We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

By replacing the inner optimizer in RAPP we rediscover the family of Lookahead algorithms for which we establish convergence in cohypomonotone problems even when the base optimizer is taken to be gradient descent ascent.

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ identified by Foulkes. Using this dataset, we train a neurosymbolic classifier that integrates invariant-theoretic features with supervised learning, yielding improved accuracy in detecting rare solvable groups compared to coefficient-based models. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into group distribution under height constraints. The methodology extends to higher-degree cases and illustrates the utility of hybrid symbolic-numeric techniques in computational algebra.

Grassroots platforms are distributed applications run by\linebreak cryptographically-identified people on their networked personal devices, where multiple disjoint platform instances emerge independently and coalesce when they interoperate. Their foundation is the grassroots social graph, upon which grassroots social networks, grassroots cryptocurrencies, and grassroots democratic federations can be built. Grassroots platforms have yet to be implemented, the key challenge being faulty and malicious participants: without secure programming support, correct participants cannot reliably identify each other, establish secure communication, or verify each other's code integrity. We present Grassroots Logic Programs (GLP), a secure, multiagent, concurrent, logic programming language for implementing grassroots platforms. GLP extends logic programs with paired single-reader/single-writer (SRSW) logic variables, providing secure communication channels among cryptographically-identified people through encrypted, signed and attested messages, which enable identity and code integrity verification. We present GLP progressively: logic programs, concurrent GLP, multiagent GLP, augmenting it with cryptographic security, and providing smartphone implementation-ready specifications. We prove safety properties including that GLP computations are deductions, SRSW preservation, acyclicity, and monotonicity. We prove multiagent GLP is grassroots and that GLP streams achieve blockchain security properties. We present a grassroots social graph protocol establishing authenticated peer-to-peer connections and demonstrate secure grassroots social networking applications.

Transformer-based deep learning methods have become the standard approach for modeling diverse data such as sequences, images, and graphs. These methods rely on self-attention, which treats data as an unordered set of elements. This ignores the neighborhood structure or graph topology of the data and requires inductive biases--such as position embeddings in sequences and images, or random walks in graphs--to incorporate topology. However, designing such task-specific biases requires significant effort and can introduce side effects that hinder generalization. We introduce Chimera, a unified model that directly incorporates data topology in a principled way, removing the need for domain-specific biases. The key idea is that state space models--which naturally do not require position embeddings--can be generalized to capture any graph topology. Our experiments show that Chimera achieves strong performance across language, vision, and graph domains, outperforming BERT on GLUE by 0.7 points, ViT on ImageNet-1k by 2.6%, and all baselines on the Long Range Graph Benchmark. We further propose algorithmic optimizations to improve Chimera's efficiency: (1) for Directed Acyclic Graphs, Chimera can be implemented as a linear-time recurrence; (2) for general graphs, a simple mathematical relaxation achieves Transformer's quadratic complexity without domain-specific heuristics. These results validate Chimera's core contribution and support the idea that data topology is a powerful inductive bias across modalities.