We compute a momentum space version of the entanglement spectrum and entanglement entropy of general Young tableau states, and one-point functions on Young tableau states. These physical quantities are used to measure the topology of the dual spacetime geometries in the context of gauge/gravity correspondence. The idea that Young tableau states can be obtained by superposing coherent states is explicitly verified. In this quantum superposition, a topologically distinct geometry is produced by superposing states dual to geometries with a trivial topology. Furthermore we have a refined bound for the overlap between coherent states and the rectangular Young tableau state, by using the techniques of symmetric groups and representations. This bound is exponentially suppressed by the total edge length of the Young tableau. It is also found that the norm squared of the overlaps is bounded above by inverse powers of the exponential of the entanglement entropies. We also compute the overlaps between Young tableau states and other states including squeezed states and multi-mode entangled states which have similarities with those appeared in quantum information theory.

Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kahler. We prove that if |Rm(g(t))| is bounded by a/t for some a > 0, then g(t) is Kahler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kahler-Ricci flow on a non-compact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and |Rm(g(t))| ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to C^n; and (ii) there is ε(n) > 0 depending only on n such that if (M^n,g_0) is a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+ε(n))^{−1}h ≤ g_0 ≤ (1+ε(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C^n.

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.

In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main tech- nical tools are local-in-space regularity estimates near the initial time, which are of independent interest.