Scientists / Open questions

Shing-Tung Yau

Differential geometry; geometric analysis

1982 Fields Medal

57 questions

Fast factorization-free eigensolvers for large directed graph Laplacians: extending the iSIRA framework beyond symmetric Laplacians
Computability of the simple-homotopy distance $\ell(K,L)$ and vertex-free expansion to a simplex for complexes simple-homotopy equivalent to a point
Reversible weighted graphs: symmetric hitting times and the weighted-cycle conjecture of Chang, Xu and Yau
Energy of harmonic maps on the Kuranishi space of a compact Kähler–Einstein manifold of general type: smooth harmonic-map families, nonsmooth Kuranishi spaces, and energy asymptotics (the deferred problems of Cao–Sun–Yau–Zhang)
Automatic optimal selection of the time-dependent scaling and translating factors in Hermite–Galerkin spectral methods for evolution equations on unbounded domains
Complex structures on higher-dimensional connected sums with a compact ball quotient summand: the case left open by the Jost–Yau rigidity theorem for Hermitian harmonic maps
Rigorous validation and generalization of the adiabatic isometric mapping (AIM) algorithm for embedding two-surface metrics in Euclidean 3-space
Two-fermion spherically symmetric Einstein-Dirac solitons: general spinor ansatz, limiting configuration of the mass-energy spirals, strong-coupling stability, and rigorous global existence
Effective torus actions on exotic spheres that do not bound spin manifolds: symmetry rank and degree of symmetry
Quantum realization and mathematical classification of the non-Abelian higher-rank tensor gauge theories obtained by gauging higher-moment polynomial symmetries
Combinatorial realization of the finite-N matrix-harmonic tridiagonal matrices on the fuzzy sphere as intersection matrices of a distance-regular graph or association scheme
Computing the Vasquez invariant $n(G)$ of a finite group and the sharpness of the index-sum bound for toral fibrations of compact flat manifolds
Virtual splitting of the center of the fundamental group for complete finite-volume manifolds of nonpositive curvature without finite generation
Quantum completion and low-energy phase structure of non-Abelian gauged fracton matter field theories and their sigma models
The degree–rank conjecture for Sasaki–Einstein volumes
Closed-form enumeration of c = 2 rational toroidal conformal field theories with non-primitive discriminants, and the Gauss-product classification for higher-dimensional tori
Open problems for the nonlocal integro-differential COVID-19 model of Liu–Wang–Zhang–Yau: parameter identification from outbreak data, turning-point uniqueness, and asymptotic theory beyond Weibull delay kernels
Extending the Cheeger–Yau lower bound for the heat kernel: Hodge Laplacians on forms, generalized comparison spaces, and curvature bounds without domination
k-positive symmetrized curvature operators on compact Kähler manifolds and rigidity to complex projective space (Wang–Yang Conjecture 2.4)
Concrete experimental realization of the Lin–Yau doubled topological band structures: two nearly degenerate double bands with second Chern numbers $(m+n,-n)$ , interpenetrating insulator lattices, 3D photonic crystals, and Weyl/Bogoliubov–Weyl heterostructure signatures
Limiting distributions and critical-window behavior of the Lin–Lu–Yau Ricci curvature of Erdős–Rényi random graphs
Realizing finite subgroups of Out(π₁(M)) by isometries of a nonpositively curved metric: the Schoen–Yau problem
Sampling-density-only accuracy bounds for discrete surface approximation: extension to general triangulations, off-surface samples, and practical surface Delaunay constructions
Certified silhouette interpolation between sampled views via aspect surfaces, and extension of silhouette-topology evolution to surfaces with flat regions

Back to open problems

Fast factorization-free eigensolvers for large directed graph Laplacians: extending the iSIRA framework beyond symmetric Laplacians

Computability of the simple-homotopy distance ℓ(K,L)\ell(K,L)ℓ(K,L) and vertex-free expansion to a simplex for complexes simple-homotopy equivalent to a point

Reversible weighted graphs: symmetric hitting times and the weighted-cycle conjecture of Chang, Xu and Yau

Energy of harmonic maps on the Kuranishi space of a compact Kähler–Einstein manifold of general type: smooth harmonic-map families, nonsmooth Kuranishi spaces, and energy asymptotics (the deferred problems of Cao–Sun–Yau–Zhang)

Automatic optimal selection of the time-dependent scaling and translating factors in Hermite–Galerkin spectral methods for evolution equations on unbounded domains

Complex structures on higher-dimensional connected sums with a compact ball quotient summand: the case left open by the Jost–Yau rigidity theorem for Hermitian harmonic maps

Rigorous validation and generalization of the adiabatic isometric mapping (AIM) algorithm for embedding two-surface metrics in Euclidean 3-space

Two-fermion spherically symmetric Einstein-Dirac solitons: general spinor ansatz, limiting configuration of the mass-energy spirals, strong-coupling stability, and rigorous global existence

Effective torus actions on exotic spheres that do not bound spin manifolds: symmetry rank and degree of symmetry

Quantum realization and mathematical classification of the non-Abelian higher-rank tensor gauge theories obtained by gauging higher-moment polynomial symmetries

Combinatorial realization of the finite-N matrix-harmonic tridiagonal matrices on the fuzzy sphere as intersection matrices of a distance-regular graph or association scheme

Computing the Vasquez invariant n(G)n(G)n(G) of a finite group and the sharpness of the index-sum bound for toral fibrations of compact flat manifolds

Virtual splitting of the center of the fundamental group for complete finite-volume manifolds of nonpositive curvature without finite generation

Quantum completion and low-energy phase structure of non-Abelian gauged fracton matter field theories and their sigma models

The degree–rank conjecture for Sasaki–Einstein volumes

Closed-form enumeration of c = 2 rational toroidal conformal field theories with non-primitive discriminants, and the Gauss-product classification for higher-dimensional tori

Open problems for the nonlocal integro-differential COVID-19 model of Liu–Wang–Zhang–Yau: parameter identification from outbreak data, turning-point uniqueness, and asymptotic theory beyond Weibull delay kernels

Extending the Cheeger–Yau lower bound for the heat kernel: Hodge Laplacians on forms, generalized comparison spaces, and curvature bounds without domination

k-positive symmetrized curvature operators on compact Kähler manifolds and rigidity to complex projective space (Wang–Yang Conjecture 2.4)

Concrete experimental realization of the Lin–Yau doubled topological band structures: two nearly degenerate double bands with second Chern numbers (m+n,−n)(m+n,-n)(m+n,−n), interpenetrating insulator lattices, 3D photonic crystals, and Weyl/Bogoliubov–Weyl heterostructure signatures

Limiting distributions and critical-window behavior of the Lin–Lu–Yau Ricci curvature of Erdős–Rényi random graphs

Realizing finite subgroups of Out(π₁(M)) by isometries of a nonpositively curved metric: the Schoen–Yau problem

Sampling-density-only accuracy bounds for discrete surface approximation: extension to general triangulations, off-surface samples, and practical surface Delaunay constructions

Certified silhouette interpolation between sampled views via aspect surfaces, and extension of silhouette-topology evolution to surfaces with flat regions

Computability of the simple-homotopy distance $\ell(K,L)$ and vertex-free expansion to a simplex for complexes simple-homotopy equivalent to a point

Computing the Vasquez invariant $n(G)$ of a finite group and the sharpness of the index-sum bound for toral fibrations of compact flat manifolds

Concrete experimental realization of the Lin–Yau doubled topological band structures: two nearly degenerate double bands with second Chern numbers $(m+n,-n)$ , interpenetrating insulator lattices, 3D photonic crystals, and Weyl/Bogoliubov–Weyl heterostructure signatures