19 problems

The Weil–Petersson (WP) metric is an incomplete Riemannian metric on the moduli space of Riemann surfaces, and WP geodesic rays have ending laminations. Ending-lamination parametri…

Let $M$ be a Weil–Petersson variety, and let $X$ be a Weil–Petersson subvariety of dimension $q$ in $M$. Let $c_k(\omega_{WP})$ denote the $k$-th elementary polynomial of the curva…

Let $M$ be a Weil–Petersson variety equipped with its Weil–Petersson metric and Hodge metric. The volumes of these two metrics are finite, and the Weil–Petersson varieties need not…

Let $P(\Theta,R^{\dagger})$ be the space of circle patterns with prescribed intersection angles $\Theta$ and reference radii $R^{\dagger}$, and let $\mathcal{T}_{\mathrm{WP}}$ deno…

For a hyperbolic surface, let $\lambda_1$ be the first nonzero eigenvalue of its Laplace–Beltrami operator. Equip the moduli space of genus-$g$ surfaces with Weil–Petersson probabi…

The DSSYK matrix integral is studied in the conventional 't Hooft limit, with $\lambda$ denoting its double-scaling parameter. Okuyama's conjecture concerns the pruned matrix corre…

Let the moduli space of a Calabi–Yau threefold be equipped with its Weil–Petersson geometry, and consider its geodesics. Stability conjecture. All Weil–Petersson geodesics on the m…

Conjectural Weil–Petersson volume current formula. For all $d$, $\omega_{g,n,d}(a)$ is the current defined by integration of

Let $\mathbb{P}_g^{\mathrm{\scriptsize{WP}}}$ denote the Weil–Petersson probability measure on compact hyperbolic surfaces of genus $g$, and let $\lambda_1>0$ be the smallest non-z…

Weil–Petersson random prime geodesic conjecture. Theorem 1.1 still holds after replacing the exponent $3/4$ by $1/2$; equivalently, there is a universal constant $c>0$ such that fo…

Let $M$ be a convex co-compact hyperbolic $3$-manifold, and let $cphi$ be the Schwarzian quadratic differential for the projective boundary of $M$. Write $cpartial_c M$ for the bou…

Let $\lambda(t,\cdot)$ be a continuous vector field of Sobolev class $H^{\frac{3}{2}}$ on the unit circle $S^1$, and let $g(t,\cdot)=\eta(t,\cdot)$ be the flow of … The flow maps a…

Let $X$ be a Fano variety, and let $C$ be the base of a test configuration with central point $0$ and punctured base $C^{o}$. The Weil–Petersson distance between $C^{o}$ and $0$ is…

Let $X$ be a projective Calabi–Yau $n$-fold, let $\mathcal{M}_{\mathrm{Kah}}(X)$ be its stringy Kähler moduli space, and let $\mathcal{D}_X=\mathrm{D^bCoh}(X)$. Let…

Let ${\mathcal{Q}}(S)$ be the space of quadratic differentials on the once-punctured torus, and let $\mu$ be the $\Phi^t_{\mathcal{T}}$-invariant Borel probability measure on…

A Weil–Petersson (WP) geodesic in the moduli space has end invariants, together with associated subsurface coefficients. Symbolic coding conjecture. End invariants and the associat…

Short Curve conjecture. If $g$ is a Weil–Petersson geodesic with end invariant $(\nu^{-},\nu^{+})$, then:

Let $V$ be a Calabi–Yau threefold and consider the Weil–Petersson metric on its complex moduli space near points at infinity, in particular near a large complex structure limit poi…

Weil's conjecture. The resulting Hermitian inner product on the Teichmüller space should be Kähler and have negative curvature.