The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^2$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a ``Big Bang" singularity at $\{t = 0\}$. We place initial data for the linearized system along $\{t = 1\}\simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t\downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lemaître-Robertson-Walker (FLRW) solution. In particular, we show that as $t\downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\{t = 0\}$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction $t\downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t\downarrow 0$ under small perturbations of its data at $\{t = 1\}$.

Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi--Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified ``no exotics" type theorem for cohomological Donaldson--Thomas invariants, which we discuss and prove in the appendix.

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\ge 3$, and let $\varphi_\lambda = \Delta_M\varphi_\lambda + \lambda\varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that$$H^{n-1}(\{\varphi_\lambda = 0\}) \le C\lambda^\alpha,$$where $\alpha > 1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$. This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \ge 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that$$ H^{n-1}(\{u=0\} \cap B) \ge c.$$We prove Nadirashvili's conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$, there exists $c>0$ such that for any Laplace eigenfunction $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$ the following inequality holds: $c\sqrt{\lambda} \le H^{n-1}(\{\varphi_\lambda = 0\})$.

We present a new curvature condition that is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere.

Our result is sharp in many interesting situations. For example, the curvature tensors of $\mathbb{C}\mathbb{P}^{n/2}$, $\mathbb{H}\mathbb{P}^{n/4}$, $S^{n-k} × S^k(2\le k\le n-2)$, $S^{n-2} × \mathbb{H}^2$, all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface that is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} × \mathbb{R}$ lies in the interior of our curvature cone.