We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for κ-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity.

A forced solution v of the axially symmetric Navier-Stokes equation in a finite cylinder D with suitable boundary condition is constructed. The forcing term, whose order of scaling is slightly worse than the critical order −2, is in the mildly super critical space L^q_t L¹_x for all q>1. The velocity, which is smooth until its final blow up moment, is in the energy space throughout.

In this paper, we establish a decay estimate for the discrete four-order Schrödinger equation on the hexagonal triangulation with γ=0. The proof is based on the uniform estimates of oscillatory integrals, as developed by Karpushkin, along with a key result by Varchenko. Our result is to show the l¹→l^∞ dispersive decay rate is ⟨t⟩^−σ for any 0<σ< 1/2. Additionally, we provide estimates for the inhomogeneous discrete fourth-order Schrödinger equation with γ=0.

Degenerate complex Monge-Ampère equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and W^3,p estimate for a class of degenerate complex Monge-Ampère equations.

In this paper, we give an exposition of our recent work on nonlinear potential theory in conformal geometry. We apply nonlinear potential theory to study p-Laplace equations arising from conformal geometry and, in particular, the problems related to the asymptotic behavior near and the size of singularities in conformal geometry.

We survey basic properties of the geometric flow for immersions within a Hamiltonian isotopy class and propose a definition for Type I singularities.

We show that by applying a set of existing analytical arguments, a more robust effective uniqueness result for blowups can be obtained, with multiple implications following therefrom.

In this paper, we investigate the one-dimensional parabolic equation. Using blow-up analysis, we prove that the zero set at each time slice is discrete and that the number of zeros decreases with respect to time. Moreover, it strictly declines at the singularity. This provides a new and simplified proof of the main results in existing literature, where the original proof relies on spectral theory. Besides, our results repre-sent a localized version of those existing results with weaker conditions.

Fukaya and Yamaguchi conjectured that if Mⁿ is a manifold with nonnega-tive sectional curvature, then the fundamental group is uniformly virtually abelian. In this short note we observe that the conjecture holds in dimensions up to four.

One of the significant motivations for studying the Kähler-Ricci flow is its relation to the Analytic Minimal Model Program as initiated by Gang Tian. Carrying out this classification program requires careful analysis of the flow metric, particularly when it encounters singularities. In this note, we survey some results pertaining to the limit for the Kähler-Ricci flow.

In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.

Let E be a smooth cubic in the projective plane P². Nobuyoshi Takahashi formulated a conjecture that expresses counts of rational curves of varying degree in P² \E as the Taylor coefficients of a particular period integral of a pencil of affine plane cubics after reparametrizing the pencil using the exponential of a second period inte-gral.

The intrinsic mirror construction introduced by Mark Gross and the third author as-sociates to a degeneration of (P²,E) a canonical wall structure from which one con-structs a family of projective plane cubics that is birational to Takahashi’s pencil in its reparametrized form. By computing the period integral of the positive real locus explicitly, we find that it equals the logarithm of the product of all asymptotic wall functions. The coefficients of these asymptotic wall functions are logarithmic Gromov-Witten counts of the central fiber of the degeneration that agree with the algebraic curve counts in (P²,E) in question. We conclude that Takahashi’s conjecture is a natu-ral consequence of intrinsic mirror symmetry. Our method generalizes to give similar results for log Calabi-Yau varieties of arbitrary dimension.