It Would Blow Up

Feng Zechun saw the thick stack of papers and was slightly taken aback. He took them and glanced at the title.

"Research on Blow-Up Phenomena in Supercritical Defocusing Nonlinear Wave Equations... You're still doing research after graduating?"

Feng Zechun asked in surprise. "Looking at this stack, you must have put in a lot of work."

Huang Tian said, "It's just something I'm interested in. I was pretty fond of mathematics as an undergrad and thought I'd pursue graduate studies, devote myself to research, and put down roots in academia. Later, my family came under financial pressure, so I had no choice but to find a job straight out of college.

"The job wasn't exactly unbearable. It was just tedious and repetitive, and of course, I hated having to work overtime so often. As time went by, I started thinking about mathematics again. I realized I still couldn't let it go. If I didn't try walking this path, I'd never be fully satisfied..."

Feng Zechun listened, deeply moved, as if he could see before him the image of a scholar who was driven to learn and would never give up, no matter how many setbacks he faced. He nodded repeatedly in approval.

"Students with your passion for learning are rare these days. Now, I'm happy if my students just diligently finish the assignments I give them. Back in my day, everyone I knew went mad for it, studying and digging into the subject with all they had. You can't learn anything real unless you push yourself to the limit..."

He reminisced for a while, then suddenly slapped his forehead and said sheepishly, "I'm getting old. I just love reminiscing."

He took a cloth from the glasses case on the glass table beside him and wiped his reading glasses, then asked, "Huang Tian, can you tell me about the results and innovations in your paper?"

Huang Tian answered without hesitation, "Based on my calculations, I've reached a conclusion: supercritical defocusing nonlinear wave equations can blow up!"

If you tossed a pebble into a still lake, it would send ripples spreading outward in all directions. The energy would spread with them, eventually returning to a state of calm. We can call the process of energy gathering together focusing, and the process of energy dispersing defocusing.

The mathematical equations used to study the former are called focusing equations. For this type of equation, its solution becomes infinite at a certain moment. This is called "blow-up," because all the energy converges at a single point and bursts outward with a bang.

But Huang Tian's paper studied defocusing equations. The mathematical community generally agreed that defocusing equations could not blow up, because they represented energy dispersing in all directions. Since the energy was being released, intuition alone told us that blow-up was impossible; their solutions should exist forever and remain well-behaved.

Yet Huang Tian had overturned decades of consensus in the field, proving with mathematical formulas that they could blow up!

Sure enough, the moment Huang Tian said this, Feng Zechun was startled. "You're saying it can blow up?!"

Huang Tian nodded. "Yes, if mathematics hasn't deceived us."

Feng Zechun hurriedly asked, "What method did you use? What tools?"

Huang Tian replied unhurriedly, "Usually, when people study defocusing equations, they focus almost entirely on real-valued solutions. After all, how could an outward-pushing force possibly blow up?

"But I tried complex-valued solutions and introduced special phases and oscillation modes. That led me to discover that changes in phase and oscillation could create complex interactions between them, which in turn caused the energy to focus and blow up.

"The most important part of the paper was introducing the relativistic Euler equations and turning the wave equation into a fluid mechanics problem."

Feng Zechun's eyes widened. "You introduced the relativistic Euler equations?"

"Yes. During my research, I found a self-similar implosion solution in the relativistic Euler equations. Then I found a similar self-similar structure in the defocusing equation, so I tried transferring the structure, mapping the former into the framework of the latter. That let me turn the latter into a fluid mechanics problem."

Feng Zechun's brow furrowed deeply as he murmured, "Self-similar, complex-valued solutions..."

At one point, he suddenly looked up. His cloudy eyes lit up, and he stared at Huang Tian in delight. "I haven't read your paper carefully yet, so I can't say whether your results are correct. But your approach alone is extraordinarily original! Transferring the relativistic Euler equations is a stroke of genius!"

Huang Tian smiled, neither boastful nor modest.

Feng Zechun said excitedly, "I'll have to take a good look at your paper. But I have a lecture in half an hour..."

He took out his phone, found the number for Xu Kaizhi, dean of the School of Mathematics, and called him.

A few seconds later, Xu Kaizhi answered. His voice was deep and booming. "Hello, Old Feng. What made you call me all of a sudden?"

Feng Zechun got straight to the point. "I've got something good here. Come take a look. I'm waiting for you in the lounge on the fifth floor of the Life Sciences Building. Also, find me a substitute lecturer. I'm skipping my next class."

"Something good? Did you come across a paper you can't make up your mind about?" Xu Kaizhi immediately caught on.

"That's right. Just come over. It's hard to explain on the phone."

"Okay, give me a few minutes. I'll arrange a substitute lecturer first."

After hanging up, Feng Zechun waited for Xu Kaizhi to arrive. Instead of rushing to read the paper, he chatted with Huang Tian, his warmth and kindness practically spilling from every word.

Before long, Xu Kaizhi appeared in the doorway of the lounge, wearing a gray jacket. He was in his fifties, with most of his hair still black. He had a square face, a lean build, and a lively air about him.

"Old Feng, what kind of paper is so good that you had to summon me in such a hurry?"

Xu Kaizhi walked into the room with a smile. When he saw Huang Tian, he paused. "And who's this?"

Feng Zechun introduced him proudly. "One of my former students, Huang Tian. He's the one who wrote the paper I wanted you to look at!"

"Dean." Huang Tian greeted him. He knew Xu Kaizhi, who had been appointed dean of the School of Mathematics at Chang University during Huang Tian's third year and had held the position for several years now.

Xu Kaizhi looked surprised. "You're from our university? You look like a fine young man, the scholarly type. What year are you in for your doctorate?"

Feng Zechun said, "He graduated with his bachelor's degree last June."

Xu Kaizhi was astonished. "A bachelor's?"

An undergraduate had written a paper that Feng Zechun couldn't make up his mind about?

Feng Zechun nodded. "I'm telling you, I really failed to recognize his talent back then. Huang Tian's mathematical intuition and insight are absolutely top-notch..."

He gave Xu Kaizhi a rough overview of Huang Tian's groundbreaking ideas in the paper. Xu Kaizhi's eyes grew brighter and brighter as he listened. At last, he drew a quiet breath. "That really is a brilliant idea. If the results prove correct, this would absolutely be a top-four journal paper..."

The four top journals in mathematics were Annals of Mathematics, Inventiones Mathematicae, Acta Mathematica, and Journal of the American Mathematical Society. Of course, people often spoke of a fifth and even a sixth journal as well, but that was another matter.

Publishing a paper in one of the top four was enough to let you swagger through most of academia. At Chang University, at least, you could swagger through the whole place. The university might not publish even one paper in those journals in a decade...