. Unlike approaches that focus on a single correlation channel, such as the electron-hole channel in $GW$ or the particle-particle channel in $T$-matrix theory, parquet theory treats all two-body scattering channels on an equal footing. We present the formal structure of the parquet equations, which couple the one-body Green's function, the self-energy, and the two-body vertex. We discuss the approximations necessary to solve this set of equations, the advantages and limitations of this approach, outline its implementation for molecular systems, and assess its accuracy for principal ionization potentials of small molecular systems.

This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive). We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for the positivity of the broad class of covariant maps. We provide a way to lift matrices from the cone of COP to PCOP, thereby creating a powerful bridge between the theory of copositive forms and the positive maps. We develop an analogous framework for decomposable maps, introducing the cone PDEC. As a primary application of this framework, we define a novel family of linear maps $\Phi_t^G$ parameterized by a graph $G$ and a real parameter $t$. We derive exact thresholds on $t$ that determine when these maps are positive or decomposable, linking these properties to fundamental graph-theoretic parameters. This construction yields vast new families of positive indecomposable maps, for which we provide explicit examples derived from infinite classes of graphs, most notably rank 3 strongly regular graphs such as Paley graphs. On the dual side, we investigate the entanglement properties of large classes of (symmetric) states. We prove that the SOS hierarchies used in polynomial optimization to approximate the cone of copositive matrices correspond precisely to dual cones of witnesses for different levels of the PPT bosonic extendibility hierarchy}-. In the setting of the DPS hierarchy for separability, we construct a large family of optimal entanglement witnesses that are not certifiable by any level of the PPT bosonic extendibility hierarchy, answering a long standing open question from [DPS04]. Leveraging the duality, we also provide an explicit construction of (mixture of) bipartite Dicke states that are simultaneously entangled and $K_r$-PPT bosonic extendible for any desired hierarchy level $r \geq 2$ and local dimension $n \geq 5$.

We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group classifications are performed. Classes of invariant DODEs for each Lie subgroup are written out. The symmetries allow us to construct invariant solutions to such equations. The application of variational methods to functionals with one delay yields DODEs with two delays. The Lagrangian and Hamiltonian approaches are reviewed. The delay analog of the Legendre transformation, which relates the Lagrangian and Hamiltonian approaches, is also analysed. Noether-type operator identities relate the invariance of delay functionals with the appropriate variational equations and their conserved quantities. These identities are used to formulate Noether-type theorems that give first integrals of second-order DODEs with symmetries. Finally, several open problems are formulated in the Conclusion.

In this paper, we consider the Q-tensor model of nematic liquid crystals, which couples the Navier-Stokes equations with a parabolic-type equation describing the evolution of the directions of the anisotropic molecules, in the half-space. The aim of this paper is to prove the global well-posedness for the Q-tensor model in the $L_p$-$L_q$ framework. Our proof is based on the Banach fixed point argument. To control the higher-order terms of the solutions, we prove the weighted estimates of the solutions for the linearized problem by the maximal $L_p$-$L_q$ regularity. On the other hand, the estimates for the lower-order terms are obtained by the analytic semigroup theory. Here, the maximal $L_p$-$L_q$ regularity and the generation of an analytic semigroup are provided by the R-solvability for the resolvent problem arising from the Q-tensor model. It seems to be the first result to discuss the unique existence of a global-in-time solution for the Q-tensor model in the half-space.

We develop a new bijective framework for the enumeration of bipartite planar maps with control on the degree distribution of black and white vertices. Our approach builds on the blossoming-tree paradigm, introducing a family of orientations on bipartite maps that extends Eulerian and quasi-Eulerian orientations and connects the bijection of Bousquet-Mélou and Schaeffer to the general scheme of Albenque and Poulalhon. This enables us to generalize the Bousquet-Mélou and Schaeffer's bijection to several families of bipartite maps. As an application, we also derive a rational and Lagrangian parametrization with positive integer coefficients for the generating series of quartic maps equipped with an Ising model, which is key to the probabilistic study of these maps.