Chapter 1660 Invisible Storm

While the world was in an uproar and Wall Street analysts were frantically adjusting their models, the Torch Laboratory, one of the centers of the storm, was in a completely different state of silence.

Chang Haonan sat in a cramped workstation that was completely inconsistent with his status, his surroundings already filled with scrap paper he'd been using for calculations.

On the terminal computer screen in front of him was a series of flow field simulation data.

To him at this moment, everything in the outside world is just distant background noise.

The inspiration in Chang Haonan's mind was sown at the moment when "Zou Yu" broke through the hypersonic speed, and now it has come to fruition.

"What if... the instantaneous value of the velocity field is just an appearance?" He muttered, tearing off another piece of manuscript paper and drawing a slightly twisted trajectory on it with his gel pen. "Perhaps... the shape of the flow field and the evolution of its topological structure are the true language of turbulence?"

Almost at the moment this thought appeared, the originally blocked thoughts suddenly became clear!

Chang Haonan knew that this was the system's response to the "feasible idea".

He immediately threw himself into a grand abstract construction.

"Treating the physical solution to the Navier-Stokes equations, that is, {velocity field u(x, t), pressure field p(x, t)}} as an abstract "point," we construct a set called the solution space S..."

But soon, the definition on paper was revised:

"No, it should be a topological space..."

"The key lies in defining the topology on S. Two solutions are "adjacent" if and only if the "skeleton morphology" and dynamic behavior of the entire flow field they describe are extremely similar under some global metric."

"This means that even if the local velocity is rapidly changing and chaotic and unpredictable, as long as the 'macro-configuration' and evolutionary trends of the flow field are similar, they are neighbors in S. The Navier-Stokes equations themselves are reinterpreted as a dynamical system defined on this abstract topological space S, a geometric flow that guides the flow of 'solution points' over time..."

"..."

The goal has completely shifted: no longer seeking an explicit universal formula, but to understand how this geometric flow drives the "solution point" to move through the "landscape" of S, especially how it changes the topological structure of the flow field corresponding to the solution.

This is to transform the most core chaotic problem of fluid mechanics into a search for order within a geometric and topological framework.

After determining the basic route, Chang Haonan pushed the calculation paper aside, straightened the keyboard, and re-entered the calculation terminal...

Defining the topology of S so that it has good mathematical properties will be the first serious challenge.

To establish a strict equivalent mapping between the NS operator and this abstract dynamical system, it is necessary to introduce difficult tools from modern geometric analysis.

Analyzing the long-term behavior of this dynamical system—the existence of attractors, stability, and bifurcations—is like walking on thin ice at every step.

He mobilized all his mathematical reserves, from fiber bundles in differential geometry to the ergodic theory of dynamical systems, and even the homology group tools of algebraic topology, all of which were incorporated into his deductions...

Modern mathematical problems have basically reached a level where it is difficult to construct the questions using paper, pencil, and the human brain alone.

For example, one of the important reasons why the Goldbach conjecture is famous is that it is one of the very few problems that ordinary people can understand the description of.

The NS equations are obviously not included in this list.

Therefore, even with the help of a cheating system, there are still many parts that need to be solved by the computer.

Of course, the good side is that Chang Haonan has long since transcended the constraints of any academic achievement system.

Furthermore, his research on the NS equations is not merely theoretical, but will directly impact the realization of magnetic confinement controlled nuclear fusion.

In other words, there is no need to prove the correctness of this achievement itself to anyone.

The light from the fusion reactor will speak for itself.

Therefore, a lot of red tape was saved.

……

At the same time, at the National Convention Center dozens of kilometers away.

Preparations for the International Congress on Industrial and Applied Mathematics (ICIAM) are in full swing.

This is the first time that ICIAM is held in Asia and in a developing country.

That’s right, at least in the official definition, China is still a “developing country”.

With the experience of preparing for the International Mathematical Congress ten years ago, China naturally invested great enthusiasm and expectations in this conference.

However, when it comes to mathematics, expectations alone are often not enough.

"Yuan, I must say that the organization of the conference is impressive and the agenda is very full."

Sir John Bohr, Chairman of the International Program Committee of the conference and former President of the Royal Society of London, gently closed the agenda book in his hand and praised sincerely.

But then he showed a hint of worry: "As for the results..."

The person who contacted Sir Bohr was an old acquaintance of Chang Haonan, Vice President of the Chinese Mathematical Society, Academician Yuan Xiangya.

"We've already planned seven award-winning presentations and 7 invited presentations, covering the most active areas of applied mathematics. In addition, there will be over 27 poster presentations and nearly 200 group presentations..." The latter took a surprisingly thick agenda summary from the side and introduced, "This will be an unprecedented conference in the history of ICIAM."

"I know, I know." Sir Pol raised his hands and pressed them down. "Actually, I'm not referring to just this meeting."

Yuan Xiangya showed some curiosity: "Then..."

Boll sighed and explained:

"What I'm trying to say is that, looking back at the past decade in mathematics, while there has been no shortage of solid progress and ingenious applications, the truly groundbreaking, historic, and groundbreaking achievements have been somewhat quiet...and applied mathematics research ultimately relies on mathematical theory, so it's inevitable that this has been affected..."

After hearing this, Yuan Xiangya shook his head helplessly.

He certainly understood why the other party defined the scope as "nearly ten years."

"Sir, a breakthrough like solving the Poincaré conjecture is something that only happens by chance. Otherwise, it wouldn't be listed as a 'Millennium Problem.'"

After finishing speaking, perhaps because I felt that the previous sharp review was a bit negative, I made up for it:

"But I believe that the next game-changing achievement may be brewing in some unknown corner."

Sir Pol nodded and looked out the window:

“That’s true. But as mathematicians, we always look forward to witnessing great moments that can reshape the discipline.”

In fact, the second half of Yuan Xiangya's sentence was purely to save face.

Boll's response was purely out of politeness.

Neither of them took this exchange seriously and quickly put it out of their minds, returning to the busy work of preparing for the meeting.

however……

One early morning half a month later.

After finishing another round of high-intensity work, Chang Haonan took a short break.

The four originally planned proof parts have all been successfully completed.

All that remains is to find the elements that connect them.

Although he said he was taking a break, Chang Haonan's eyes were still fixed on several seemingly unrelated late-stage turbulence simulations.

In traditional velocity field or vortex field analysis, they belong to completely different physical categories.

But at this moment, from the topological perspective of the S-space he constructed, the "attractor forms" of the two sets of flow fields in the later stages of evolution actually showed an astonishing equivalence in some high-order homotopy invariants that characterize the overall entanglement and connectivity!

A key lemma is like a bridge that suddenly appears, instantly connecting the subsequent chain of proofs.

Chang Haonan sat up straight suddenly, and his pen tip moved rapidly on the last draft paper, completing the final logical loop.

Under reasonable physical boundary conditions, the overall topological structure of the solution space S that was carefully constructed before was proved to be extremely simple and almost "trivial".

"what does that mean?"

Chang Haonan took a deep breath, slowly leaned back in his chair, and asked himself:

This means that the 'general solution' to the NS equations is not an explicit formula filled with partial derivatives, but rather the ultimate fate of all physically possible turbulent solutions, which, after a long evolution, can be divided into a finite number of equivalence classes based on the 'ultimate topological form' of the flow field.

“Although the solutions within each class exhibit infinite chaotic details during transient processes, their core dynamical behavior and ultimately stable manifold can be effectively described, predicted, and even controlled by a relatively low-dimensional parameter space containing several key global modes…”

The corners of Chang Haonan's mouth, which had been tense for a long time, finally curved up.

Then, the smile on his face became more and more obvious, until a row of white teeth was exposed.

"I see..." He looked at the two parts of the results on the monitor and the draft paper in front of him, and breathed a sigh of relief.

Just at this moment, the cell phone on the corner of the table suddenly rang.

In order to avoid being bothered by trivial matters, Chang Haonan turned on the do not disturb mode early in the morning.

Only numbers in the address book whitelist can call in.

He picked up his phone and unlocked the screen.

Above is a text message from Du Yishan:

Tomorrow we'll vote on Yabo's nomination for the Academy of Sciences. Do you think you can show up?

A master's school is similar to a family, and often there is a sense of "intergenerational kinship".

As Du Yishan's disciple, Li Yabo naturally received a lot of love.

Even the latter considered coming out in person to show their support.

But since Chang Haonan's current work has been completed, there is definitely no reason to let the octogenarian academician go through all the trouble again...