A Little Test Before Dinner!

Dewitt sat in a wheelchair while Deng Puhe pushed him along, looking every bit like a pair of good brothers.

The three young mathematicians hurriedly rose to greet them, their faces carrying the excitement of an audience with an emperor—or rather, with two emperors at once!

"Although I'm late, I brought Deng Puhe with me. Buy one, get one free!" Dewitt joked. "This is the highest-caliber DeWitt Dinner in history. I came fully prepared!"

"I'm here to mooch a meal!" Deng Puhe laughed. "When someone else is paying, you'd be a fool not to eat!"

Dewitt and Deng Puhe took their seats, and the three young mathematicians sat down as well, somewhat stiffly.

"I've already ordered the dishes. They'll be served soon, so please trust my taste." Dewitt was visibly animated. "I'm not one of those fake foodies who eats General Tso's Chicken. Every year, I spend a month as a Visiting Professor at Tsinghua University or Peking University. Deng often takes me to fine restaurants, and I'm familiar with all the major Chinese cuisines! In a few years, when my mind starts failing, I'll switch careers and become a food critic!"

The three young mathematicians quickly expressed their anticipation. Since someone else was treating them, there was no need to be picky. Whatever was served, they would eat. In such an upscale place, surely they would not bring out two bowls of instant noodles.

"Over the past few days, several of my doctoral students have been preparing to choose directions for their graduation theses." Dewitt abruptly changed the subject and grew serious. "This morning, they sent me some possible topics. I'd like to hear what you think. Once we've briefly discussed them, you can ask me about any directions that interest you."

As he spoke, Dewitt reached behind his back, pulled out a black leather bag, and took out several sheets of paper.

The three young mathematicians hurriedly stood, each taking a sheet and laying it on the table to study carefully.

Song He immediately understood what the old man was up to. Who knew whether these were really doctoral thesis topics, but they were definitely being used for a surprise test. Dewitt also wanted to probe the abilities of those present.

Put bluntly, it was a little test paper!

The atmosphere at the dining table immediately grew tense. The three young mathematicians desperately read the questions while Dewitt and Deng Puhe whispered to each other, just like two proctors.

"Don't be too nervous. I merely want to hear your thoughts. Any ideas are fine. I might even steal some of your viewpoints and pass them on to my doctoral students." Dewitt smiled. "Any thoughts on the first question?"

Galois's gaze wavered, but he still spoke first.

"For the first question... I think we could construct a space as a k-fold Covering Space and give a representation through monodromy action. We would only need to add finitely many points to the conditions and... and..."

Dewitt held a copy of the little test paper as well. After listening carefully, he nodded and looked kindly at Galois. "And what?"

"Extend the holomorphic mapping to G!" Katerina suddenly added. "Galois's approach is to first regard it as a local problem, representing the elements of the corresponding permutation group as products of disjoint cycles!"

"Yes, yes!" Galois shot Katerina a grateful look. "I think approaching it this way could solve the problem."

"Very good!" Dewitt beamed.

"I think we could approach it from another direction." Katerina had gotten into the rhythm. "Since J is known to be a branched cover, then j can have an inverse map, which means the Covering Space is trivial."

"The Covering Space might also be disconnected," Galois suddenly added.

"Right. In any case, that contradicts the facts given in the question, so there's no way to obtain an inverse map of J on the punctured plane Ω." Katerina spoke faster. "Why not introduce a new variable and specifically consider the algebraic variety inside L? Restricting the projection map a should give us the branched cover we need."

Dewitt held the little test paper, one brow lifting slightly. "Hmm, that makes sense! And then?"

"And then..." Katerina suddenly got stuck.

"How about this?" Galois hesitated before speaking. "Since the finite covering map here is smoothly homotopic to the standard map, we can try assuming c1 is given by H, then glue together the standard map and attempt to obtain an n-fold branched cover jn. If we do this for every j, we can obtain..."

"A branched covering map!" Katerina exclaimed in delight. "Right! That satisfies the required conditions. Once all the local issues are resolved, the global branched cover is resolved too, and the rest becomes easy!"

"There's one detail to watch out for here..." Galois picked up the thread again.

Galois and Katerina had completely gotten into the zone, constantly advancing the problem as they spoke rapidly back and forth.

They were like runners in a relay race on a track, passing the baton to each other after every lap. The other would choose a new lane, sprint around it, then return to hand over the baton again. They had somehow developed a quick rapport.

Dewitt studied the little test paper, occasionally murmuring assent and maintaining a kindly smile, as though admiring promising juniors.

Deng Puhe did not look at the questions. He silently observed the expressions of the three young people, eventually focusing on Tang Jiang.

Tang Jiang sat off to the side, staring at the test paper without a word. From time to time, he looked up at the two companions who were talking at breakneck speed. His expression was blank, and he carried an air of detached calm. He seemed wholly uninterested in Katerina and Galois's solution process, as though he were simply waiting for the food to arrive.

After a long while, Dewitt looked up as well and turned to Tang Jiang.

"I more or less understand Katerina and Galois's lines of thought. Tang Jiang, why haven't you said anything? Do you have anything to add?"

Song He spoke, his smile somewhat awkward. "I couldn't keep up."

Katerina froze for a moment, then turned to look at him suspiciously.

At that, the tension on Galois's face eased considerably. It seemed someone had taken last place.

"I have another angle of approach," Song He added. "Regarding multivalued holomorphic functions, we might as well equivalently consider the corresponding algebraic curve J."

The dining table suddenly fell silent.

Galois and Katerina stared at the test papers in their hands, frowning with bewildered expressions.

Dewitt frowned slightly as well, looking thoughtful. "And then?"

"Considering that the multivalued function may have branch points at infinity, we can, based on the situation at infinity, find an equivalent compactification x on the algebraic curve, sort out the projective space in the problem statement, and homogenize its corresponding polynomial." After speaking, Song He picked up his teacup and took a sip.

Galois frowned more deeply. "That doesn't seem workable, does it?"

Katerina hesitated for a moment without saying anything, then looked up at Dewitt.

"Based on what Tang Jiang just said, it indeed won't work," Dewitt said. "There's one fatal flaw."

"Is there?" Tang Jiang was stunned.