arXiv subjects
Publications and source records attributed to Shing-Tung Yau.
We classify compact Hermitian manifolds with flat Levi-Civita connection. This is equivalent to the classification of all orthogonal complex structures on flat tori. It generalized the work of Khan, Yang, and Zheng in 2017 where they solved the case in complex dimension three.
We identify a boundary-geometry-driven mechanism for the dynamical formation of marginally outer trapped surfaces (MOTSs) in vacuum general relativity. Mild three-dimensional anisotropies evolve inside a compact Cauchy domain whose effective isotropic thickness remains controlled, while the generalized boundary mean curvature can increase during either contracting or expanding boundary evolution. This drives the boundary across Yau's geometric threshold, forcing MOTS formation from initially untrapped data. We further interpret the characteristic shear construction of Ref.~\cite{MondalYau2026} as the null manifestation of the same anisotropic vacuum dynamics. The result provides a purely vacuum physical realization of MOTS formation through global geometric effects, without invoking a short-pulse concentration mechanism for gravitational radiation.
Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem \[ \kappa_{\min}^{\mathrm{LLY}}(G)>\frac12 \quad\Longrightarrow\quad \PathH_1(G;\R)=0. \] Equivalently, nonzero first GLMY path homology forces an edge of Lin--Lu--Yau curvature at most $1/2$. The threshold $1/2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin--Lu--Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and $\kappa_{\min}^{\mathrm{LLY}}(G)>0$, then $\pi_1(\Xshort{5}(G),o)$ is finite, where $\Xshort{5}(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in $\pi_1^{\mathrm{GLMY}}(G,o)$. In higher degrees the situation is different: for each integer $r\geq1$, the Cartesian product $T_r=C_5^{\square r}$ has curvature $1/(2r)$ on every edge and, for every field $\F$, \[ \PathH_p(T_r;\F)\cong\F^{\binom rp}\qquad(0\leq p\leq r), \] so strict positivity of Lin--Lu--Yau curvature does not force higher-dimensional GLMY path homology to vanish.
Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius $R\to\infty$ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as $N\to\infty$ flattens the remaining curvature, yielding \[ \mathrm{SMG}\xrightarrow{\;R\to\infty\;}\mathrm{IG}\xrightarrow{\;N\to\infty\;}\mathrm{CS}. \] The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.
This paper establishes the global proof of the Second Edge Theorem: as sample size tends to infinity, sample-dependent information-geometric manifolds---formed by the parameter space, sample-scaled Fisher metric, and dual alpha-connections---undergo metric-topological collapse onto the flat tangent space of Conventional Statistics at the true parameter. We first prove a universal tensor valence scaling law, under which tensor fields of valence one through four degenerate at rates determined by sample size. Score fluctuations stabilize, the Fisher metric freezes to its true-parameter value, affine connections dissolve, and Riemann curvature is annihilated. We then bridge geometric collapse with statistical decision theory, showing that Cheeger--Gromov flattening and Le Cam risk condensation are dual projections of the same asymptotic phase transition. A Fisher-compatible Ehresmann connection extends these results to over-parameterized and singular models, yielding horizontal leaf-space collapse and uniform local asymptotic normality. Unifying the First and Second Edge Theorems yields a nested dual-edge hierarchy: Conventional Statistics is the boundary of Information Geometry, which is itself the boundary of Statistical Mechanics and Geometry. Thus, Conventional Statistics is not a heuristic approximation but the unique zero-curvature thermodynamic attractor of regular parametric information manifolds. This redefines modern statistics as a dynamic non-equilibrium field theory of finite-sample fluctuations, phase transitions, and gauge-invariant interactions.
Large-scale formalization of advanced mathematics requires more than translating individual statements: it must reconstruct a coherent theory distributed across heterogeneous sources. This process raises four challenges: discovering implicit dependencies, correcting source defects, preserving semantic fidelity, and reconciling cross-source misalignments. We present FormaTheoria, an end-to-end, AI-assisted workflow that coordinates source acquisition, formalization, proof construction, recursive dependency discovery, independent review, and reconciliation, while preserving provenance and protecting approved declarations. A shared agent framework supports long-horizon execution through tool use, context compaction, review-gated termination, section-level source context, and dependency-aware batch parallelization. Applying FormaTheoria to major components of the Classification of Finite Simple Groups (CFSG), we construct a machine-checked Lean development extending through the Bender--Suzuki theorem and encompassing the Feit--Thompson Odd Order Theorem, Glauberman's $Z^*$ theorem, and the Brauer--Suzuki theorem. This development verifies an extensive body of deeply interdependent finite-group theory while providing a foundation for continuing the CFSG formalization. An empirical analysis of the code and recorded construction process supports the practical relevance of the identified challenges and illustrates the roles of the corresponding workflow components. Together, these results demonstrate how AI-assisted workflows can reconstruct mathematically significant formal theories from distributed literature by combining language-model agents with formal verification, structured review, and explicit dependency management.
We establish a trapped surface formation theorem for the spherically symmetric Einstein Yang Mills equations in a double-null gauge. The theorem concerns characteristic initial data posed on a pair of transversely intersecting null hypersurfaces and allows nontrivial incoming data. The proof extends the singular characteristic method of An and Lim for the Einstein Maxwell Charged Scalar Field System to the non-abelian Yang Mills setting, where the curvature coupling and gauge field nonlinearities introduce new structural difficulties. This paper constitutes the first part of a program toward weak cosmic censorship for the Einstein Yang Mills system.
We prove a sharp volume-stability theorem for closed hyperbolic three-manifolds. Let \((M,h)\) be closed hyperbolic with \(\operatorname{Ric}_h=-2h\), and let \(g_i\) be smooth metrics on \(M\) satisfying $R(g_i)\geq -6, \operatorname{Vol}_{g_i}(M)\longrightarrow \operatorname{Vol}_h(M)$. After passing to a subsequence, there exist \(Z_i\subset M\), smooth domains \(K_i\subset M\), and diffeomorphisms $\psi_i:K_i\longrightarrow M\setminus Z_i $ such that $\operatorname{Vol}_{g_i}(Z_i)\longrightarrow0, \operatorname{Vol}_h(M\setminus K_i)\longrightarrow0, $ and $ \|\psi_i^*g_i-h\|_{C^0(K_i,h)}\longrightarrow0. $ Thus near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric outside regions of vanishing volume. As an application, we establish stability of the Fischer--Moncrief reduced Hamiltonian at the Lorentz-cone ground state: after CMC normalization, near-minimizing compact vacuum data in the hyperbolic topological class converge, modulo sets of vanishing volume, to the hyperbolic Lorentz-cone geometry in tensorial \(C^0\). This provides a rigorous volume-dominance formulation of the Fischer--Moncrief asymptotic picture.
The explosive growth of complex data has catalyzed the emergence of graph statistics as a fundamentally new discipline in data science. Unlike traditional statistics, which operates primarily within the comfortable confines of Euclidean spaces, graph statistics confronts the reality that modern data naturally organize themselves as dynamic networks composed of complex interconnections. In this article, we present an overview of graph statistics as an emerging discipline, tracing its theoretical foundations, methodological innovations, and transformative applications. We examine how the integration of evolutionary game theory, ecological niche theory, topological data analysis, and graph theory through quasi-dynamic nonlinear modeling has created a new norm of statistical thinking capable of analyzing non-Euclidean data. We show how graph statistics is poised to revolutionize fields ranging from quantitative genetics and systems biology to materials science and artificial intelligence, offering a principled framework for transforming big data into practical knowledge.
Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models, such as large language transformers and biological sequence networks. With near-infinite unconstrained internal degrees of freedom, their optimization landscapes develop flat vertical gauge valleys, rendering classical generalization metrics vacuous and inducing severe pathologies, specifically generative hallucination and catastrophic forgetting. We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting deterministic parametric models into infinite-dimensional non-parametric Orlicz statistical manifolds. Modeling the total state space as a differential fiber bundle ($\mathcal{M}, \mathcal{B}, \pi, \mathcal{V}, \mathcal{H}, \omega$), we establish a Two-Fold Inference Paradigm. We formalize an Ehresmann connection 1-form $\omega$ as a dynamic geometric filter that strips away vertical gauge noise (Structural Internal Directions, or SID) and isolates learning trajectories along the strictly non-degenerate horizontal distribution (Statistical Variational Directions, or SVD$\chi$). We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable quotient base manifold $\mathcal{B}$, establishing a hard geometric containment of generative hallucinations. By projecting downstream updates onto the orthogonal complement of the historical horizontal carriage, we formalize the SMG Sequential Adaptation Flow, proving the total non-asymptotic elimination of catastrophic forgetting. SMG replaces empirical fine-tuning heuristics with coordinate-free topological constraints, bridging advanced differential geometry with structural reliability in AI.
The rapid scaling of over-parameterized machine learning architectures, particularly LLMs, raises a profound crisis: do these systems exhibit genuine intelligence, or are they merely sophisticated statistical pattern matchers? Classical flat Euclidean statistics cannot differentiate continuous interpolation from the autonomous discovery of novel causal laws. To resolve this, we introduce Statistically Meaningful Geometry (SMG), a framework modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. We prove that under persistent out-of-distribution (OOD) stimuli governed by unmodeled causal mechanisms, continuous optimization fails. Unmodeled variance is rejected by the visible horizontal base manifold, leaking into the unobservable vertical fiber space and generating an accumulation of Active Acausal Tension. Driven by the statistical manifold's non-linear curvature, this tension inevitably strikes a conjugate focal boundary ($T_{\text{crit}} = \pi^2 / K_{\text{max}}$), triggering localized volumetric collapse and a catastrophic matrix singularity ($[G_f]^{-1} \to \infty$). We demonstrate this geometric breakdown acts as the strict non-equilibrium trigger for a Gauge Symmetry Break (GSB). The system purges hidden tension from unobservable gauge redundancies, spontaneously crystallizing a new, mathematically independent horizontal coordinate axis. This non-parametric phase transition registers as a discrete $+1.0$ integer step-jump in observable Structural G-Entropy. By decoupling parameter charts and subjecting emergent axes to a Minimal Energy Path Criterion and a Causal Invariance Filter, we distinguish genuine discovery from malignant hallucinations. Ultimately, SMG provides a parameter-free, falsifiable dashboard to mathematically certify true intelligence, transforming AI for Science into an engine of autonomous paradigm shifts.
Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,ω_g)$. We show that for any real number $μ>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence $\{h_m\}$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{m+1}}\right) =(λ_E-μ)h_{m+1}+μh_m, $$ and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{\infty}}\right) =λ_Eh_\infty, $$ where $λ_E\in \mathbb R$ is the stability constant. A key feature of this proof is that it is independent of Donaldson's variational framework and applies to non-Kähler manifolds.
Classical nonabelian Hodge theory identifies Dolbeault and de Rham moduli spaces by providing a real-analytic isomorphism. In this paper, motivated by the Kapustin--Witten theory, we study this correspondence in the more general framework of perfect complexes on proper varieties, paying special attention to the surface case. We establish a Lagrangian correspondence which relates the shifted symplectic geometries by Pantev--To\"en--Vaqui\'e--Vezzosi (PTVV) between the derived stacks of flat and Higgs perfect complexes. We investigate the existence of derived twistor structures of hyperk\"ahler type on the moduli stack of perfect complexes endowed with $\lambda$-connections by Deligne--Hitchin--Simpson. We establish a version of the AKSZ/PTVV transgression, Lagrangian intersection, and (hyperk\"ahler) symplectic reduction theorems in this context. Moreover, we prove that the derived Riemann--Hilbert correspondence of Porta and Holstein--Porta, which states an equivalence of derived analytic stacks of perfect complexes on $X_{\mathrm{Betti}}$ and $X_{\mathrm{DR}}$, is compatible with the natural shifted--symplectic structures. We then study the relation between the shifted (pre-)twistor structures and the shifted symplectic forms on the fibers, and prove that the analytic Deligne--Hitchin--Simpson moduli stack on a smooth projective variety $X$ has a canonical $2(1-\dim X)$ shifted pretwistor structure over $\mathbb{P}^1_{\mathbb{C}}$, a result which has been anticipated for some time. In particular, the moduli stack of solutions to the Kapustin--Witten equations modulo gauge equivalence on a smooth proper complex algebraic surface exibits a $(-2)$-shifted (pre)twistor structure as a family over $\mathbb{P}^1_{\mathbb{C}}$.
We construct a sheaf theoretic and derived geometric machinery to study nonlinear partial differential equations and their singular supports. We establish a notion of derived microlocalization for solution spaces of non-linear equations and develop a formalism to pose and solve singular non-linear Cauchy problems globally. Using this approach we estimate the domains of propagation for the solutions of non-linear systems. It is achieved by exploiting the fact that one may greatly enrich and simplify the study of derived non-linear PDEs over a space $X$ by studying its derived linearization which is a module over the sheaf of functions on the $S^1$-equivariant derived loop stack $\mathcal{L}X$.
In this paper, we introduce a broad class of flows, including the prescribed Hermitian-Yang-Mills flow: $$\frac{\partial h}{\partial t}=-\Lambda_{\omega_g}\left(\sqrt{-1} R^h\right)+P$$ where $P\in\Gamma(M,E^*\otimes\bar{E}^*)$ is a prescribed Hermitian tensor associated with a holomorphic vector bundle $E$ over a K\"ahler (or Hermitian) manifold $(M,\omega_g)$. We establish the long-time convergence of the flow to a limiting metric $h_{\infty}$ and use it to solve the prescribed Hermitian-Yang-Mills tensor equation $$\Lambda_{\omega_g}\left(\sqrt{-1} R^{h_\infty}\right)=P, $$ for a general class of prescribed Hermitian tensors $P$. The crucial uniform $C^0$-estimate of $\{h(t)\}$ along the flow is obtained via a parabolic comparison principle.
We prove an analogue of the classical Donaldson-Uhlenbeck-Yau theorem by using the prescribed Hermitian-Yang-Mills flow. Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(M,ω_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{ω_g}(F)<deg_{ω_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in Γ(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -Λ_{ω_g}\left(\sqrt{-1}\, R^h\right) + P, $$ admits a global smooth solution on $[0,\infty)$. Moreover, as $t\rightarrow\infty$, the flow converges smoothly to a Hermitian metric $h_\infty$ on $E$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $ω$ and any positive-definite Hermitian tensor $P\inΓ(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ Λ_ω\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.
We prove the dynamical formation of a marginally outer trapped surface in pure vacuum spacetime from smooth asymptotically flat Cauchy data which initially contain no MOTS. The mechanism is a boundary effect rather than a collapse mechanism. We work in a Cauchy--double-null framework and use Yau's boundary criterion \cite{yau}, which gives the existence of an interior MOTS from a lower bound for the generalized boundary mean curvature relative to the Schoen--Yau radius of the domain. We construct an explicit class of vacuum initial data for which this criterion is strictly subcritical on the initial hypersurface, while the Einstein evolution drives the same domain into the supercritical regime. More precisely, a mild incoming gravitational radiation field increases the generalized boundary mean curvature of an isotropically large interior region sufficiently to force the formation of a MOTS in its future development. A characteristic feature of the initial data is a large interior anisotropic curvature component: the trace-free Ricci curvature is of larger order than the scalar curvature, which is balanced at the vacuum constraint scale. Thus the MOTS forms not from matter concentration or standard gravitational collapse, but from the interaction between boundary geometry, large-scale interior geometry, and the vacuum Einstein dynamics. This gives a rigorous realization of a long-suspected physical idea that apparent horizons may form from global geometric effects in vacuum general relativity.
In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact Kähler manifold $(M,ω_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such that the Hermitian-Yang-Mills tensor $ Λ_{ω_g}\left(\sqrt{-1} R^{h_0}\right) $ is positive-definite. Then for any positive-definite Hermitian tensor $ P\in Γ\left(M,E^*\otimes \overline E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$Λ_{ω_g} \left(\sqrt{-1} R^h\right)=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact Kähler manifolds.