Inventiones mathematicae最新文献

We deal, for the classical (N)-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions.

In this paper, we first generalize the work of Bourgain (Geom. Funct. Anal. 1(4):321–374, 1991) and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain’s condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain’s condition, they satisfy the same (L^{p}) bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain’s condition fails, then the (L^{infty } to L^{q}) boundedness always fails for some (q= q(n) > frac{2n}{n-1}), extending Bourgain’s three-dimensional result (Geom. Funct. Anal. 1(4):321–374, 1991). On the other hand, if Bourgain’s condition holds, then we prove (L^{p}) bounds for Hörmander-type oscillatory integral operators for a range of (p) that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl (A note on Fourier restriction and nested polynomial wolff axioms, 2020, arXiv:2010.02251). This gives new progress on the Fourier restriction problem, the Bochner-Riesz problem on (mathbb{R}^{n}), the Bochner-Riesz problem on spheres (S^{n}), etc.

Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and (2times 2) canonical systems). We prove a number of Szegő-type theorems for generalized indefinite strings and related spectral problems (including Krein strings, canonical systems and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert–Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa–Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform.

We prove that the ((2,1))-cable of the figure-eight knot is not smoothly slice by showing that its branched double cover bounds no equivariant homology ball.

We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to establish a number of results that reveal a form of irreversibility in the Euler equations governing the motion of an incompressible and inviscid fluid. In particular, we show that nearby general stable steady states (i) all fluid flows exhibit indefinite twisting (ii) vorticity generically exhibits gradient growth and wandering. We also give examples of infinite time gradient growth for smooth solutions to the SQG equation and of smooth vortex patches that entangle and develop unbounded perimeter in infinite time.

In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians.

We prove a sharp (up to (C_{varepsilon }R^{varepsilon })) (L^{7}) square function estimate for the moment curve in (mathbb{R}^{3}).

For (dgeq 3), we study the Ising model on (mathbb{Z}^{d}) with random field given by ({epsilon h_{v}: vin mathbb{Z}^{d}}) where (h_{v})’s are independent normal variables with mean 0 and variance 1. We show that for any (T < T_{c}) (here (T_{c}) is the critical temperature without disorder), long range order exists as long as (epsilon ) is sufficiently small depending on (T). Our work extends previous results of Imbrie (1985) and Bricmont–Kupiainen (1988) from the very low temperature regime to the entire low temperature regime.

For various 2-Calabi–Yau categories (mathscr{C}) for which the classical stack of objects (mathfrak{M}) has a good moduli space (pcolon mathfrak{M}rightarrow mathcal{M}), we establish purity of the mixed Hodge module complex (p_{!}underline{{mathbb{Q}}}_{{mathfrak {M}}}). We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism (p) is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Via the integrality theorem in cohomological Donaldson–Thomas theory we then prove purity of (p_{!}underline{{mathbb{Q}}}_{{mathfrak {M}}}). It follows that the Beilinson–Bernstein–Deligne–Gabber decomposition theorem for the constant sheaf holds for the morphism (p), despite the possibly singular and stacky nature of ({mathfrak {M}}), and the fact that (p) is not proper. We use this to define cuspidal cohomology for ({mathfrak {M}}), which conjecturally provides a complete space of generators for the BPS algebra associated to (mathscr{C}). We prove purity of the Borel–Moore homology of the moduli stack (mathfrak{M}), provided its good moduli space ℳ is projective, or admits a suitable contracting ({mathbb{C}}^{*})-action. In particular, when (mathfrak{M}) is the moduli stack of Gieseker semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. Without the usual assumption that (r) and (d) are coprime, we prove that the Borel–Moore homology of the stack of semistable degree (d) rank (r) Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

This article is dedicated to the asymptotic geometry of wreath products (Fwr H := left ( bigoplus _{H} F right ) rtimes H) where (F) is a finite group and (H) is a finitely generated group. Our first main result says that a coarse map from a finitely presented one-ended group to (Fwr H) must land at bounded distance from a left coset of (H). Our second main result, building on the later, is a very restrictive description of quasi-isometries between two lamplighter groups on finitely presented one-ended groups. Third, we obtain a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups (F_{1}), (F_{2}) and two finitely presented one-ended groups (H_{1}), (H_{2}), we show that (F_{1} wr H_{1}) and (F_{2} wr H_{2}) are quasi-isometric if and only if either (i) (H_{1}), (H_{2}) are non-amenable quasi-isometric groups and (|F_{1}|), (|F_{2}|) have the same prime divisors, or (ii) (H_{1}), (H_{2}) are amenable, (|F_{1}|=k^{n_{1}}) and (|F_{2}|=k^{n_{2}}) for some (k,n_{1},n_{2} geq 1), and there exists a quasi-((n_{2}/n_{1}))-to-one quasi-isometry (H_{1} to H_{2}). This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of (H=mathbb{Z}). Our approach is however fundamentally different, as it crucially exploits the assumption that (H) is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces.