ALBERT

Description: We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain \({\Omega \subset \mathbb{R}^3}\) . In particular, we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case \({\Omega=\mathbb{R}^3}\) , we show that the set of possible singularities is compact.

Description: We consider a magnetic Schrödinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (the Aharonov–Bohm effect, Phys Rev (2) 115:485–491, 1959 ). Moreover, the numerical computations performed in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014 ; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010 ) show a rather complex behavior of the eigenvalues as the pole varies in a planar domain. In this paper, in continuation of the analysis started in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014 ; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010 ), we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions. We deal with planar domains with Dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain: then, the operator loses its singular character and the k -th magnetic eigenvalue converges to that of the standard Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the k -th eigenfunction of the Laplacian. The proof relies on the variational characterization of eigenvalues, together with a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.

Description: We consider monopoles with singularities of Dirac type on quasiregular Sasakian three-folds fibering over a compact Riemann surface \({\Sigma}\) , for example the Hopf fibration \({S^3 \longrightarrow S^2}\) . We show that these correspond to holomorphic objects on \({\Sigma}\) , which we call twisted bundle triples. These are somewhat similar to Murray’s bundle gerbes. A spectral curve construction allows us to classify these structures, and, conjecturally, monopoles.

Description: In the context of formal deformation quantization, we provide an elementary argument showing that any universal quantization formula necessarily involves graphs with wheels.

Description: A family of discontinuous symplectic maps arising naturally in the study of nonsmooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general description of the map and a construction of nontrivial unbounded solutions for the special case of the pinball transformation. An asymptotic expansion of the pinball map in the limit of large energy is derived and used for the construction of unbounded solutions. For the generic values of the parameters, in the large energy limit, the map behaves similarly to another one considered earlier by Kesten (Acta Arith 1966 ).

Description: We construct approximate transport maps for non-critical \({\beta}\) -matrix models, that is, maps so that the push forward of a non-critical \({\beta}\) -matrix model with a given potential is a non-critical \({\beta}\) -matrix model with another potential, up to a small error in the total variation distance. One of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models.

Description: In this paper, we consider a system of homogeneous algebraic equations in complex variables and their conjugates, which arise naturally from the range criterion for separability of PPT states. We examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions. This gives us possible upper bounds of ranks of PPT entangled edge states and their partial transposes. We will focus on the multi-partite cases, which are much more delicate than the bi-partite cases. We use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion.

Description: We prove that the Fourier–Laplace–Nahm transform for connections with finitely many logarithmic singularities and a double pole at infinity on the projective line, all with semi-simple singular parts, is a hyper-Kähler isometry.

Description: We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo–Werner channels \({W_n^-}\) are factorizable, for all odd integers \({n\neq 3}\) . Furthermore, we investigate factorizability of convex combinations of \({W_3^+}\) and \({W_3^-}\) , a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.

Description: We consider continuous one-dimensional multifrequency Schrödinger operators, with analytic potential, and prove Anderson localization in the regime of positive Lyapunov exponent for almost all phases and almost all Diophantine frequencies.

Description: In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the Camassa–Holm equation for the shallow water waves. It is proved that these new isospectral deformations result in evolution equations on the mass density whose form depends on how the string is tied at the endpoints. Moreover, it is shown that the evolution equations in this class linearize on the spectral side and hence can be solved by the inverse spectral method. In particular, the problem involving a mass density given by a discrete finite measure and arbitrary boundary conditions is shown to be solvable by Stieltjes’ continued fractions.

Description: We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra A 0 , any faithful normal state \({\varphi_0}\) and any discrete group \({\Gamma}\) , the associated Bernoulli crossed product von Neumann algebra \({M=(A_0,\varphi_0)^{\overline{\otimes} \Gamma} \rtimes \Gamma}\) is solid relatively to \({\mathcal{L}(\Gamma)}\) . In particular, if \({\mathcal{L}(\Gamma)}\) is solid then M is solid and if \({\Gamma}\) is non-amenable and \({A_0 \neq \mathbb{C}}\) then M is a full prime factor. This gives many new examples of solid or prime type III factors. Following Chifan and Ioana, we also obtain the first examples of solid non-amenable type III equivalence relations.

Description: We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinic, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.

Description: We prove the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices. The proof is based on fermionic cluster expansion techniques combined with lattice Ward identities, and on a reconstruction theorem that allows us to compute the Kubo conductivity as the analytic continuation of its imaginary time counterpart.

Description: We prove that local random quantum circuits acting on n qubits composed of O ( t 10 n 2 ) many nearest neighbor two-qubit gates form an approximate unitary t -design. Previously it was unknown whether random quantum circuits were a t -design for any t 〉  3. The proof is based on an interplay of techniques from quantum many-body theory, representation theory, and the theory of Markov chains. In particular we employ a result of Nachtergaele for lower bounding the spectral gap of frustration-free quantum local Hamiltonians; a quasi-orthogonality property of permutation matrices; a result of Oliveira which extends to the unitary group the path-coupling method for bounding the mixing time of random walks; and a result of Bourgain and Gamburd showing that dense subgroups of the special unitary group, composed of elements with algebraic entries, are ∞-copy tensor-product expanders. We also consider pseudo-randomness properties of local random quantum circuits of small depth and prove that circuits of depth O ( t 10 n ) constitute a quantum t -copy tensor-product expander. The proof also rests on techniques from quantum many-body theory, in particular on the detectability lemma of Aharonov, Arad, Landau, and Vazirani. We give applications of the results to cryptography, equilibration of closed quantum dynamics, and the generation of topological order. In particular we show the following pseudo-randomness property of generic quantum circuits: Almost every circuit U of size O ( n k ) on n qubits cannot be distinguished from a Haar uniform unitary by circuits of size O ( n ( k -9)/11 ) that are given oracle access to U .

Description: Recently, we introduced T-duality in the study of topological insulators. In this paper, we study the bulk-boundary correspondence for three phenomena in condensed matter physics, namely, the quantum Hall effect, the Chern insulator, and time reversal invariant topological insulators. In all of these cases, we show that T-duality trivializes the bulk-boundary correspondence.

Description: Particle states transforming in one of the infinite spin representations of the Poincaré group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher–Haag–Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincaré group containing infinite spin, then it has infinite statistics. These results hold under the natural assumption of the Bisognano–Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s  + 1 where infinite spin representations exist, namely \({s\geq 2}\) .

Description: The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\) . The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\) -module with central charge \({c \in \mathbb{C}}\) , and its irreducible quotient \({\mathcal{V}_c}\) , possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\) . We show that for each integer \({n 〉 0}\) , \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\) -algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\) , respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\) , respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\) , and we prove this for \({n = 1}\) . As an application, we show that invariant subalgebras of \({\beta\gamma}\) -systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

Description: We consider isotropic XY spin chains whose magnetic potentials are quasiperiodic and the effective one-particle Hamiltonians have absolutely continuous spectra. For a wide class of such XY spin chains, we obtain lower bounds on their Lieb–Robinson velocities \({\mathfrak{v}}\) in terms of group velocities of their effective Hamiltonians: $$\mathfrak{v}{\geqslant} {\mathop {\rm ess sup}_{[0,1]}}\frac{2}{\pi}\frac{dE}{dN}.$$ where E is considered as a function of the integrated density of states.

Description: In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighbourhoods of sub-manifolds of L 2 -norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations.

Description: We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak- \({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\) . We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

Description: In this paper, we study a restricted four-body problem called the planar two-center-two-body problem. In the plane, we have two fixed centers Q 1 and Q 2 of masses 1, and two moving bodies Q 3 and Q 4 of masses \({\mu\ll 1}\) . They interact via Newtonian potential. Q 3 is captured by Q 2 , and Q 4 travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions that lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all earlier collisions. This problem is a simplified model for the planar four-body problem case of the Painlevé conjecture.

Description: We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.

Description: Let M be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure m F associated with a potential F . We compute the Hausdorff dimension of the conditional measures of m F . We study the m F -almost sure asymptotic penetration behaviour of locally geodesic lines of M into small neighbourhoods of closed geodesics, and of other compact (locally) convex subsets of M . We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objects. As an arithmetic consequence, we give almost sure Diophantine approximation results of real numbers by quadratic irrationals with respect to general Hölder quasi-invariant measures.

Description: The Shale–Stinespring Theorem (J Math Mech 14:315–322, 1965 ) together with Ruijsenaar’s criterion (J Math Phys 18(4):720–737, 1977 ) provide a necessary and sufficient condition for the implementability of the evolution of external field quantum electrodynamics between constant-time hyperplanes on standard Fock space. The assertion states that an implementation is possible if and only if the spatial components of the external electromagnetic four-vector potential \({A_\mu}\) are zero. We generalize this result to smooth, space-like Cauchy surfaces and, for general \({A_\mu}\) , show how the second-quantized Dirac evolution can always be implemented as a map between varying Fock spaces. Furthermore, we give equivalence classes of polarizations, including an explicit representative, that give rise to those admissible Fock spaces. We prove that the polarization classes only depend on the tangential components of \({A_\mu}\) w.r.t. the particular Cauchy surface, and show that they behave naturally under Lorentz and gauge transformations.

Description: A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations ( R -matrices). For a torus action on cotangent bundles over flag varieties the resulting R -matrices are the standard rational solutions of the Yang-Baxter equation, well known in the theory of quantum integrable systems. The torus action on the instanton moduli space leads to more complicated R -matrices, depending additionally on two equivariant parameters t 1 and t 2 . In this paper we derive an explicit expression for the R -matrix associated with the instanton moduli space. We study its matrix elements and its Taylor expansion in the powers of the spectral parameter. Certain matrix elements of this R -matrix give a generating function for the characteristic classes of tautological bundles over the Hilbert schemes in terms of the bosonic cut-and-join operators. In particular we rederive from the R -matrix the well known Lehn’s formula for the first Chern class. We explicitly compute the first several coefficients for the power series expansion of the R -matrix in the spectral parameter. These coefficients are represented by simple contour integrals of some symmetrized bosonic fields.

Description: We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d  + 1, with d the space dimension, this happens for all values of J smaller than a critical value J c ( p ), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p  〉 2 d and J in a left neighborhood of J c ( p ). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes ( d  = 2) or slabs ( d  = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.

Description: We introduce a class of gapped Hamiltonians on quantum spin chains, which allows asymmetric edge ground states. This class is an asymmetric generalization of the class of Hamiltonians (Fannes et al. Commun Math Phys 144:443–490, 1992 ). It can be characterized by five qualitative physical properties of ground state structures. In this Part I, we introduce the models and investigate their properties.

Description: We determine a substantial part of the unitary representation theory of the Drinfeld double of a q -deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every q -deformation of a higher rank compact Lie group has central property (T). We also determine the unitary dual of \({SL_q(n,\mathbb{C})}\) .

Description: We analyze 2-dimensional Ginzburg–Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large area limit.

Description: We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface \({\Sigma_0}\) in the interior of the black hole region, tangent to the singular hypersurface \({\{r = 0\}}\) at a single sphere, we study the problem of perturbing the Schwarzschild data on \({\Sigma_0}\) and solving the Einstein vacuum equations backwards in time. We obtain a local backwards well-posedness result for small perturbations lying in certain weighted Sobolev spaces. No symmetry assumptions are imposed. The perturbed spacetimes all have a singularity at a “collapsed” sphere on \({\Sigma_0}\) , where the leading asymptotics of the curvature and the metric match those of their Schwarzschild counterparts to a suitably high order. As in the Schwarzschild backward evolution, the pinched initial hypersurface \({\Sigma_0}\) ‘opens up’ instantly, becoming a regular spacelike (cylindrical) hypersurface. This result thus yields classes of examples of non-symmetric vacuum spacetimes, evolving forward-in-time from regular initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which turns out to be at the threshold that our energy methods can handle.

Description: We study the synchronization properties of the random double rotations on tori. We give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher.

Description: The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.

Description: We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as \({x\to\pm\infty}\) . The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n -th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N -wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.

Description: We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman’s condition) on the rotation number, must have a critical point on their boundaries.

Description: We investigate the asymptotic structure of (possibly type III) crossed product von Neumann algebras \({M = B \rtimes \Gamma}\) arising from arbitrary actions \({\Gamma \curvearrowright B}\) of bi-exact discrete groups (e.g. free groups) on amenable von Neumann algebras. We prove a spectral gap rigidity result for the central sequence algebra \({N' \cap M^\omega}\) of any nonamenable von Neumann subalgebra with normal expectation \({N \subset M}\) . We use this result to show that for any strongly ergodic essentially free nonsingular action \({\Gamma \curvearrowright (X, \mu)}\) of any bi-exact countable discrete group on a standard probability space, the corresponding group measure space factor \({L^\infty(X) \rtimes \Gamma}\) has no nontrivial central sequence. Using recent results of Boutonnet et al. (Local spectral gap in simple Lie groups and applications, 2015 ), we construct, for every \({0 〈 \lambda \leq 1}\) , a type \({{\rm III_\lambda}}\) strongly ergodic essentially free nonsingular action \({\mathbf{F}_\infty \curvearrowright (X_\lambda, \mu_\lambda)}\) of the free group \({{\mathbf{F}}_\infty}\) on a standard probability space so that the corresponding group measure space type \({{\rm III_\lambda}}\) factor \({L^\infty(X_\lambda, \mu_\lambda) \rtimes \mathbf{F}_\infty}\) has no nontrivial central sequence by our main result. In particular, we obtain the first examples of group measure space type \({{\rm III}}\) factors with no nontrivial central sequence.

Description: We give a quantitative version of Vainberg’s method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size \({\tau}\) with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate \({\tau}\) . Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate \({\tau}\) , then there are resonances in logarithmic strips whose width is given by \({\tau}\) . As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.

Description: We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel–Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

Description: This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta–Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we prove that the considered energy \({\Gamma}\) -converges to an energy functional of the limit “homogenized” measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplet per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

Description: The stability of asymptotic profiles of solutions to the Cauchy–Dirichlet problem for fast diffusion equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Łojasiewicz–Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Łojasiewicz–Simon inequality in a different way.

Description: Multiple Schramm–Loewner Evolutions (SLE) are conformally invariant random processes of several curves, whose construction by growth processes relies on partition functions—Möbius covariant solutions to a system of second order partial differential equations. In this article, we use a quantum group technique to construct a distinguished basis of solutions, which conjecturally correspond to the extremal points of the convex set of probability measures of multiple SLEs.

Description: In this paper, we present an explicit formula that connects the Kontsevich-Witten tau-function and the Hodge tau-function by differential operators belonging to the \({\widehat{GL(\infty)}}\) group. Indeed, we show that the two tau-functions can be connected using Virasoro operators. This proves a conjecture posted by Alexandrov in (From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators, Letters in Mathematical physics, doi: 10.1007/s11005-013-0655-0 , 2014 ).

Description: We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah–Singer index theorem and another term involving the \({\eta}\) -invariant of the Cauchy hypersurfaces.

Description: The interface problem for the linear Korteweg–de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas’s Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.

Description: We propose a new theory of higher spin gravity in three spacetime dimensions. This is defined by what we will call a Nambu–Chern–Simons (NCS) action; this is to a Nambu 3-algebra as an ordinary Chern–Simons (CS) action is to a Lie (2-)algebra. The novelty is that the gauge group of this theory is simple ; this stands in contrast to previously understood interacting 3D higher spin theories in the frame-like formalism. We also consider the N  = 8 supersymmetric NCS-matter model (BLG theory), where the NCS action originated: Its fully supersymmetric M2 brane configurations are interpreted as Hopf fibrations, the homotopy type of the (infinite) gauge group is calculated and its instantons are classified.

Description: We prove a conjecture by Bravyi on an upper bound on entanglement rates of local Hamiltonians. We then use this bound to prove the stability of the area law for the entanglement entropy of quantum spin systems under adiabatic and quasi-adiabatic evolutions.

Description: We consider Carleson’s problem regarding convergence for the Schrödinger equation in dimensions \({d\ge 2}\) . We show that if the solution converges almost everywhere with respect to \({\alpha}\) -Hausdorff measure to its initial datum as time tends to zero, for all data \({H^{s}(\mathbb{R}^{d})}\) , then \({s\ge \frac{d}{2(d+2)}(d+1-\alpha)}\) . This strengthens and generalises results of Bourgain and Dahlberg–Kenig.

Description: In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H : $$d\mathfrak{u}(t) +[{\rm A}\mathfrak{u}(t)+{\rm B}(\mathfrak{u}(t),\mathfrak{u}(t))] dt = dL(t), \quad\mathfrak{u}(0)=x\in {\rm H}.$$ Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a \({D({\rm A}^{-1/2})}\) -valued lévy process satisfying several technical assumptions. By using a density transformation theorem type for lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein–Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain \({\gamma \in [0,1/4]}\) B is continuous on \({D({\rm A}^\gamma)\times D({\rm A}^\gamma)}\) with values in \({D({\rm A}^ {-1/2})}\) . Using a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on \({{\rm H} \times{\rm H}}\) . While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier–Stokes, Magnetohydrodynamics equations, the 3D Leray- \({{\varvec{\alpha}}}\) model, the GOY and Sabra shell models.

Description: We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an “effective hyperbolicity” condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.

Description: We determine a \({q \to 1}\) limit of the two-dimensional q -Whittaker driven particle system on the torus studied previously in Corwin and Toninelli (Electron. Commun. Probab. 21(44):1–12, 2016 ). This has an interpretation as a (2 + 1)-dimensional stochastic interface growth model, which is believed to belong to the so-called anisotropic Kardar–Parisi–Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the (2 + 1)-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in (2 + 1)-dimension is irrelevant.

Description: We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge’s strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L 2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved \({L^\infty}\) estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767–781, 2011 ). This estimate improves the L 4 restriction estimate of Burq et al. (Duke Math J 138:445–486, 2007 ) and Hu (Forum Math 6:1021–1052, 2009 ) by a power of \({(\log\log\lambda)^{-1}}\) . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of \({(\log\lambda)^{-1}}\) by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014 ) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014 ) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover \({\mathbb{H}^2}\) .

Description: The local well-posedness for the Cauchy problem of a system of semirelativistic equations in one space dimension is shown in the Sobolev space H s of order s ≥ 0. We apply the standard contraction mapping theorem by using Bourgain type spaces X s , b . We also use an auxiliary space for the solution in L 2  =  H 0 . We give the global well-posedness by this conservation law and the argument of the persistence of regularity.

Description: The p–q duality is a relation between the ( p , q ) model and the ( q , p ) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of Fukuma-Kawai-Nakayama. Already in the duality between the (2, 3) model and the (3, 2) model the formulas are long. In this work a new approach to p–q duality is given: It can be realized in a precise sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac–Schwarz operators. Therefore, since these operators correspond to Virasoro constraints, this allows us to view the p–q duality as a consequence of the duality of the relevant Virasoro constraints.

Description: Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU ( N ). The crossing matrices directly relate to 6j-symbols of the quantum group \({\mathcal{U}_{q}su(N)}\) . We present powerful methods to calculate such quantum 6j-symbols for general N . This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or—in the limit to classical groups—to determine color factors for Yang Mills amplitudes.

Description: We consider a rigid body colliding with a continuum of particles. We assume that the body is moving at a velocity close to an equilibrium velocity \({V_{\infty}}\) and that the particles colliding with the body reflect diffusely, that is, probabilistically with some probability distribution K . We find a condition that is sufficient and almost necessary that the collective force of the colliding particles reverses the relative velocity V ( t ) of the body, that is, changes the sign of \({V(t)-V_{\infty}}\) , before the body approaches equilibrium. Examples of both reversal and irreversal are given. This is in strong contrast with the pure specular reflection case in which only reversal happens.

Description: We study non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of classical information that can be transmitted when a quantum channel is used a finite number of times and a fixed, non-vanishing average error is permissible. In this work we consider the classical capacity of quantum channels that are image-additive, including all classical to quantum channels, as well as the product state capacity of arbitrary quantum channels. In both cases we show that the non-asymptotic fundamental limit admits a second-order approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the blocklength tends to infinity. The behavior is governed by a new channel parameter, called channel dispersion, for which we provide a geometrical interpretation.

Description: We prove \({|x|^{-2}}\) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^{d}}\) , in the upper critical dimension d = 4. This is a statement that the critical exponent \({\eta}\) exists and is equal to zero. Results of this nature have been proved previously for dimensions \({d \ge 5}\) using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

Description: In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich–Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac–Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.

Description: We prove that translationally invariant Hamiltonians of a chain of n qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit \({n \rightarrow \infty}\) we show that any translationally invariant Hamiltonian of a chain of n qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e., Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly, in the limit \({n \rightarrow \infty}\) we show that any nearest-neighbour Hamiltonian of a chain of n qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented that suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains.

Description: A multiparametric family of 2D Toda \({\tau}\) -functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of S n . The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) in their series expansion over products \({P_{\mu}P^{'}_{\nu}}\) of power sum symmetric functions in the two sets of Toda flow parameters and powers of the l  +  m auxiliary parameters are shown to enumerate \({|\mu|=|\nu|=n}\) fold branched covers of the Riemann sphere with specified ramification profiles \({ \mu}\) and \({\nu}\) at a pair of points, and two sets of additional branch points, satisfying certain additional conditions on their ramification profile lengths. The first group consists of l branch points, with ramification profile lengths fixed to be the numbers \({(n-c_{1}, . . . , n-c_{l})}\) ; the second consists of m further groups of “coloured” branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers \({(d_{1}, . . . , d_{m})}\) . The latter are counted with signs determined by the parity of the total number of such branch points. The coefficients \({{F^{c_{1}, . . . , c_{l}}_{d_{1}, . . . , d_{m}}}(\mu, \nu)}\) are also shown to enumerate paths in the Cayley graph of the symmetric group S n generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type \({\mu}\) and ending in the class of type \({\nu}\) , with the first l consecutive subsequences of \({(c_{1}, . . . , c_{l})}\) transpositions strictly monotonically increasing, and the subsequent subsequences of \({(d_{1}, . . . , d_{m})}\) transpositions weakly increasing.

Description: We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems.

Description: Self-similarity of Burgers’ equation with stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers’ equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.

Description: We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo (Forum Math, 2014 ) about regions \({\mathcal{R}_g \subset \mathbb{C}}\) for which there can be uniform \({L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}}\) resolvent estimates for \({\Delta_g + \zeta}\) , \({\zeta \in \mathcal{R}_g}\) , where \({\Delta_g}\) is the Laplace-Beltrami operator with metric g on a given compact boundaryless Riemannian manifold of dimension \({n \geq 3}\) . This is related to earlier work of Kenig, Ruiz and the third author (Duke Math J 55:329–347, 1987 ) for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, S n , the resolvent estimates in (Dos Santos Ferreira et al. in Forum Math, 2014 ), involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from \({\zeta}\) to the spectrum of \({\Delta_{S^n}}\) . In the other direction, we also show that the bounds in (Dos Santos Ferreira et al. in Forum Math, 2014 ) can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, \({\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n}\) , with the flat metric. The latter improves earlier bounds of Shen (Int Math Res Not 1:1–31, 2001 ). The work of (Dos Santos Ferreira et al. in Forum Math, 2014 ) and (Shen in Int Math Res Not 1:1–31, 2001 ) was based on Hadamard parametrices for \({(\Delta_g + \zeta)^{-1}}\) . Ours is based on the related Hadamard parametrices for \({\cos t \sqrt{-\Delta_g}}\) , and it follows ideas in (Sogge in Ann Math 126:439–447, 1987 ) of proving L p -multiplier estimates using small-time wave equation parametrices and the spectral projection estimates from (Sogge in J Funct Anal 77:123–138, 1988 ). This approach allows us to adapt arguments in Bérard (Math Z 155:249–276, 1977 ) and Hlawka (Monatsh Math 54:1–36, 1950 ) to obtain the aforementioned improvements over (Dos Santos Ferreira et al. in Forum Math, 2014 ) and (Shen in Int Math Res Not 1:1–31, 2001 ). Further improvements for the torus are obtained using recent techniques of the first author (Bourgain in Israel J Math 193(1):441–458, 2013 ) and his work with Guth (Bourgain and Guth in Geom Funct Anal 21:1239–1295, 2011 ) based on the multilinear estimates of Bennett, Carbery and Tao (Math Z 2:261–302, 2006 ). Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of \({\sqrt{-\Delta_g}}\) , and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals, as opposed to those in (Sogge in J Funct Anal 77:123–138, 1988 ) for unit-length intervals. We show that the resolvent estimates are sensitive to clustering within the spectrum, which is not surprising given Sommerfeld’s original conjecture (Sommerfeld in Physikal Zeitschr 11:1057–1066, 1910 ) about these operators.

Description: Using the Griffiths–Simon construction of the \({\varphi^4}\) model and the lace expansion for the Ising model, we prove that, if the strength \({\lambda\ge0}\) of nonlinearity is sufficiently small for a large class of short-range models in dimensions d  〉 4, then the critical \({\varphi^4}\) two-point function \({\langle{\varphi_o\varphi_x \rangle}_{\mu_c}}\) is asymptotically \({|x|^{2-d}}\) times a model-dependent constant, and the critical point is estimated as \({\mu_c = \hat{\fancyscript{J}} -\frac{\lambda}{2} \langle {\varphi_o^2}\rangle_{\mu_c} + O (\lambda^2)}\) , where \({\hat{\fancyscript{J}}}\) is the massless point for the Gaussian model.

Description: Benguria and Loss have conjectured that, amongst all smooth closed curves in \({\mathbb{R}^2}\) of length 2 π , the lowest possible eigenvalue of the operator \({L=-\Delta+\kappa^2}\) is 1. They observed that this value was achieved on a two-parameter family, \({\mathcal{O}}\) , of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in \({\mathcal{O}}\) as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems.

Description: We define exotic twisted \({\mathbb{T}}\) - equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut–Chern character form, a loop space refinement of the twisted Chern character form in Bouwknegt et al. (Commun Math Phys 228:17–49, 2002 ) and Mathai and Stevenson (Commun Math Phys 236:161–186, 2003 ), which represents classes in the completed periodic exotic twisted \({\mathbb{T}}\) -equivariant cohomology of LZ .We establish a localisation theorem for the completed periodic exotic twisted \({\mathbb{T}}\) -equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.

Description: Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V -modules are equivalent.

Description: We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena, such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on \({\mathbb{Z}}\) with periodic parameters an explicit strictly positive lower bound for the Lieb–Robinson velocity.

Description: In this paper, we study complex valued branching Brownian motion in the so-called glassy phase, also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so-called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian motion context.

Description: We develop a general instability index theory for an eigenvalue problem of the type \({\mathcal{L} u=\lambda u'}\) , for a class of self-adjoint operators \({\mathcal{L}}\) on the line R 1 . More precisely, we construct an Evans-like function to show (a real eigenvalue) instability in terms of a Vakhitov–Kolokolov type condition on the wave. If this condition fails, we show by means of Lyapunov–Schmidt reduction arguments and the Kapitula–Kevrekidis–Sandstede index theory that spectral stability holds. Thus, we have a complete spectral picture, under fairly general assumptions on \({\mathcal{L}}\) . We apply the theory to a wide variety of examples. For the generalized Bullough–Dodd–Tzitzeica type models, we give instability results for travelling waves. For the generalized short pulse/Ostrovsky/Vakhnenko model, we construct (almost) explicit peakon solutions, which are found to be unstable, for all values of the parameters.

Description: Motivated by recent developments on solvable directed polymer models, we define a ‘multi-layer’ extension of the stochastic heat equation involving non-intersecting Brownian motions. By developing a connection with Darboux transformations and the two-dimensional Toda equations, we conjecture a Markovian evolution in time for this multi-layer process. As a first step in this direction, we establish an analogue of the Karlin-McGregor formula for the stochastic heat equation and use it to prove a special case of this conjecture.

Description: We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green’s function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

Description: We consider solutions of the scalar wave equation \({\Box_g\phi=0}\) , without symmetry, on fixed subextremal Reissner-Nordström backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ϕ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\) . Using this, we show here that ϕ is in fact uniformly bounded, \({|\phi| \leq C}\) , in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\) , to which ϕ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.

Description: In this paper we study radial solutions for the following equation $$\Delta u(x)+f (u(x), |x|) = 0,$$ where \({x \in {\mathbb{R}^{n}}}\) , n  〉 2, f is subcritical for r small and u large and supercritical for r large and u small, with respect to the Sobolev critical exponent \({2^{*} = \frac{2n}{n-2}}\) . The solutions are classified and characterized by their asymptotic behaviour and nodal properties. In an appropriate super-linear setting, we give an asymptotic condition sufficient to guarantee the existence of at least one ground state with fast decay with exactly j zeroes for any j ≥ 0. Under the same assumptions, we also find uncountably many ground states with slow decay, singular ground states with fast decay and singular ground states with slow decay, all of them with exactly j zeroes. Our approach, based on Fowler transformation and invariant manifold theory, enables us to deal with a wide family of potentials allowing spatial inhomogeneity and a quite general dependence on u . In particular, for the Matukuma-type potential, we show a kind of structural stability.

Description: Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erdős–Renyi random graph G ( n , d / n ), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k -colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007 ). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k 0 .

Description: We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.

Description: A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.

Description: In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K k , t . We also showed that the set K k , t is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of \({\ell^p}\) norms on the set K k , t and prove that the maximum is attained on a vector of shape ( a , b , . . . , b ) where a 〉  b . In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any \({\varepsilon 〉 0}\) , it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as \({\log 2 -\varepsilon}\) . Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

Description: In this paper we analyze the Feynman wave equation on Lorentzian scattering spaces. We prove that the Feynman propagator exists as a map between certain Banach spaces defined by decay and microlocal Sobolev regularity properties. We go on to show that certain nonlinear wave equations arising in QFT are well-posed for small data in the Feynman setting.

Description: We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres are uniquely isoperimetric in manifolds that are Schwarzschild–anti-deSitter at infinity. These results have important repercussions for our understanding of spacelike hypersurfaces in Lorentzian space-times which are asymptotic to null infinity. In fact, as an application of our results, we verify the asymptotically hyperbolic Penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes.

Description: We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime \({\beta 〈 \beta_c}\) , and the mean-field lower bound \({\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}\) for \({\beta 〉 \beta_c}\) . For finite-range models, we also prove that for any \({\beta 〈 \beta_c}\) , the probability of an open path from the origin to distance n decays exponentially fast in n . For the Ising model, we prove finiteness of the susceptibility for \({\beta 〈 \beta_c}\) , and the mean-field lower bound \({\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}\) for \({\beta 〉 \beta_c}\) . For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \({\beta 〈 \beta_c}\) .

Description: We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities. Using this, for the systems started in step initial data, we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar–Parisi–Zhang universality class, as well as leads to many new examples of such models. In particular, asymmetric simple exclusion process, the stochastic six-vertex model, q -totally asymmetric simple exclusion process and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the R -matrix related to the six-vertex model. One of the key tools used here is the fusion of R -matrices and we provide a probabilistic proof of this procedure.

Description: We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric \({|\omega|^2}\) , where \({\omega}\) is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form \({\omega}\) . As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

Description: Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\) ) is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein–Gordon equation in dimension n = 2, which is captured by constructing an adapted systems of coordinate frames.

Description: The homogeneous coordinate ring of the Grassmannian Gr k , n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Gr k , n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plücker coordinate. We also relate the twist map to a maximal green sequence.

Description: We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call skeleton , that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measures bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each other—through explosions of their basins—as the dynamics varies. This theory also allows us to prove that, in the absence of collapses, the basins are continuous functions of the diffeomorphism.

Description: A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang–Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.

Description: We give explicit formulas for the elements of the center of the completed quantum affine algebra in type A at the critical level that are associated with the fundamental representations. We calculate the images of these elements under a Harish-Chandra-type homomorphism. These images coincide with those in the free field realization of the quantum affine algebra and reproduce generators of the q -deformed classical \({{\mathcal W}}\) -algebra of Frenkel and Reshetikhin.

Description: We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix \({\mathcal{T}_0}\) . Focusing on the eigenvalues of \({\mathcal{T}_0}\) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required \({\mathcal{T}_0}\) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.

Description: 〈h3〉Abstract〈/h3〉 〈p〉For each 〈span〉 〈span〉\(\lambda 〉0\)〈/span〉 〈/span〉 and under necessary conditions on the parameters, we construct normalized waves for second order PDE’s with mixed power non-linearities, with 〈span〉 〈span〉\(\Vert u\Vert _{L^2({{\mathbf {R}}}^n)}^2=\lambda , n\ge 1\)〈/span〉 〈/span〉. We show that these are bell-shaped smooth and localized functions, and we compute their precise asymptotics. We study the question for the smoothness of the Lagrange multiplier with respect to the 〈span〉 〈span〉\(L^2\)〈/span〉 〈/span〉 norm of the waves, namely the map 〈span〉 〈span〉\(\lambda \rightarrow \omega _\lambda \)〈/span〉 〈/span〉, a classical problem related to its stability. We show that this is intimately related to the question for the non-degeneracy of the said solitons. We provide a wide class of non-linearities, for which the waves are non-degenerate. Under some minimal extra assumptions, we show that a.e. in 〈span〉 〈span〉\(\lambda \)〈/span〉 〈/span〉, the map 〈span〉 〈span〉\(\lambda \rightarrow f_{\omega _\lambda }\)〈/span〉 〈/span〉 is differentiable and the waves 〈span〉 〈span〉\(e^{i \omega _\lambda t} f_{\omega _\lambda }\)〈/span〉 〈/span〉 are spectrally (and in some cases orbitally) stable as solutions to the NLS equation. Similar results are obtained for the same waves, as traveling waves of the Zakharov–Kuznetsov system.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer–Picard ring for the fusion category 〈span〉 〈span〉\({{\bf Vec} (\mathbb{Z}/p\mathbb{Z})}\)〈/span〉 〈/span〉 . Moreover, we provide a physical interpretation of all indecomposable bimodule categories in terms of domain walls in the associated topological phase. We show how this interpretation can be used to compute the Brauer–Picard ring from a physical perspective.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of 〈em〉N〈/em〉-particle systems. We establish an equation governing the evolution of our quantum analogue of the 〈em〉N〈/em〉-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the 〈em〉N〈/em〉-particle Schrödinger equation include an 〈span〉 〈span〉\({O(1/\sqrt{N})}\)〈/span〉 〈/span〉 convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the 〈em〉N〈/em〉-particle density operator, uniform in 〈span〉 〈span〉\({\hbar\in(0,1]}\)〈/span〉 〈/span〉 provided that 〈em〉V〈/em〉 and 〈span〉 〈span〉\({(-\Delta)^{3/2+d/4}V}\)〈/span〉 〈/span〉 have integrable Fourier transforms.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We investigate the propagation of the scalar waves in the FLRW universes beginning with a Big Bang and ending with a Big Crunch, a Big Rip, a Big Brake, or a Sudden Singularity. We obtain the sharp description of the asymptotics for the solutions of the linear Klein–Gordon equation, and similar results for the semilinear equation with a subcritical exponent. We prove that the number of cosmological particle creation is finite under general assumptions on the initial Big Bang and the final Big Crunch or Big Brake.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We obtain the first results on convergence rates in the Prokhorov metric for the weak invariance principle (functional central limit theorem) for deterministic dynamical systems. Our results hold for uniformly expanding/hyperbolic (Axiom A) systems, as well as nonuniformly expanding/hyperbolic systems such as dispersing billiards, Hénon-like attractors, Viana maps and intermittent maps. As an application, we obtain convergence rates for deterministic homogenization in multiscale systems.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Makeenko and Migdal (Phys Lett B 88(1):135–137, 〈span〉1979〈/span〉) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and Kostov (Nucl Phys B 176(1):199–215, 〈span〉1980〈/span〉) reformulated the Makeenko–Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of Lévy (〈span〉2017〈/span〉). Subsequently Driver, Kemp, and Hall Commun. Math. Phys. 351(2), 741–774 (〈span〉2017〈/span〉) gave a simplified proof of Lévy’s result and then with Driver, Gabriel, Kemp, and Hall Commun. Math. Phys. 352(3), 967–978 (〈span〉2017〈/span〉) we showed that these simplified proofs extend to the Yang–Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko–Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains “why” the Makeenko–Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang–Mill’s fields.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of 〈em〉N〈/em〉, 〈span〉 〈span〉\({N\ge2}\)〈/span〉 〈/span〉, symbols and with 〈em〉C〈/em〉〈sup〉1〈/sup〉 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of 〈span〉 〈span〉\({2\times 2}\)〈/span〉 〈/span〉 matrix cocycles and our results apply to an open and dense subset of elliptic 〈span〉 〈span〉\({\mathrm{SL}(2,\mathbb{R})}\)〈/span〉 〈/span〉 cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉Stochastic growth processes in dimension (2 + 1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian 〈span〉 〈span〉\({H_\rho}\)〈/span〉 〈/span〉 of the speed of growth 〈span〉 〈span〉\({v(\rho)}\)〈/span〉 〈/span〉 as a function of the average slope 〈span〉 〈span〉\({\rho}\)〈/span〉 〈/span〉 satisfies 〈span〉 〈span〉\({{\rm det} H_\rho 〉 0}\)〈/span〉 〈/span〉 (“isotropic KPZ class”) or 〈span〉 〈span〉\({{\rm det} H_\rho \le 0}\)〈/span〉 〈/span〉 (“anisotropic KPZ (AKPZ)” class). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class fluctuations are logarithmic in time and space. It is natural to ask (a) if one can exhibit interesting growth models with “smooth” stationary states, i.e., with 〈em〉O〈/em〉(1) fluctuations (instead of logarithmically or power-like growing, as in Wolf’s picture) and (b) what new phenomena arise when 〈span〉 〈span〉\({v(\cdot)}\)〈/span〉 〈/span〉 is not differentiable, so that 〈span〉 〈span〉\({H_\rho}\)〈/span〉 〈/span〉 is not defined. The two questions are actually related and here we provide an answer to both, in a specific framework. We define a (2 + 1)-dimensional interface growth process, based on the so-called shuffling algorithm for domino tilings. The stationary, non-reversible measures are translation-invariant Gibbs measures on perfect matchings of 〈span〉 〈span〉\({\mathbb{Z}^2}\)〈/span〉 〈/span〉 , with 2-periodic weights. If 〈span〉 〈span〉\({\rho\ne0}\)〈/span〉 〈/span〉 , fluctuations are known to grow logarithmically in space and to behave like a two-dimensional GFF. We prove that fluctuations grow at most logarithmically in time and that 〈span〉 〈span〉\({{\rm det} H_\rho 〈 0}\)〈/span〉 〈/span〉 : the model belongs to the AKPZ class. When 〈span〉 〈span〉\({\rho=0}\)〈/span〉 〈/span〉 , instead, the stationary state is “smooth”, with correlations uniformly bounded in space and time; correspondingly, 〈span〉 〈span〉\({v(\cdot)}\)〈/span〉 〈/span〉 is not differentiable at 〈span〉 〈span〉\({\rho=0}\)〈/span〉 〈/span〉 and we extract the singularity of the eigenvalues of 〈span〉 〈span〉\({H_\rho}\)〈/span〉 〈/span〉 for 〈span〉 〈span〉\({\rho\sim 0}\)〈/span〉 〈/span〉 .〈/p〉

Description: 〈h3〉Abstract〈/h3〉 〈p〉The global existence issue in critical spaces for compressible Navier–Stokes equations, was addressed by Danchin (Invent Math 141:579–614, 〈span〉2000〈/span〉) and then developed by Charve and Danchin (Arch Rational Mech Anal 198:233–271, 〈span〉2010〈/span〉), Chen et al. (Commun Pure Appl Math 63:1173–1224, 〈span〉2010〈/span〉) and Haspot (Arch Rational Mech Anal 202:427–460, 〈span〉2011〈/span〉) in more general 〈em〉L〈/em〉〈sup〉〈em〉p〈/em〉〈/sup〉 setting. The main aim of this paper is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical regularity framework. To the best of our knowledge, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 〈span〉1980〈/span〉) in the 〈span〉 〈span〉\({L^1(\mathbb{R}^d)}\)〈/span〉 〈/span〉 space. We now claim 〈em〉a new low-frequency assumption〈/em〉 for barotropic compressible Navier–Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space 〈span〉 〈span〉\({\dot{B}^{-\sigma_1}_{2,\infty}(\mathbb{R}^d)}\)〈/span〉 〈/span〉 with the regularity 〈span〉 〈span〉\({\sigma_1\in (1-d/2, 2d/p-d/2]}\)〈/span〉 〈/span〉, then a 〈em〉sharp〈/em〉 time-weighted inequality including enough time-decay information can be available, where 〈em〉optimal decay exponents for the high frequencies〈/em〉 are exhibited. The proof mainly depends on some non standard Besov product estimates. As a by-product, those optimal time-decay rates of 〈em〉L〈/em〉〈sup〉〈em〉q〈/em〉〈/sup〉–〈em〉L〈/em〉〈sup〉〈em〉r〈/em〉〈/sup〉 type are also captured in the critical framework.〈/p〉