Night, like ungrindable thick ink, heavily covered the valley housing Stringlight Research Institute. Unlike the brightly lit, equipment‑humming, personnel‑foot‑stepping laboratories downstairs, the private study‑cum‑meditation room belonging to Yue'er in the east wing of the institute's top floor was immersed in near‑absolute silence. Only an austerely‑desk‑shaped desk lamp cast a warm, concentrated halo of light on the broad raw‑wood desk, like a solitary lighthouse in a dark ocean, guarding thought's deepest voyage.
Yue'er curled in a wide armchair, wrapped in a soft cashmere shawl against late‑night chill. Before her on the desk lay several heavy mathematics monographs, page edges studded with colorful index tabs like peculiar wings. But her gaze rested long on the holographic projection area floating above the desktop. There, no concrete, minute chip‑structure diagrams or spectral data like in Xiuxiu's labs, only countless light‑formed mathematical symbols and geometric figures—constantly flowing, transforming, interweaving. Sometimes they coalesced into complex manifold structures; sometimes scattered into abstract algebraic expressions; sometimes collapsed into probability‑distribution‑representing cloud‑like light points.
This was the "scaffolding" she had erected for her envisioned "Information‑Geometric Field Theory"—a grand mathematical framework attempting to unify information, geometry, and physics, ultimately touching consciousness and computation's essence. Since sketching this theory's vague contours to the world at the Fields Medal ceremony, shocking academia with its geometrization breakthrough regarding the P versus NP problem, she knew deeply that was merely a beginning—a lighthouse pointing far away. The real voyage was constructing solid, elegant mathematical bridges reaching that lighthouse.
Yet building these bridges proved far harder than she initially imagined. She was trying to fuse three seemingly disparate mathematical branches—each possessing vast systems and profound connotations—into one crucible: differential geometry, probability theory, and representation theory.
**Differential geometry** is language describing curved spaces, studying structures and changes on manifolds. It provides a powerful toolkit for characterizing "space's" own form. In her conception, every computational problem—whether simple sorting or complex NP‑complete problems—has all its possible "solutions" not as scattered, chaotic points but forming a specific "solution space" with inherent geometric structure. This space's "curvature," "connectivity," and other geometric properties perhaps encode the problem's "computational complexity." An easily‑computable problem might have a flat, simple solution space; a hard problem's solution space might be highly curved, tangled, filled with "canyons" and "peaks."
**Probability theory** mathematically describes uncertainty and randomness. In her view, the computational process itself—especially facing complex problems—is not a deterministic, straight path to an answer but more like a random walk within the solution space, continuously sampling, adjusting direction based on feedback. Algorithms, whether classical or quantum, can be regarded as probability evolution processes defined on this geometric space. Algorithm efficiency relates closely to how quickly this probability process converges to the correct (or approximate) solution.
**Representation theory** studies how abstract algebraic structures (like groups, Lie algebras) "act" on vector spaces through linear transformations (i.e., "representations"). It resembles a bridge connecting abstract symmetries (algebra) with concrete transformations (geometry or linear algebra). Yue'er keenly sensed that computational problems might hide profound "symmetries" or "invariants," and representation theory was key to revealing and utilizing such symmetries. Perhaps different computational‑complexity‑class problems correspond to different symmetry groups, and these groups' representational properties determine their solution spaces' geometric forms and probability evolution laws.
Fusing these three meant she needed to create a new mathematical language capable of simultaneously describing space's geometry, process's randomness, and inherent symmetry. This wasn't mere tool cobbling but required rethinking these concepts' fundamental connections at the most basic level.
Recently, her work's core was introducing and re‑interpreting "fiber bundle"—a core concept from differential geometry.
In the holographic projection, a smooth base manifold slowly rotated; this could represent the space formed by all possible "problem instances." At each "problem instance" point, an additional geometric structure was "attached"—a fiber. This fiber was that specific problem instance's "solution space." All these problem instances together with their respective solution spaces constituted a vast "fiber bundle."
This image was powerful. It organized countless isolated computational problems into a geometric object with overall structure. But the more crucial step was defining "connection."
In standard differential geometry, connection defines rules for "connecting" fibers at different points when moving along the fiber bundle's base manifold. It tells us how corresponding solution spaces "change" when sliding from one problem instance to a neighboring one. This is a precise mathematical tool describing "change."
Yue'er boldly equated this "connection" with the concept of "information flow."
In her constructed framework, computational processes—whether human‑brain thinking, computer operation, or more abstract algorithm execution—essentially were processes of information transmission and transformation on the solution‑space fiber bundle, along some path (defined by a sequence of problem instances), through rules prescribed by the "connection." Each algorithmic step could be seen as using this "information connection" to "parallel‑transport" current state (a point on some fiber) to the next state.
An efficient algorithm might correspond to a "flat" connection, making information move in the solution‑space bundle with minimal loss, clearest direction, quickly reaching the correct solution on the target fiber. A hard problem's corresponding connection might have large "curvature," causing information to easily "lose direction" during transmission, fall into local‑optima "valleys," or need extremely tortuous paths to reach the destination.
This conception excited her, also made her deeply feel her own insignificance. She was trying to use geometric language to establish a solid mathematical foundation for "computation"—this so fundamental yet mysterious concept. This wasn't merely solving the P versus NP problem but attempting to answer more fundamental questions: "What is computation?" "Where are computation's boundaries?"
She extended slender fingers, lightly traced in air; the holographic projection's fiber‑bundle structure twisted, deformed accordingly; connection expressions like golden streams meandered between fibers. She attempted to introduce a "group action" from representation theory, letting some symmetry group act on the entire fiber bundle, hoping this would help her classify computational problems more essentially.
Symbols danced, combined, scattered at her fingertips. An seemingly elegant equation just emerged; she derived a few steps deeper, only to find hidden internal contradictions, or conclusions contradicting known computational facts. The entire structure collapsed like a sandcastle under logic's tide.
She didn't lose heart, just lightly waved to disperse the failed structure, took a deep breath, started anew. This was who‑knows‑how‑many‑times of demolishing and rebuilding tonight.