Beneath the dome of the Collège de France in Paris, the air itself seemed to solidify with the light of centuries of reason. Heavy carpets absorbed footsteps; ancient walls had witnessed countless collisions and births of thought. Here, a small but exceptionally high‑level international mathematics conference was taking place, themed "The Unification of Mathematics in the Twenty‑First Century." Most attendees were white‑haired academic giants whose names were already written into textbooks, along with a few young scholars like Yue'er who, through remarkable talent and groundbreaking ideas, had broken into this inner circle.
Yue'er sat in a forward row, her palms slightly damp with sweat. It wasn't stage fright but a near‑sacred sense of reverence, mixed with unease about presenting her not‑yet‑fully‑mature thinking. Today she would give a talk titled "A Dialogue between Number Theory and Geometry: Computational Complexity in the Light of the Langlands Program." This was not only an opportunity to explain her research direction to top peers but also an adventure in placing herself under the most rigorous scrutiny.
Her turn came. She took a deep breath, straightened the hem of her somewhat formal suit, and walked steadily to the podium. The spotlight fell on her; below were pairs of deep, sharp eyes. Adjusting the microphone, her gaze swept over those familiar‑yet‑unfamiliar faces, and she began.
"Ladies and gentlemen," her voice, amplified, rang clearly in the ancient hall, "the mathematics we discuss today is, in many eyes, a highly fragmented field. Number theory, studying the mysterious properties of integers, is like exploring the universe's most basic atoms; geometry, studying the structure of space and shape, is like sketching the contours of the macroscopic world. They seem to use different languages, care about different problems."
The large screen behind her lit up. On the left appeared the seemingly patternless distribution of prime numbers; on the right, the geometric image of a complex algebraic variety, with winding curves and distributed singularities.
"Yet," Yue'er pivoted, her voice infused with firm passion, "the Langlands program, that grand conjecture proposed by Professor Robert Langlands more than half a century ago, reveals to us a stunning picture: profound, precise correspondences exist between number theory and geometry, and indeed across broader mathematical domains. They are not isolated islands, but different coastlines of the same mathematical continent, connected by a hidden strait."
She paused briefly, letting the concept settle in the listeners' minds.
"Then how to cross this strait? How to 'translate' the language of number theory into the language of geometry, or vice versa?" Yue'er operated the controller; on the screen's center emerged a prominent mathematical symbol: **L(s, π)**. "Today, I would like to introduce a crucial 'translator,' a bridge linking the two shores—the **L‑function**."
To make the core idea comprehensible to mathematicians from different fields, even to listeners with strong mathematical background, she chose an elegant analogy.
"You may imagine it as a **mathematical codebook**." Yue'er's voice carried a guiding clarity. "In this codebook, the complete 'secret information' of a certain mathematical object is recorded. This object could be a number‑theoretic object, such as a modular form—a function on the complex plane with high symmetry, holding endless information; or it could be a geometric object, such as an elliptic curve—a smooth curve defined by a simple cubic equation, endowed with rich algebraic structure."
On the screen, intricate periodic graphs representing modular forms and the smooth outline of an elliptic curve appeared side by side.
"Now imagine we have two spies from different countries," she continued with the analogy. "One understands only 'number‑theory language'; observing the modular form, he sees its Fourier coefficients, its symmetries. The other understands only 'geometry language'; studying the elliptic curve, he sees its rational‑point count, its conductor. They cannot communicate directly."
"But they possess the same **L‑function codebook**!" Yue'er's tone rose with the excitement of revealing a secret. "One of the core conjectures of the Langlands program is that for every modular form (π) described in 'number‑theory language,' there exists a corresponding algebraic object (say, a Galois representation ρ) described in 'geometry language,' such that their **L‑functions L(s, π) and L(s, ρ) are completely equal**!"
On the screen, the images representing modular form and elliptic curve were connected by a large, faintly glowing "L(s)" symbol, signifying that their L‑functions were the same.
"What does this mean?" Yue'er looked around the hall. "It means that although modular forms and elliptic curves appear unrelated—one concerns infinite series and symmetry, the other concerns solutions of polynomial equations—they are proven, via the L‑function codebook, to describe the same deeper, essential mathematical reality! The Fourier coefficients of the modular form (number‑theoretic information), encoded by the L‑function and translated, might correspond to the number of points of the elliptic curve over finite fields (geometric information). And vice versa."
"This 'codebook'—the L‑function—possesses a magical ability: it can transform complex, obscure information from one mathematical domain into a form that may be easier to understand and handle in another domain. It is like the Rosetta Stone, providing us with a key concordance table for deciphering different mathematical 'languages.'"
She began steering the topic toward her more central exploration: "My work attempts to probe whether this profound unity can provide new perspective for understanding another fundamental conundrum—computational complexity, especially the P versus NP problem. If such exquisite 'translation' mechanisms exist between number theory and geometry, could 'computability' and 'non‑computability,' 'easy' and 'hard,' also be linked and interpreted by some analogous 'L‑function' within a grander mathematical framework? Can we find 'codebooks' for complexity classes as well, thereby revealing the symmetry and invariance of their underlying structure?"
In the following time, Yue'er presented her preliminary yet highly suggestive thoughts in this direction. She sketched how computational problems might be connected to specific algebraic‑geometric objects, and attempted to outline the concept of a "complexity L‑function." She candidly pointed out the immense difficulties and immaturity of the approach, yet clearly articulated the revolutionary prospects this path might bring.
When the talk concluded, the hall fell into a brief silence, then erupted in warm, lasting applause. This applause didn't mean everyone endorsed her specific vision; rather, it was appreciation for the grand scope, deep insight, and fearless spirit of exploration she displayed. Several eminent scholars raised extremely sharp and profound questions during the Q&A; Yue'er responded one by one—some with clear answers, others with frank acknowledgment that these were exactly the directions she was striving to conquer. The clash of ideas was intense yet pure, leaving her with a feeling of exhilarating intellectual stimulation.
After the conference, she was surrounded by many colleagues continuing to discuss related issues, exchanging contact information. Only as night deepened could she extricate herself and return to her hotel room.
Fatigue surged like a tide, yet her mind remained in a peculiar state of excitement. Pouring herself a glass of water, she stood by the window, gazing at Paris's glittering nightscape—the Seine's ripples like flowing golden threads. This talk felt like a tempering of thought, forging her scattered sparks of inspiration into a more holistic research direction. Though the road ahead was long, that call named "Langlands" grew clearer and stronger within her.
She opened her laptop to check emails, handling backlogged matters from the conference. Among the many messages from academic journals, collaborators, and conference organizers, an email from a sender named "Mozi" caught her attention.
The subject was simple: "Reflections on 'Order' and 'Codebooks.'"
Yue'er paused a moment, recalling that late‑night phone call from Shanghai discussing "order." With a touch of curiosity, she clicked the email.
The body held no polite greetings, diving straight to the point:
"Ms. Yue'er, apologies for disturbing again. Through a live‑stream link on an academic forum, I had the privilege of listening to your talk today in Paris. I was deeply moved. Your analogy of the L‑function as a 'mathematical codebook' is brilliant.
"This reminds me of my own field. In the sea of data that is financial markets, there also exist various 'languages'—price series, trading volume, news text, social‑media sentiment… They appear chaotic, yet perhaps there exists some deep, unified 'codebook' capable of 'translating' these different dimensions of information into a more essential description of market state. The models we currently use may only be attempting to decipher scattered fragments of this codebook, far from grasping its full picture.
"What you pursue is the unified codebook of the underlying structure of the mathematical universe. What I attempt is to search for the possibly existing, dynamically evolving 'order codebook' in chaotic markets. The two fields differ greatly, yet on the point of 'seeking a unified explanation,' there seems to be a curious resonance. I wonder what your view on this might be.
"Also, your mention of the Langlands program attempting to connect number theory and geometry makes me think: between the 'number theory' of capital markets (micro‑trading behavior) and the 'geometry' of macro‑market morphology, might there exist a similar, undiscovered 'Langlands correspondence'?"