For each partition \(\underline{p}\) of an integer \(N\ge 2\), consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the \(\underline{p}\)-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction

We explain how dispersionless integrable hierarchy in 2d topological field theory arises from the Kodaira–Spencer gravity (BCOV theory). The infinitely many commuting Hamiltonians are given by the current observables associated to the infinite abelian symmetries of the Kodaira–Spencer gravity. We describe a BV framework of effective field theories that leads to the B-model interpretation of dispersionless

We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on Kerr black hole spacetimes, both in the extremal and subextremal case. We also give a quantitative refinement of mode stability. As an immediate application, we show that the transmission and reflection coefficients of the scattering problem are bounded, independently of the specific angular

Let (M, g) be a closed Riemannian manifold. The RG-2 flow is defined by $$\begin{aligned} \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm {Ric}(t) \, -\, \frac{\alpha }{2} \mathrm {Rm}^2(t), \end{aligned}$$ where \( g = \mathrm {Riemannian \ metric}, \mathrm {Ric} = \mathrm {Ricci \ curvature, } \ \mathrm {Rm}^2_{ij}:=\mathrm {R}_{irmk}\mathrm {R}_j^{rmk},\) and \(\alpha \ge 0\) is a parameter

This article is the second part of our study of the spectrum \(\sigma (L_n;\tau )\) of the Lamé operator $$\begin{aligned} L_n=\frac{d^2}{dx^2}-n(n+1)\wp ( x+z_0;\tau )\quad \text {in}\;\;L^2(\mathbb {R}, \mathbb {C}), \end{aligned}$$ where \(n\in \mathbb {N}\), \(\wp (z;\tau )\) is the Weierstrass elliptic function with periods 1 and \(\tau \), and \(z_0\in \mathbb {C}\) is chosen such that \(L_n\)

Gersho’s conjecture in 3D asserts the asymptotic periodicity and structure of the optimal centroidal Voronoi tessellation. This relatively simple crystallization problem remains to date open. We prove bounds on the geometric complexity of optimal centroidal Voronoi tessellations as the number of generators tends to infinity. Combined with an approach of Gruber in 2D, these bounds reduce the resolution

We study defect modes in a one-dimensional periodic medium perturbed by an adiabatic dislocation of amplitude \(\delta \ll 1\). If the periodic background admits a Dirac point—a linear crossing of dispersion curves—then the dislocated operator acquires a gap in its essential spectrum. For this model (and its honeycomb analog) Fefferman et al. (Proc Natl Acad Sci USA 111(24):8759–8763, 2014, Mem Am

For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem. This vector equilibrium problem is defined for three measures with an upper constraint on

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are obtained, and applications to enumeration of ribbon graphs

We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on ‘tensor hierarchies’, which describe towers of p-form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low

Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects

We study a class of close-packed dimer models on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point, and the dimer model with plaquette interaction previously analyzed in previous works. By tuning the edge weights, we can impose a non-zero average tilt for the height function

We introduce a new family of Poisson vertex algebras \(\mathcal {W}(\mathfrak {a})\) analogous to the classical \(\mathcal {W}\)-algebras. The algebra \(\mathcal {W}(\mathfrak {a})\) is associated with the centralizer \(\mathfrak {a}\) of an arbitrary nilpotent element in \(\mathfrak {gl}_N\). We show that \(\mathcal {W}(\mathfrak {a})\) is an algebra of polynomials in infinitely many variables and

Continuing recent efforts in extending the classical singularity theorems of General Relativity to low regularity metrics, we give a complete proof of both the Hawking and the Penrose singularity theorem for \(C^1\)-Lorentzian metrics—a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. The proofs make use of careful estimates of the curvature of approximating

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special

We correct an error in the Appendix of [1].

We study the stability of the Couette flow \((y,0,0)^T\) in the presence of a uniform magnetic field \(\alpha (\sigma , 0, 1)\) on \({{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}\) using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit \(\mathbf{Re} ^{-1}\), \(\mathbf{R }_m^{-1} \ll 1\) and prove that for strong and suitably oriented

This article is devoted to the study of the Hele–Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the

We study the Yang–Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that

We study the quantum mechanical many-body problem of \(N \ge 1\) non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and \(K \ge 0\) static nuclei. We model the dynamics of the electrons and their self-generated electromagnetic field with the so-called many-body Maxwell–Pauli equations. Here we construct time global, finite-energy, weak solutions

We consider the asymmetric simple exclusion process (ASEP) on \({\mathbb {Z}}\) with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t)

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