Stephen Shing-Toung Yau

Genome mining typically begins with known genes or protein families, which can make systems defined primarily by genomic organization difficult to detect. We developed a DNA-first framework combining asymmetric-ovariance natural-vector representations with generic positional recurrence, joining separately calibrated signals through genomic coordinates. In benchmarks of array-associated reverse transcriptase (ART) loci, the target-free mathematical arm localized seven of eight predeclared loci, including all seven with exact-array coordinates. Dual-arm signals occurred in none of three non-array RT controls and in one of five UG27 systems, indicating selectivity within these small control sets. A separate profile-guided search of 6.12 Gb identified five candidate array-associated RT-like systems. In an independent 54.38-Gb evaluation of this profile-guided arm, structural compatibility was 24/32 under a historical-rule mirror, 12/32 after novelty-only selection, and 0/17 for architecture-selected complete ORFs; the two 32-member panels shared six exact sequences. Together, these results suggest that mathematical features of raw DNA can complement protein-directed searches in nominating genomic architectures for experimental RNA-to-DNA studies.

A central task in population genetics is to identify genetic diversity in a population containing a number of individuals. In recent years, with the development of the third generation sequencing (TGS) technology, pan-genome research has become a hot topic. Although graphical representation has been a popular way to represent the pangenome, few works have attempted to describe it in a more mathematical way. In this paper, we used 79 high-quality assembly data of third-generation sequencing in yeast (including Saccharomyces cerevisiae and Saccharomyces paradoxus ) to construct the graph pangenome, and introduced the Primitive GLMY (Grigor’yan-Lin-Muranov-Yau) Homology in algebraic topology to quantitatively represent the pan-genome. We further made an intriguing attempt to conduct a population genetic analysis of this resulting dataset from the topological features of the graph pangenome. We found that there was good agreement between the obtained results and the biological context. We believe this study has developed a method for population genetic analysis of the genetic diversity of genome structural variation.

The natural vector method is an important method for the analysis of biological sequences. In this study, we applied this method to population genetic analysis, with the core purpose of using it to evaluate the characteristics of a set of sequences rather than just pairwise comparison. We used the mitochondrial genome dataset from the human 1000 Genomes Project as a dataset to verify the feasibility of this improved natural vector method. The results showed that the modified natural vector method could be used for various population genetic approaches at least in the sense of population average, including the calculation of principal component analysis, population structure analysis and genetic diversity parameters. The results were in good agreement with those based on traditional molecular genetic markers such as SNP. The new method validates the feasibility of natural vector method for population genetic analysis and provides a framework for the application of matchless pair method to population genomic analysis on a wider scale.

Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. When facing nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. To address this issue, this paper reconstructs the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein's lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step's objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we derive its natural gradient that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new iterative filter for nonlinear systems called Natural Gradient Gaussian Approximation filter, or NANO filter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. Furthermore, the estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like property across consecutive time steps. Real-world experiments demonstrate that, compared to popular Gaussian filters such as EKF, UKF, iterated EKF, and posterior linearization filter, NANO filter reduces the average root mean square error by approximately 45% while maintaining a comparable computational burden.

Single-cell RNA sequencing (scRNA-seq) enables high resolution characterization of cellular heterogeneity but poses significant challenges for cross institutional collaboration due to privacy constraints and distributional heterogeneity. To address this problem, we propose a Federated Distillation framework with Knowledge Sharing (scKSFD) for privacy-preserving cell type classification. Unlike conventional federated learning approaches that exchange model parameters, scKSFD performs knowledge aggregation in prediction space by sharing probability-level soft label outputs on a reference dataset, thereby reducing privacy risks. To better accommodate domain specific characteristics of scRNA-seq data, scKSFD integrates stratified proxy sampling to preserve rare cell populations and employs probability-level aggregation to mitigate batch specific expression shifts without explicit feature level correction. Comprehensive evaluations across 42 clinical single-cell transcriptome datasets demonstrate that scKSFD achieves higher or comparable F1 scores relative to centralized and existing federated baselines under heterogeneous settings, with statistically significant improvements in paired comparisons. In a multiple hospital COVID-19 case study, federated collaboration using scKSFD improved classification performance compared with local-only training while avoiding direct sharing of patient level expression data. Overall, scKSFD provides a federated distillation framework that balances predictive performance, robustness, and data-sharing constraints for multiple institutional single-cell transcriptomic analysis.

One of the most important breakthroughs was the propose of the Kalman filter in the 1960 s, which motivated a large number of studies on nonlinear filtering. Among these efforts, a special class of systems known as Yau filters has attracted attention. For the finite-dimensional case, the Direct Method (DM) was proposed to address systems with quadratic potential quantity [1, 2]. To generalize the DM, we shall focus on a class of infinite-dimensional filtering systems in a statistical sense. Our approach is capable of handling the potential quantity of arbitrary polynomial degree of filtering systems, meanwhile accommodating systems with nonlinear observation terms expressed as arbitrary-degree polynomials. The core idea is to apply several transformations to simplify the fundamental kernel corresponding to the filtering system. By leveraging a series expansion, we derive an explicit series solution from which optimal state estimates are obtained via convolution. Building on this series solution, we introduce a truncation technique and develop the Extended Direct Method (EDM) filtering algorithm which is supported by strong theoretical guarantees of uniqueness and existence through Euler operator theory. Numerical experiments demonstrate that our method not only generalizes the classical Direct Method to a wider class of systems but also enables fast, real-time implementation. It achieves near-optimal performance in challenging filtering scenarios compared to the Galerkin-Spectral method with less computational cost time, significantly outperforming conventional algorithms, including the Particle Filter, Extended Kalman Filter etc.

Let On=C{x1,…,xn} be the ring of convergent power series. An isolated hypersurface singularity (V, 0) is defined by f∈On. Its moduli algebra A(V)≔On/(f,∂f/∂x1,…,∂f/∂xn) is a finite-dimensional algebra that determines the complex analytic structure of (V, 0). The Yau algebra L(V)≔DerC(A(V),A(V)) is a solvable Lie algebra associated with the singularity. As the singularity deforms, the structure of L(V) also varies, leading to a Torelli-type problem: can the Yau algebra distinguish the different analytic structures within a deformation family? This paper investigates a crucial subalgebra, the liftable Yau subalgebra L̃(V), which consists of derivations that can be lifted along the deformation. We focus on a one-parameter (μ, τ)-invariant deformation family Vt generated by f0 = x4 + y4 + z4 (whose projectivization in CP2 defines a smooth plane quartic curve of genus 3). We compute the 37-dimensional liftable Yau subalgebra family L̃(Vt) for this deformation. A key technical breakthrough of this paper is the development of new computational tools and algorithms to calculate a series of Lie algebra isomorphism invariants (namely, cross-ratios) for its high-dimensional (36-dim) nilradical Nt=[L̃(Vt),L̃(Vt)]. This extends the invariant-based method, previously used for lower-dimensional algebras, to a significantly more complex case. By applying this new computational framework, we prove that L̃(Vt)≅L̃(Vs) implies t2 = s2 or t2 = −324(s2 − 36)/(1015s2 + 324). Comparing these two results, we demonstrate that although the liftable Yau subalgebra L̃(Vt) is itself a well-behaved family of solvable Lie algebras constructed geometrically, it fails to be a complete invariant for the singularity isomorphism. Indeed, it fails to be even a basic invariant, as there exist cases where Vt ≅ Vs (isomorphic singularities) but L̃(Vt)≇L̃(Vs) (non-isomorphic Lie algebras). This contrasts sharply with the known cases of Ẽ7 and Ẽ8 singularities. An important part of this work is the development of computational algorithms that make the analysis of these high-dimensional Lie algebras feasible on a computer.