arXiv subjects

Shing-Tung Yau

Publications and source records attributed to Shing-Tung Yau.

At least 55 records · Page 3Linked to original sources

arXiv2026

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

In this paper, we demonstrate that twisted Hermitian-Einstein metrics on holomorphic vector bundles exist without obstruction. More precisely, for an arbitrary holomorphic vector bundle $E$ over a compact Kähler manifold $(M,ω_g)$, we prove that the twisted Hermitian-Einstein equation $$Λ_{ω_g}\left(\sqrt{-1}R^h\right) = λh + P$$ admits a unique smooth solution $h$, provided that $P\inΓ(M,E^*\otimes\bar{E}^*)$ is positive-definite and $λ<λ_E^-$. The constant $λ_E^-$ is intrinsically associated with the stability constant of $E$. This result extends the classical Donaldson-Uhlenbeck-Yau (DUY) theorem for stable bundles and, in the limit $P\rightarrow0$, gives a new proof of the DUY theorem. As an application, we obtain an intrinsic Chern number inequality for unstable vector bundles: $$\int_M \left((r-1)c_1(E)^2 - 2rc_2(E)\right) \wedge ω_g^{n-2} \leq \Bigl\lfloor \frac{r^2}{4} \Bigr\rfloor \frac{(λ_E^+-λ_E^-)^2}{4π^2 n^2} \int_M ω_g^n.$$

math.DG↗

arXiv2026

The Critical LYZ Equation in Kähler Geometry

We establish the existence of smooth solutions for the LYZ equation at the critical phase $θ=(n-2)\fracπ{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $θ\leq (n-2)\fracπ{2}$. As applications, we solve the 3D Hessian equation $σ_2 = 1$ and the 4D Hessian quotient equation $σ_3 = σ_1$ under weaker assumptions than previously required.

math.DG↗

arXiv2026

Yau's Affine-Normal Descent for Large-Scale Unrestricted Higher-Moment Portfolio Optimization

Unrestricted mean-variance-skewness-kurtosis portfolio optimization can capture asymmetry and tail risk, but sample-moment formulations become computationally impractical when the asset universe is large: they produce dense nonconvex quartic objectives with prohibitive coskewness and cokurtosis tensors and anisotropic, ill-conditioned level sets. We develop a structure-exploiting algorithm based on Yau's affine-normal descent that follows affine-normal directions of the current level set while working directly with the return matrix. The method avoids explicit higher-order tensors and exploits the quartic structure for exact sample oracles, derivative evaluation, and exact line search. We also provide theory for the reduced simplex formulation, including regularity and convexity conditions that separate data-map geometry from investor preference coefficients. Computational results show a clear implementation split: a direct configuration is effective on the standard small benchmark, whereas a preconditioned conjugate-gradient configuration with stall recovery becomes the preferred large-scale implementation by the upper end of the hundreds and remains competitive as the asset universe moves into the thousands. On a 5-minute A-share panel with 5,440 stocks, the method makes direct full-universe comparisons with exact mean-variance portfolios feasible and shows on the baseline split that the incremental value of higher moments is strongest at moderate return targets.

q-fin.PM↗

arXiv2026

Polylab: A MATLAB Toolbox for Multivariate Polynomial Modeling

Polylab is a MATLAB toolbox for multivariate polynomial scalars and polynomial matrices with a unified symbolic-numeric interface across CPU and GPU-oriented backends. The software exposes three aligned classes: MPOLY for CPU execution, MPOLY_GPU as a legacy GPU baseline, and MPOLY_HP as an improved GPU-oriented implementation. Across these backends, Polylab supports polynomial construction, algebraic manipulation, simplification, matrix operations, differentiation, Jacobian and Hessian construction, LaTeX export, CPU-side LaTeX reconstruction, backend conversion, and interoperability with YALMIP and SOSTOOLS. Versions 3.0 and 3.1 add two practically important extensions: explicit variable identity and naming for safe mixed-variable expression handling, and affine-normal direction computation via automatic differentiation, MF-logDet-Exact, and MF-logDet-Stochastic. The toolbox has already been used successfully in prior research applications, and Polylab Version 3.1 adds a new geometry-oriented computational layer on top of a mature polynomial modeling core. This article documents the architecture and user-facing interface of the software, organizes its functionality by workflow, presents representative MATLAB sessions with actual outputs, and reports reproducible benchmarks. The results show that MPOLY is the right default for lightweight interactive workloads, whereas MPOLY-HP becomes advantageous for reduction-heavy simplification and medium-to-large affine-normal computation; the stochastic log-determinant variant becomes attractive in larger sparse regimes under approximation-oriented parameter choices.

cs.MS↗

arXiv2026

Routine Blood Biomarkers Reveal a Preclinical Continuum of Multiple Myeloma Risk

Multiple myeloma (MM) is preceded by a long preclinical phase spanning decades, yet scalable, non-specialist tools to identify individuals at elevated risk before end-organ damage are lacking. In a prospective analysis of 299,035 cancer-free UK Biobank participants followed for a median of 12.4 years, during which 768 developed incident MM, we conducted a biomarker-wide association scan across 61 routinely measured blood analytes spanning hematological, protein metabolism, renal, and immune categories. Markers of protein dysregulation-elevated total protein, depressed albumin, and a low albumin-to-globulin (A/G) ratio-showed the strongest preclinical associations (hazard ratios 0.61-1.54 per SD), consistent with progressive monoclonal immunoglobulin accumulation and suppression of normal polyclonal synthesis years before diagnosis. These signals were accompanied by indicators of erythropoietic suppression, morphological red cell dysregulation, and a shift toward lower neutrophil and higher lymphocyte fractions, reflecting coordinated perturbations across hematopoietic and immune compartments. Longitudinal trajectory analyses showed that these multi-system deviations emerge more than a decade before diagnosis and intensify as clinical onset approaches. Dose-response modelling revealed pronounced nonlinear associations for protein and erythrocytic markers, with risk concentrated among individuals with extreme values. Incorporating significant biomarkers into a clinical risk model improved 10-year MM discrimination from a C-index of 0.684 to 0.744, with the high-risk decile accumulating 0.79% cumulative incidence versus 0.47% under the clinical model alone. These findings provide a practical framework for biomarker-guided MM risk stratification and targeted surveillance using routinely available clinical tests.

stat.AP↗

arXiv2026

Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis

We propose Yau's Affine Normal Descent (YAND), a geometric framework for smooth unconstrained optimization in which search directions are defined by the equi-affine normal of level-set hypersurfaces. The resulting directions are invariant under volume-preserving affine transformations and intrinsically adapt to anisotropic curvature. Using the analytic representation of the affine normal from affine differential geometry, we establish its equivalence with the classical slice-centroid construction under convexity. For strictly convex quadratic objectives, affine-normal directions are collinear with Newton directions, implying one-step convergence under exact line search. For general smooth (possibly nonconvex) objectives, we characterize precisely when affine-normal directions yield strict descent and develop a line-search-based YAND. We establish global convergence under standard smoothness assumptions, linear convergence under strong convexity and Polyak-Lojasiewicz conditions, and quadratic local convergence near nondegenerate minimizers. We further show that affine-normal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations. Numerical experiments illustrate the geometric behavior of the method and its robustness under strong anisotropic scaling.

math.OC↗

arXiv2026

Affine Normal Directions via Log-Determinant Geometry: Scalable Computation under Sparse Polynomial Structure

Affine normal directions provide intrinsic affine-invariant descent directions derived from the geometry of level sets. Their practical use, however, has long been hindered by the need to evaluate third-order derivatives and invert tangent Hessians, which becomes computationally prohibitive in high dimensions. In this paper, we show that affine normal computation admits an exact reduction to second-order structure: the classical third-order contraction term is precisely the gradient of the log-determinant of the tangent Hessian. This identity replaces explicit third-order tensor contraction by a matrix-free formulation based on tangent linear solves, Hessian-vector products, and log-determinant gradient evaluation. Building on this reduction, we develop exact and stochastic matrix-free procedures for affine normal evaluation. For sparse polynomial objectives, the algebraic closure of derivatives further yields efficient sparse kernels for gradients, Hessian-vector products, and directional third-order contractions, leading to scalable implementations whose cost is governed by the sparsity structure of the polynomial representation. We establish end-to-end complexity bounds showing near-linear scaling with respect to the relevant sparsity scale under fixed stochastic and Krylov budgets. Numerical experiments confirm that the proposed MF-LogDet formulation reproduces the original autodifferentiation-based affine normal direction to near machine precision, delivers substantial runtime improvements in moderate and high dimensions, and exhibits empirical near-linear scaling in both dimension and sparsity. These results provide a practical computational route for affine normal evaluation and reveal a new connection between affine differential geometry, log-determinant curvature, and large-scale structured optimization.

math.OC↗

arXiv2026

Hodge rigidity of Chern classes

In this paper, we study the homogeneous components of the Chern--Schwartz--MacPherson (CSM) classes of Schubert cells. We prove that, under suitable conditions, each such component is represented by an irreducible subvariety. In particular, our result extends Huh's result \cite{Huh} by relaxing the regularity assumption on log resolutions. As a consequence, the conclusion holds for all cominuscule Schubert cells of classical type and for a large family of exceptional cases. We also obtain analogous results for certain Schubert varieties in symplectic Grassmannians and flag varieties.

math.AG↗

arXiv2026

A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

We explain how the spectrum of a closed embedded surface $Σ\subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $Ω\subset \mathbb{R}^3$ with $\partial Ω= Σ$. We prove that there exists a positive constant $K_g$, depending only on the genus $g$ of $Σ$, such that $λ_k^D(Ω)^{3/2}/(λ_k(Σ)\sqrt{λ_1(Σ)}) \ge K_g$, where $λ_k(Σ)$ denotes the $k$-th nonzero eigenvalue of the Laplace-Beltrami operator on $Σ$ and $λ_k^D(Ω)$ denotes the $k$-th eigenvalue of the Laplacian on $Ω$ with Dirichlet boundary conditions. Moreover, we explicitly obtain the dependence of $K_g$ on the genus, showing that $K_g \propto (g+1)^{-1}$, and we determine the optimal constant $K_0$ for $k=1$ in the genus-zero case. A generalized version of this result in arbitrary dimension is also provided for domains whose boundaries have nonnegative Ricci curvature.

math.DG↗

arXiv2026

Microlocal index theorems and analytic torsion invariants in the geometric theory of partial differential equations

We develop a microlocal and derived-geometric framework for index theory and analytic torsion of nonlinear PDEs. By integrating Spencer hypercohomology, microlocal sheaf theory, and factorization algebras, we establish new connections between classical index theorems, BCOV invariants of Calabi-Yau manifolds, and the geometry of configuration spaces. We prove sheaf-theoretic index formulas for families of formally integrable PDEs, a microlocal index theorem for D-algebras generalizing Atiyah-Singer, and a mixed-type index theorem via microlocal stratification. We construct Ray-Singer analytic torsion for involutive systems and show that the BCOV invariant equals the Spencer torsion of the de Rham system. A categorical trace interpretation leads to a virtual index theory for derived moduli spaces of solutions. Finally, we extend the theory to configuration spaces using factorization algebras, with applications to renormalization and QFT. These results unify geometric perspectives on PDEs, torsion invariants, and moduli theory, with implications for mirror symmetry, quantum fields, and Calabi-Yau degenerations.

math.AG↗

arXiv2026

Szczarba's twisted shuffle and equivariant path homology of directed graphs

To a marked simplicial set one can associate its path chain complex, and define its homology to be the homology of this complex, inspired by path homology theories for directed graphs, quivers, and marked categories. Given a marked simplicial set with a simplicial group action preserving the markings and degenerate 1-simplices, together with a twisting function, we define a marked twisted Cartesian product using the box product. Classically, Szczarba's twisted shuffle provides a quasi-isomorphism between the chain complex of a twisted Cartesian product and the corresponding twisted tensor product. In this paper, we prove that in the marked setting, this map restricts to a chain isomorphism on path chain complexes. As an application, for directed graphs with group actions, we obtain a natural Borel construction as a special case of marked twisted Cartesian products. Equivariant path homology is defined as the homology of this construction and is computed by an explicit twisted tensor product.

math.AT↗

arXiv2025

Distinguishing Calabi-Yau Topology using Machine Learning

While the earliest applications of AI methodologies to pure mathematics and theoretical physics began with the study of Hodge numbers of Calabi-Yau manifolds, the topology type of such manifold also crucially depend on their intersection theory. Continuing the paradigm of machine learning algebraic geometry, we here investigate the triple intersection numbers, focusing on certain divisibility invariants constructed therefrom, using the Inception convolutional neural network. We find $\sim90\%$ accuracies in prediction in a standard fivefold cross-validation, signifying that more sophisticated tasks of identification of manifold topologies can also be performed by machine learning.

math.AG↗

arXiv2025

A Neural Symbolic Model for Space Physics

In this study, we unveil a new AI model, termed PhyE2E, to discover physical formulas through symbolic regression. PhyE2E simplifies symbolic regression by decomposing it into sub-problems using the second-order derivatives of an oracle neural network, and employs a transformer model to translate data into symbolic formulas in an end-to-end manner. The resulting formulas are refined through Monte-Carlo Tree Search and Genetic Programming. We leverage a large language model to synthesize extensive symbolic expressions resembling real physics, and train the model to recover these formulas directly from data. A comprehensive evaluation reveals that PhyE2E outperforms existing state-of-the-art approaches, delivering superior symbolic accuracy, precision in data fitting, and consistency in physical units. We deployed PhyE2E to five applications in space physics, including the prediction of sunspot numbers, solar rotational angular velocity, emission line contribution functions, near-Earth plasma pressure, and lunar-tide plasma signals. The physical formulas generated by AI demonstrate a high degree of accuracy in fitting the experimental data from satellites and astronomical telescopes. We have successfully upgraded the formula proposed by NASA in 1993 regarding solar activity, and for the first time, provided the explanations for the long cycle of solar activity in an explicit form. We also found that the decay of near-Earth plasma pressure is proportional to r^2 to Earth, where subsequent mathematical derivations are consistent with satellite data from another independent study. Moreover, we found physical formulas that can describe the relationships between emission lines in the extreme ultraviolet spectrum of the Sun, temperatures, electron densities, and magnetic fields. The formula obtained is consistent with the properties that physicists had previously hypothesized it should possess.

astro-ph.SR↗

arXiv2025

Bridgeland/Weak Stability Conditions under Spherical Twist Associated to A Torsion Sheaf

In this paper, we study the action of an autoequivalence, the spherical twist associated to a torsion sheaf, on the standard Bridgeland stability conditions and a generalized weak stability condition on the derived category of a K3 surface. As a special case, we construct a Bridgeland stability condition associated to a non-nef divisor, which conjecturally lies in the geometric component but outside the geometric chamber. We also discuss the destabilizing objects and stability of certain line bundles at the weak stability condition associated to a nef divisor.

math.AG↗

arXiv2025

Mirror symmetry for singular double cover Calabi--Yau varieties: quantum test

We continue our study on the pairs of singular Calabi--Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of $\mathbb{P}^{3}$ branched along (1) a union of eight hyperplanes in general position, and (2) a union of four hyperplanes and a quartic in generation. Our previous construction produces hypothetical singular mirror partners. We prove that they are mirror pairs in the sense that the $B$-model of one (variation of Hodge structure) is equivalent to the $A$-model of another (the untwisted part of the genus zero orbifold Gromov--Witten invariants). The technique can be generalized and applied to the case when the nef-partition is trivial. As a byproduct, we also verify Morrison's conjecture in certain circumstances.

math.AG↗

arXiv2025

A D-brane fantasy on noncommutative mirror symmetry, prelude: Noncommutative ringed spaces from local noncommutative crepant resolutions of a singular Calabi-Yau space, dynamical D-branes thereupon, and questions beyond

In contrast to the world-sheet of a fundamental string, the world-volume of stacked D-branes carries an Azumaya noncommutative structure ([L-Y1: Sec.\ 2] (D(1))), allowing it to directly serve as a probe into noncommutative target-spaces. This feature leads to a D-brane fantasy: {\it Noncommutative Mirror Symmetry between noncommutative Calabi-Yau spaces may be realized as different realizations of a supersymmetric D-brane world-volume quantum field theory exactly like the string world-sheet aspect for Mirror Symmetry between (commutative) Calabi-Yau manifolds}. Driven by this fantasy, in the current notes a class of noncommutative ringed spaces shadowing over a $C^\infty$-manifold with corners are constructed from gluing local noncommutative crepant resolutions of Gorenstein isolated singularities. Dynamical D-branes on such noncommutative target-spaces are realized as maps/morphisms from an Azumaya manifold with a fundamental module with a connection $\nabla$ thereto. The notion of $\nabla$-adjusted kinetic energy for such a map is given via the basic noncommutative differential calculus developed earlier in [L-Y4] (D(11.1)). This provides an action functional for dynamical D-branes on such noncommutative spaces in parallel to the Polyakov action functional for fundamental bosonic strings on a commutative target-space. This sets up a basic stage to begin with for the realization of the D-brane fantasy on Noncommutative Mirror Symmetry. Questions beyond are sampled along the discussion.

math.AG↗

arXiv2025

An Improved Yau-Yau Algorithm for High Dimensional Nonlinear Filtering Problems

Nonlinear state estimation under noisy observations is rapidly intractable as system dimension increases. We introduce an improved Yau-Yau filtering framework that breaks the curse of dimensionality and extends real-time nonlinear filtering to systems with up to thousands of state dimensions, achieving high-accuracy estimates in just a few seconds with rigorous theoretical error guarantees. This new approach integrates quasi-Monte Carlo low-discrepancy sampling, a novel offline-online update, high-order multi-scale kernel approximations, fully log-domain likelihood computation, and a local resampling-restart mechanism, all realized with CPU/GPU-parallel computation. Theoretical analysis guarantees local truncation error $O(Δt^2 + D^*(n))$ and global error $O(Δt + D^*(n)/Δt)$, where $Δt$ is the time step and $D^*(n)$ the star-discrepancy. Numerical experiments, spanning large-scale nonlinear cubic sensors up to 1000 dimensions, highly nonlinear small-scale problems, and linear Gaussian benchmarks, demonstrate sub-quadratic runtime scaling, sub-linear error growth, and excellent performance that surpasses the extended and unscented Kalman filters (EKF, UKF) and the particle filter (PF) under strong nonlinearity, while matching or exceeding the optimal Kalman-Bucy filter in linear regimes. By breaking the curse of dimensionality, our method enables accurate, real-time, high-dimensional nonlinear filtering, opening broad opportunities for applications in science and engineering.

math.OC↗

arXiv2025

C-R-T Fractionalization in the First Quantized Hamiltonian Theory

Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the $\mathbb{Z}_2^{\mathcal{C}}\times\mathbb{Z}_2^{\mathcal{R}}\times\mathbb{Z}_2^{\mathcal{T}}$ for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity, thereby forming a group extension of the $\mathbb{Z}_2$ direct product. Conventionally, a Majorana fermion is defined by one Dirac fermion with trivial charge conjugation. However, when the spacetime dimension $d+1=5,6,7\bmod8$, the real dimension of Majorana fermion (dim$_{\mathbb{R}}χ_{\mathcal{C}\ell(d,0)}$) aligns with the real dimension of Dirac fermion (dim$_{\mathbb{R}}ψ_{\mathcal{C}\ell(d)}$), rather than being half, which necessitates the introduction of a symplectic Majorana fermion, defined by two Dirac fermions with trivial charge conjugation. To include these two types of Majorana fermions, we embed the theory in $n_{\mathbb{R}}$ and define the Majorana fermion field as a representation of the real Clifford algebra with 8-fold periodicity. Within the Hamiltonian formalism, we identify the 8-fold CRT-internal symmetry groups across general dimensions. Similarly, Dirac fermion field is defined as a representation of the complex Clifford algebra with 2-fold periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an 8-fold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act non-trivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.

cond-mat.str-el↗