We show that a 2-parameter family of \(\tau \)-functions for the first Painlevé equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.

We consider the random iteration of finitely many expanding \(\mathscr {C}^{1+\epsilon }\) diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on spaces of Hölder continuous functions. As an application we introduce generalised Takagi functions on the real line and we perform a complete multifractal analysis

We associate to each Temperley–Lieb–Jones C*-tensor category \({\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )\) with parameter \(\delta \) in the discrete range \(\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}\) a certain C*-algebra \({\mathcal {B}}\) of compact operators. We use the unitary braiding on \({\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )\) to equip the category \(\mathrm {Mod}_{{\mathcal

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes

This paper is about the dynamics of non-diffusive temperature fronts evolving by the incompressible viscous Boussinesq system in \({\mathbb {R}}^3\). We provide local in time existence results for initial data of arbitrary size. Furthermore, we show global in time propagation of regularity for small initial data in critical spaces. The developed techniques allow to consider general fronts where the

We study C*-algebras generated by left regular representations of right LCM one-relator monoids and Artin–Tits monoids of finite type. We obtain structural results concerning nuclearity, ideal structure and pure infiniteness. Moreover, we compute K-theory. Based on our K-theory results, we develop a new way of computing K-theory for certain group C*-algebras and crossed products.

We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation

Let \(\gamma \in (0,2)\), let h be the planar Gaussian free field, and consider the \(\gamma \)-Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a \(\gamma \)-LQG metric ball with respect to the Euclidean (resp. \(\gamma \)-LQG) metric is \(2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group SU(n), for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas–de la Ossa on the conifold, Pando Zayas–Tseytlin on the canonical bundle over \(\mathbb {CP}^1\times \mathbb {CP}^1\), as well as the metrics on canonical bundles over flag

This second part of the series treats spin \(\pm 2\) components (or extreme components), that satisfy the Teukolsky master equation, of the linearized gravity in the exterior of a slowly rotating Kerr black hole. For each of these two components, after performing a first-order differential operator once and twice, the resulting equations together with the Teukolsky master equation itself constitute

By definition, the exterior asymptotic energy of a solution to a wave equation on \({\mathbb {R}}^{1+N}\) is the sum of the limits as \(t\rightarrow \pm \infty \) of the energy in the the exterior \(\{|x|>|t|\}\) of the wave cone. In our previous work Duyckaerts et al. (J Eur Math Soc 14(5):1389–1454, 2012), we have proved that the exterior asymptotic energy of a solution of the linear wave equation

We consider generalizations of the BC-type relativistic Calogero–Moser–Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh–Feigin–Sergeev–Veselov type deformation of these

A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators—the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states

We study the asymptotic properties of the small data solutions of the Vlasov–Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For

We prove the Bisognano–Wichmann property for asymptotically complete Haag–Kastler theories of massless particles. These particles should either be scalar or appear as a direct sum of two opposite integer helicities, thus, e.g., photons are covered. The argument relies on a modularity condition formulated recently by one of us (VM) and on the Buchholz’ scattering theory of massless particles.

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called “pockets”. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension 2. We then apply these dynamical results

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for \(d=1\) and the beam equation for \(d\le 3\). We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new

We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used

The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group \(\Gamma \) acts on a bounded geometric space X properly, isometrically

We study a class of quantum channels arising from the representation theory of compact quantum groups that we call Temperley–Lieb quantum channels. These channels simultaneously extend those introduced by Brannan and Collins (Commun Math Phys 358(3):1007–1025, 2018), Nuwairan (Int J Math 25(6):1450048, 2014) and Lieb and Solovej (Acta Math 212(2):379–398, 2014). (Quantum) Symmetries in quantum information

In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities

This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the C-field in Cohomotopy theory implies a list of expected anomaly cancellation conditions

There exists an index theory to classify strictly local quantum cellular automata in one dimension (Fidkowski et al. in Interacting invariants for Floquet phases of fermions in two dimensions, 2017. arXiv:1703.07360; Gross et al. in Commun Math Phys 310(2):419–454, 2012; Po et al. in Phys Rev B 96: 245116, 2017). We consider two classification questions. First, we study to what extent this index theory

The \({\mathcal {B}}_{p}\)-algebras are a family of vertex operator algebras parameterized by \(p\in {\mathbb {Z}}_{\ge 2}\). They are important examples of logarithmic CFTs and appear as chiral algebras of type \((A_1, A_{2p-3})\) Argyres–Douglas theories. The first member of this series, the \({\mathcal {B}}_2\)-algebra, are the well-known symplectic bosons also often called the \(\beta \gamma \)

A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional

This work develops a rigorous setting allowing one to prove several features related to the behaviour of the Heisenberg–Ising (or XXZ) spin-1/2 chain at finite temperature T. Within the quantum inverse scattering method the physically pertinent observables at finite T, such as the per-site free energy or the correlation length, have been argued to admit integral representations whose integrands are

We show that any \(3+1\)-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein–Vlasov system. For the analysis of the Einstein equations we use the constant-mean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric \(L^2\)-estimates based on the Sasaki-metric. The resulting estimates on the energy-momentum

Unfortunately, the handling editor was wrongly published in the original online publication of this article.

In many contexts one encounters Hermitian operators M on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such M is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten p-norm for all integer

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in \({S^1{\setminus}

The necklace braid group \({\mathcal {NB}}_n\) is the motion group of the \(n+1\) component necklace link \(\mathcal {L}_n\) in Euclidean \(\mathbb {R}^3\). Here \(\mathcal {L}_n\) consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group \({\mathcal {NB}}_n\), especially

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in \(\mathrm{SL}(2,\mathbb {R})\). In the case of perfect fluids, the motion is given by geodesic flow in \(\mathrm{SL}(2,\mathbb {R})\) with the Euclidean metric, while

We derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with pair-interactions. Our method is based on Bogoliubov’s approximation, which also provides the information on the fluctuations around the condensate in terms of a norm

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the \({{\mathcal {W}}}\)-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra \({\mathfrak {s}}{\mathfrak {l}}_3\) on the three-punctured

The well-known Constantin–Lax–Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities (Constantin et al. in Commun Pure Appl Math 38:715–724, 1985). De Gregorio modified the CLM model by adding a convective term (De Gregorio in Math Methods Appl Sci 19(15):1233–1255, 1996), which is known important for fluid dynamics (Hou and

Let (M, g) be a compact Riemannian manifold of dimension n and \(P_1:=-h^2\Delta _g+V(x)-E_1\) so that \(dp_1\ne 0\) on \(p_1=0\). We assume that \(P_1\) is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators \(P_2,\dots P_n\) with \([P_i,P_j]=0\), \(i,j=1,\dots n\). We study the pointwise bounds for the joint eigenfunctions, \(u_h\)

We prove convergence to a Lévy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung and Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod \({{\mathcal {J}}}_1\) topology. Our main results yield elementary geometric conditions for convergence (i) in \({{\mathcal {M}}}_1\), (ii) in

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections

For a \(C^\infty \) map f on a compact manifold M we prove that for a Lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top Lyapunov exponent of \(\Lambda \, df:\Lambda \,TM\circlearrowleft \) at x. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii’s work [23] relating physical and Sinai-Ruelle-Bowen

Abstract For a diagram automorphism of an affine Kac–Moody algebra such that the folded diagram is still an affine Dynkin diagram, we show that the associated Drinfeld–Sokolov hierarchy also admits an induced automorphism. Then we show how to obtain the Drinfeld–Sokolov hierarchy associated to the affine Kac–Moody algebra that corresponds to the folded Dynkin diagram from the invariant sub-hierarchy

Abstract We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions (x) and centered values (h) of the extreme local maxima in lattice versions of a bounded domain \(D\subset {\mathbb {C}}\) converges, as the

Abstract We define (p, q) Hermitian geometry as the target space geometry of the two dimensional (p, q) supersymmetric sigma model. This includes generalised Kähler geometry for (2, 2), generalised hyperkähler geometry for (4, 2), strong Kähler with torsion geometry for (2, 1) and strong hyperkähler with torsion geometry for (4, 1). We provide a generalised complex geometry formulation of hermitian

Abstract We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional \(\mathcal {N}=1\) supergravity coupled to a chiral non-linear sigma model and a \(\mathrm {Spin}^{c}_0\) structure. The model involves a Lorentzian metric g on a four-manifold M, a complex chiral spinor and a map \(\varphi :M\rightarrow \mathcal {M}\) from M to a complex manifold

Abstract We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation

Abstract The KP and 2D Toda \(\tau \)-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted

We test M. Berry’s ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions (“boundary-adapted arithmetic random waves”). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the

We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘\(G_2\) mode’

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With

We consider canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of \(\mathbb {R}^2\). We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called energy–enstrophy Gaussian random distributions

We consider the weakly coupled \(\phi ^4 \) theory on \(\mathbb {Z}^4 \), in a weak magnetic field h, and at the chemical potential \(\nu _c \) for which the theory is critical if \(h=0\). We prove that, as \(h\searrow 0\), the magnetization of the model behaves as \((h\log h^{-1})^{\frac{1}{3}} \), and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to

For \(0<\gamma <2\) and \(\delta >0\), we consider the Liouville graph distance, which is the minimal number of Euclidean balls of \(\gamma \)-Liouville quantum gravity measure at most \(\delta \) whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at \(\delta \rightarrow 0\). In particular

We study, using mean curvature flow methods, \(2+1\) dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spatial slices reach infinite volume, asymptotically converge on average to de Sitter

In a previous article, we introduced the first passage set (FPS) of constant level \(-a\) of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path along which the GFF is greater than or equal to \(-a\). This description can be taken as a definition of the FPS for the metric

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media

Consider the Cauchy problem of incompressible Navier–Stokes equations in \(\mathbb {R}^3\) with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data

We show that typical [in the sense of Bonatti and Viana (Ergod Theory Dyn Syst 24(5):1295–1330, 2004) and Avila and Viana (Port Math 64:311–376, 2007)] Hölder and fiber-bunched \(\text {GL}_d(\mathbb {R})\)-valued cocycles over subshifts of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding

The author wishes to rectify some oversights and, mainly, to amend an erroneous term.