Princeton's autumn night lay deep and hushed, like a vast, sound‑absorbing velvet that tenderly enveloped the day's soft murmurs within the halls of learning. Yue'er's office, nestled at the top of an old ivy‑clad building, was lit only by an antique green‑shaded desk lamp, casting a warm yet lonely circle of light on the desk piled high with books and scratch paper.
She had been sitting almost motionless at that desk for nearly six hours. Spread before her were the key steps of a derivation attempting to bridge the connection between the P/NP problem and the Langlands program. Yet her thoughts seemed mired in a viscous swamp; each step was unusually arduous, every seemingly promising path, upon deeper exploration, led to a dead end or spiraled into ever more complex, unmanageable expressions. The mathematical symbols she knew so intimately in ordinary hours now lay on the page like spells drained of magic, stiff and refusing to divulge any secrets.
A sense of frustration crept up like fine tendrils, coiling around her heart, bringing a suffocating weariness. She knew she was touching an immensely profound domain—every breakthrough here might demand years, even decades, of accumulation and a flash of inspiration. Knowing was one thing; actually being stuck before this intangible barrier made the intellectual helplessness vivid and piercing.
She pushed aside the draft, exhaled a long breath, leaned back in her chair, and closed her tired eyes. Outside the window, stars dotted the cold, distant sky, much like the space between her and that ultimate answer.
She couldn't keep grinding like this. She understood the rhythm of thought: when one forces oneself into a dead end, the harder one pushes, the deeper one sinks. Sometimes, a step back was needed—to search for a spark of inspiration from a different angle, even from a seemingly unrelated field.
Her gaze fell on a hefty, spine‑worn tome on the bookshelf—Galois Theory. Évariste Galois, that French mathematical genius who died young, had blazed across the mathematical sky like a comet in his mere twenty years of life, leaving ideas that would illuminate centuries to come. His theory, originally conceived to settle the age‑old question of solvability of polynomial equations by radicals, transcended that scope, revealing profoundly the central role of "symmetry" in mathematical structures.
Yue'er reached out and gently stroked the book's spine, as if feeling the warmth of the wisdom stored there across nearly two centuries. She took the book down; dust motes danced in the lamplight. She opened its pages—not to find a direct answer to the P/NP problem, but to let her mind immerse itself in a different kind of pure beauty, that of "structure" and "transformation."
**Symmetry.** That was the key to understanding Galois theory, and indeed a vast territory of modern mathematics.
How could one explain the essence of the **Galois group**, the core of Galois theory, to someone outside mathematics? In Yue'er's mind arose a toy almost everyone had encountered—the **Rubik's cube**.
Imagine a scrambled 3×3 Rubik's cube. Its six faces show a jumble of colors, seemingly chaotic, without pattern. Yet you know that fundamentally it is ordered; there exists a "perfectly symmetric" state—each face a solid color.
Now you start trying to solve it. Every move you make—rotating a layer ninety or a hundred eighty degrees—is a "transformation" of the cube's state. These rotations are not arbitrary; they must obey the rules allowed by the cube's mechanical structure.
Here is the key: the collection of all **legal rotations** (say, rotating the top layer clockwise, denoted U; rotating the front layer counter‑clockwise, denoted F′, etc.), together with the rules for combining them, forms a "group." More specifically, for the 3×3 cube, this is called the Rubik's cube group.
And this "Rubik's cube group" precisely captures the **symmetry** among all possible states of the cube!
A scrambled cube can be transformed from one scrambled state A to another scrambled state B by a sequence of rotations (i.e., elements of the group). Whether it can finally be restored to the perfect six‑solid‑color state (the symmetric state) depends entirely on the position and properties of the cube's current state within the structure of the "Rubik's cube group."
The idea of the Galois group is strikingly similar, except that Galois shifted the stage from a concrete cube to abstract polynomial equations.
Consider a polynomial equation in one variable, say x⁵ – x – 1 = 0. Its roots (solutions) are fixed, but may be extremely complex, not expressible by simple radicals (like square roots, cube roots, etc.). Galois stopped trying to solve the equation directly and instead studied the "symmetry" among those roots.
He regarded the roots as an abstract set. Then he considered all those "transformations" that could "permute" the roots (just as rotating the cube changes the positions of the colored stickers) while keeping unchanged all algebraic relations among the roots (determined by the equation's coefficients). This collection of "structure‑preserving transformations" constitutes the equation's **Galois group**.
This Galois group, much like the Rubik's cube group characterizes the cube's symmetry, precisely describes the symmetric structure of the polynomial equation's roots.
The question of whether the roots can be expressed by radicals—which had puzzled mathematicians for centuries—is entirely encoded in the structural properties of its Galois group! Galois proved that a polynomial equation is solvable by radicals **if and only if** its Galois group is a "solvable group" (a specific, "friendly" type of group with a layered, nested structure).
It is analogous to being able to determine, purely in theory by analyzing some deep property of the "Rubik's cube group," whether a scrambled cube **can** be restored through a sequence of legal rotations, without ever touching it! The Galois group is the equation's "symmetry DNA"; it determines the equation's fundamental nature.
Yue'er immersed herself in Galois's grand and exquisite vision, her heart filled with admiration. He had elegantly transformed a hard problem about "computation" and "solution" into a more essential, deeper question about "structure" and "symmetry." This shift from concrete calculation to the study of abstract structure was a milestone in the history of mathematics.
What about the P/NP problem? At its core it too is a question about the difficulty of "computation"—if verifying a solution is easy, does that mean finding it is also easy? The Langlands program, on the other hand, seeks to establish profound symmetric connections (through L‑functions and the duality of Galois representations/automorphic representations) between different mathematical realms (number theory vs. harmonic analysis).
Could there be a way, much as Galois recast equation solvability as a question about group structure, to transform computational complexity (P/NP) into something deeper—a question about the "symmetry" or "rigidity" of mathematical structure itself? Might one find a counterpart of computational complexity within the grand symmetric network revealed by the Langlands program?
A vague yet thrilling thought flashed like a faint firefly in the depths of her mind—too dim, too indistinct to be captured and fixed, yet it offered a hint of direction. The suffocating feeling of being trapped loosened a little. Galois theory felt like a cool beam of light shone into her mental impasse; though it didn't directly point a way out, it revealed different contours of the surroundings.
Just then, her phone, lying on the corner of the desk, lit up and buzzed with vibration, startlingly abrupt in the deep silence of the night.
Yue'er was jolted from her reverie and looked at the phone, puzzled. The screen showed an unfamiliar number from China. Her social circle was small; who could it be at this hour?
She hesitated for a moment, then out of politeness slid to answer.
"Hello?" Her voice bore a hint of hoarseness and wariness from hours of concentration.
"May I speak with… Ms. Yue'er?" came a steady, clear male voice from the other end, speaking Chinese, his tone calm and carrying a peculiar cool clarity, as if able to cut through the noise of the line.
"Speaking. Who is this?"
"Sorry to disturb you. My name is Mozi." He introduced himself, neither deferential nor overbearing. "Through some academic channels, I came across your recent preprint on the possible link between the Langlands program and computational complexity. I found it very inspiring."
Mozi? Yue'er quickly searched her memory—she had heard the name somewhere, perhaps associated with an emerging quantitative fund? But she wasn't familiar with the financial world.
"Thank you." She remained polite but still puzzled. "May I ask why you're calling?" Someone from finance calling late at night to discuss a pure‑mathematics preprint? That was most unusual.
"I work mainly in quantitative trading," Mozi seemed to sense her doubt and explained directly. "My work is largely about trying to find and exploit tiny, statistically significant 'fragments of order' within the chaos of financial markets. Your paper, with its thoughts on the possible 'complexity barriers' inherent in mathematical structures, reminded me of certain phenomena in market behavior that are hard to explain with existing models—seemingly random but perhaps subject to deep constraints."
He paused, then continued: "So, perhaps this is a bit presumptuous, but I'd very much like to hear your own view on 'order' itself. In your opinion, is there some kind of… isomorphism between the underlying order of the universe revealed by mathematics and the seemingly chaotic socioeconomic systems we live in—systems composed of countless individual decisions?"
Yue'er was taken aback. She hadn't expected such a question. Someone navigating the waves of capital markets, trying to extract signal from noise, would actually discuss with a theorist like her, immersed in pure mathematical abstraction, a fundamental philosophical‑scientific topic like "order."
Oddly, she didn't feel offended; rather, she sensed a subtle resonance. She recalled the awe she had just felt in Galois theory—the shock of grasping the essence of complex structure through "symmetry" and "invariance." Wasn't that also an effort to find eternal "order" amid change?
"Order…" Yue'er unconsciously repeated the word, her eyes returning to the open Galois Theory. "Perhaps order does not mean rigidity and immutability. Like the Galois group, it doesn't describe the roots themselves, but all possible transformations among the roots that preserve the relations. Order may be hidden precisely in the symmetry constituted by these 'allowed transformations.' It defines the possibilities and impossibilities of a system, defines what is 'structure' and what is 'noise.'"
She rarely discussed mathematics this way with others, especially with someone from a seemingly distant field. Yet now, in the quiet of the night, after having just experienced mental blockage, this strange voice from far‑off Asia felt like a stone tossed into a calm lake, stirring a desire to express.
"And in a complex system, like the market you mentioned," she went on thinking aloud, "perhaps there exists some analogous 'generalized symmetry' defined by the participants' behavioral rules, information‑flow patterns, capital constraints, and so on? It may not be as clear and eternal as a mathematical structure, but maybe there is a dynamic, statistical 'group structure' that determines the broad contours and boundaries of the system's evolution. Those hard‑to‑explain 'randomnesses' might stem precisely from our insufficient understanding of this underlying 'order structure'?"
On the other end, Mozi fell silent for a while, seemingly engaged in equally deep reflection. Then his voice returned, carrying a trace of barely detectable excitement: "A most brilliant perspective. Understanding order as the 'set of allowed symmetric operations' rather than a static pattern… This gives me a completely new framework for interpreting 'patterns' and 'anomalies' in the market. Thank you, Ms. Yue'er. You've inspired me greatly."
The call wasn't long—barely ten minutes in all. But after hanging up, Yue'er sat motionless in her chair for a long while, her mind unsettled.
Outside, Princeton still slept, starlight faint. Yet inside her, it was as if a small, luminous pebble had been dropped. That stranger named Mozi, his question, his domain, the "order" he pursued, formed a peculiar reflection of her own world.
She looked again at the formulas on the desk that had earlier left her at a loss. They hadn't become any easier, but the feeling of hopeless entrapment had largely dissipated. Galois's theory had opened a window for her, and that unexpected phone call felt like a gust of fresh air that blew away the mist before it.
The magic of symmetry existed not only in the permutations of equation roots, not only in the connections between different mathematical worlds bridged by the Langlands program, but also seemed to linger, faintly, between seemingly completely different fields of knowledge and human activity. Seeking order, understanding structure, revealing symmetry—perhaps this was the deepest, most common drive of all rational explorers.
She picked up her pen and began sketching fresh drafts on a new sheet of paper—outlines relating Galois groups and complexity classes. The path remained arduous, but directions seemed to have multiplied. The deep‑night silence no longer felt lonely; instead, it became a fertile ground for nurturing ideas. She knew the answer was still far off, but at least on this night she had regained the courage to move forward and a sliver of guiding light. And that distant conversation about "order" from a distant land had fallen like an unexpected seed in a corner of her heart, waiting perhaps for some future moment to take root and sprout.