Princeton's autumn deepened; gold‑red and ochre soaked the campus, adding a note of serene poetry to the ivory tower. Yet in Yue'er's mental world, seasonal shifts were almost entirely blocked. She remained day after day immersed in that abstract universe composed of symbols, conjectures, and grand theoretical structures, trying to find that crucial bridge between P/NP's fog and the Langlands program's call.
Feedback from the Paris conference and correspondence with Mozi, though broadening her vision, hadn't directly advanced her core breakthrough. That vision linking computational complexity with mathematics' deep structure still hung like a building suspended mid‑air, lacking solid foundations. She'd tried many paths, approaching the problem from different angles—but most, after going deeper, sank into more complex technical‑detail mires or led to known, unbreachable barriers.
Frustration hovered like persistent background noise; though not enough to make her abandon the quest, it genuinely drained her mental energy and inspiration. She knew she needed a new tool, a new language, to re‑express this problem, perhaps then glimpse a sliver of new possibility.
She turned her gaze toward **algebraic geometry**. This mathematical branch could be called one of the brightest gems on modern mathematics' crown, skillfully blending algebra's precision with geometry's intuition. And one of its core subjects was the **algebraic variety**.
How to explain the **algebraic variety**—this highly abstract concept—to someone outside mathematics? In her thinking, Yue'er habitually sought the most apt analogy. What surfaced in her mind was a **topographic‑contour map**.
Imagine a map depicting complex mountainous terrain. The map is covered with countless closed curves—contour lines. Each contour line represents the set of points at exactly the same altitude above sea level. For example, all points at 100‑meter altitude connect into one line; all points at 200‑meter altitude connect into another; and so on.
Now, "translate" this real‑world scenario into algebraic‑geometry language. That mountainous region can correspond to an abstract "space." And "altitude" can correspond to a **polynomial equation**. For instance, consider a polynomial with variables x and y: P(x, y) = x² + y² – 1.
What's the "solution" of this polynomial equation P(x, y) = 0? It's all ordered pairs (x, y) that satisfy this equation. In our familiar Cartesian coordinate system, this equation's solutions happen to form a **unit circle**!
In this analogy:
* **The polynomial equation P(x, y) = 0** is like the specific contour line defining "altitude 0."
* The set of all solutions of this equation—that **unit circle**—is the simplest **algebraic variety** (more precisely, an affine algebraic variety). It's a geometric shape defined by algebraic equations (polynomials).
Generalizing: any set of common solutions of one (or more) polynomial equations defines an **algebraic variety**. It might be a curve (like an elliptic curve), a surface, or a higher‑dimensional "shape" difficult to visualize intuitively. These "shapes" exist in the abstract space spanned by the equation variables.
The "contour‑line" analogy helps understand algebraic varieties' hierarchy and structure:
* Different polynomial equations define different "contour lines" (varieties).
* Studying an algebraic variety's geometric properties (e.g., whether it's connected, has "holes" (genus), is smooth, what singularities it has) resembles geographers studying terrain features revealed by contour lines (valleys, ridges, cliffs).
* Algebraic relationships between polynomial equations (e.g., whether one equation divides another) might correspond to geometric‑mapping relationships between varieties (e.g., whether one variety can be continuously transformed or projected onto another).
Yue'er immersed herself in the world of algebraic varieties, studying their topological invariants, singularity resolution, and moduli‑space theory. She was fascinated by this powerful ability to "geometrize" algebraic problems. A complex algebraic structure could be understood and classified through geometric properties of its corresponding algebraic variety.
While pondering varieties' "rigidity" (some varieties' structures are very stable, hard to continuously deform) versus "flexibility" (some varieties allow families of continuous deformations), an inspiration—like lightning through the dark night—suddenly shattered mental chaos!
**Could computational problems also be "geometrized"?**
A concrete computational problem, like a Boolean satisfiability problem (SAT), could be seen as searching, within a high‑dimensional, discrete "space" formed by all possible assignments, for "points" satisfying specific conditions (all clauses true).
Then, for this kind of computational problem, could one construct a specific **algebraic variety** such that certain geometric properties of this variety—for example, its **connectedness**, **dimension**, or whether it contains special "rational points" (solutions with certain nice properties)—directly encode the original computational problem's **computability** information?
For instance, a problem belonging to class P (easy to solve) might correspond to its "geometric incarnation" (some algebraic variety) having very "simple" and "regular" geometric structure, enabling us efficiently (via polynomial‑time algorithms) probe its key geometric features, thus determine solution existence or even find solutions.
And a problem being NP‑complete (extremely hard to solve, but easy to verify) might correspond to its "geometric incarnation" possessing extremely complex geometric structure, full of "singularities" and "high‑dimensional holes," such that any attempt to systematically probe its geometric properties inevitably demands exponential‑time complexity.
This thought made her tremble with excitement. If true, then the P/NP problem—the core conundrum of computational theory—would be transformed into a purely algebraic‑geometry problem! Determining whether P equals NP would equate to determining: do all those algebraic varieties corresponding to NP problems—with the property "easy to verify"—necessarily also have the property "geometric structure simple" (hence easy to solve)?
This was undoubtedly an extremely bold, even somewhat crazy conjecture. Linking discrete computational problems with continuous algebraic geometry required building countless bridges, defining countless new concepts. But behind this door might lie a secret path toward the ultimate answer. She immediately plunged into even more absorbed work, beginning to attempt constructing corresponding "algebraic‑variety models" for some simple computational problems, exploring possible correlations between their geometric properties and computational complexity.
As she immersed in excitement from the revelation of varieties, she received an email from Mozi—unexpectedly, an invitation.
The email mentioned Mozi would be traveling to the U.S. East Coast for business, with brief time in New York. Knowing Princeton wasn't far from New York, he inquired whether Yue'er might be available to meet—hoping for an opportunity to discuss in person some questions about "trends in mathematics" and her recent research progress.
Looking at the email, Yue'er hesitated. She was accustomed to solitude, accustomed to that calm, rational textual exchange via email. Face‑to‑face social interaction, especially with someone from a completely different field, merely connected by a few emails, was for her a drain on energy.
But… she recalled the mental collision in email exchanges with Mozi—those questions from entirely different perspectives often brought unexpected inspiration. Moreover, he'd just helped Xiuxiu solve the light‑source‑stabilizer problem (Xiuxiu briefly mentioned in a prior email, expressing gratitude), which gave her a kind of goodwill toward Mozi beyond pure academic curiosity. Perhaps face‑to‑face exchange could bring deeper intellectual stimulation?