Passing on Knowledge

"OK, I've got it."

Shen Qi wrote on the blackboard:

Let Xr+(T)={λ=∑m+n, i=1λiεi∣0≤λi-...}

"This is... an enhanced version of Algebraic Group Moduli?" Ralph suddenly stood up, his eyes fixed on the blackboard.

"Yes, but not entirely." Shen Qi continued his deduction. Considering the Module Category, he performed a Linear Mapping to obtain Finite-Dimensional Graded Algebra.

"I think there's a condition here that satisfies the Reciprocity Law in the Infinitesimal Group Representation Category, ultimately giving the result of the Irreducible Representation Frobenius Kernel representation of the supergroup. Let m≠n(mod p), then St1 is a projective u(g)-modulus, therefore T(μ) is a projective u(g)-modulus. Hmm, this is more like a lemma, and I need to prove it." Shen Qi turned to Ralph and said, his eyes flashing with an all-knowing wisdom.

"Then...please begin your proof, Professor Shen." Ralph was intimidated by Shen Qi's powerful aura.

"Wait a moment." Shen Qi walked to the bookshelf and took out a document to consult.

Ralph: "..."

"Lie Algebra, Complex Analysis, Group Theory —their outlines are etched in my soul, but I can't remember all the details, so sometimes I need to look them up," Shen Qi said, smiling at Ralph. "You were just thinking that Shen Qi is just an ordinary person, not an omnipotent god. Yes, that's right, I'm not a god. I'm just more focused on my work than ordinary people. Focus leads to expertise, and the depth of expertise determines the height of one's career."

"Professor Shen, you're absolutely right." Ralph was filled with respect, but also secretly astonished. How could you read my mind?

After reviewing several conclusions in the Holmes - Nakano Shigeo method, Shen Qi provided a proof of the lemma on the blackboard.

This process lasted for more than ten minutes. Shen Qi wrote, and Ralph read. The two were very focused, one imparting knowledge and the other absorbing it.

Shen Qi tapped the blackboard. "In the proof process, I used the Holmes - Nakano Shigeo method, the Brauer Model, and the Humphreys Theorem, all of which are existing theories. I didn't propose any innovative inventions. Innovation is extremely difficult. Most of the time, we did comprehensive application and theoretical reshaping. In a sense, comprehensive application is also a kind of innovation."

"Yes." Ralph nodded emphatically, a look of sudden enlightenment on his face. "The most crucial point is the application of the Weyl Group, where the point action is still defined as ωv=ω(v+ρ)-ρ, so we obtain that its weight space dimension is 1, that is, Q1(λ) is the projective cover of the restricted irreducible g-modulus L1(λ)."

"Very good." Shen Qi began erasing the blackboard.

"Wait a minute, Professor Shen!" Ralph quickly took out his phone, wanting to take a picture.

"Stop filming, Ralph, there's no need." Shen Qi continued erasing the blackboard, quickly finishing a whole board of mathematical symbols. "It seems you haven't fully grasped what I meant by comprehensive application. Its essence lies in the built-in logic, not the external form. Logic cannot be filmed."

"I..." Ralph felt ashamed.

Ralph is a down-to-earth guy who doesn't play games, but he's a little lacking in comprehension.

Yu Lei is eloquent and seems a little impetuous, but his comprehension is better than Ralph's, and his thinking is divergent.

As Shen Qi spent more time with the two students, he became increasingly familiar with their personality traits and academic strengths and weaknesses. He found that extremely meticulous people may lack rich imaginations, while people with unrestrained thinking often cannot prove the conjectures or predictions they come up with in a flash of inspiration.

No one is perfect, and teaching according to aptitude is a test for teachers.

Shen Qi said, "You can repeat my thought process, but please don't copy my proof. Ralph, I'll give you some new hints. Start from another path, relying on Exchange Superalgebra, and comprehensively utilize Matrix Elements, Frobenius Morphism, and Baby Verma Module to complete the proof of the lemma we just discussed."

"Pay attention to the order of the mathematical tools I mentioned. If you use Baby Verma Module first, you will waste your time. My expectations of you may be a bit high, but I think it is necessary because you are a Mathematical Research at Princeton, and you also did your undergraduate studies at Princeton." Shen Qi expressed his earnest hopes for Ralph. As a mathematics mentor, the most important thing is to "guide" rather than "cram".

The third condition for advancing to the Math Level 14 requires Shen Qi to mentor a student who makes significant progress under Shen Qi guidance. The day the student publishes a paper in Top Four Journals will be the day Shen Qi's condition three task is completed.

Shen Qi will naturally step in when necessary, but if he does everything for Ralph, it won't count as guidance, and the mission won't be completed.

The pressure is immense... Ralph is drenched in sweat.

In today's fast-paced society, who doesn't experience some pressure?

Pressure is a motivator.

Ralph lingered in Shen Qi's office, determined to get to the bottom of things. "Professor Shen, when studying finite-dimensional modules of typical Lie Algebra, the Baby Verma Module has a single u(g)-head. My question is, in your conception and derivation sequence, how do you arrive at the conclusion that any irreducible u(g)-submodule is a homomorphic image of some Baby Verma Module?"

"This isn't difficult. Do it this way, and this way..." Shen Qi gave him instructions.

"I see, I understand now." Ralph jotted down important information in his notebook, then asked, "Professor Shen, the use of Matrix Elements is a fundamental algebraic tool, but I feel it's not that simple. Please tell me..."

"This isn't difficult. Do it like this, and like this..." Shen Qi patiently guided Ralph.

"Then Professor Shen..."

"Then Professor Shen..."

"Ralph, you can only ask a maximum of three questions a day. Today's Q&A session is over. Now it's time for entertainment. Let's go relax."

Shen Qi invited Yu Lei, and the three of them went for a run on the Princeton campus, a relaxation method that Shen Qi taught his students.

"Ralph, you're terrible! You collapsed after running 10 kilometers!" Yu Lei yelled.

"Try to race me in a sprint." Ralph lay on the grass, panting heavily.

"We'll talk about the sprint later," Shen Qi said. "But today, there are still 10 kilometers left."

"You can run so fast!" Ralph gripped the grass tightly with both hands, refusing to get up no matter what; he was playing dead.

"Yu Lei, let's continue for another 10 kilometers," Shen Qi called to Yu Lei.

Yu Lei immediately collapsed onto the grass. "Actually, my strongest event is also sprinting."

"You guys are too weak." Shen Qi strode away, continuing to run alone.

"Professor Shen has amazing physical strength and boundless energy. Besides mathematics, he is also good at physics and long-distance running." Ralph was truly impressed.

"He won a long-distance running championship when he was studying in China. By the way, Ralph, how's your collaborative project with Yale going?" Yu Lei asked with concern.

"Not very good."

"You're keeping it a secret from me?"

"It's just not very good."

"Never mind, I don't care anyway." Yu Lei looked at Shen Qi, who was walking away. He and Ralph were Shen Qi's first batch of graduate students, and he didn't want to lose to Ralph academically.

Shen Qi ran all the way to Carnegie Lake at the southernmost point, then turned west and ran towards the faculty and staff dormitory.

Shen Qi certainly has plenty of energy, even excess energy. He needs to release this excess energy and physical strength, and recently he has resumed running.

Shen Qi returned to the Princeton staff dormitory, Ou Ye learned how to make soup. Life became simple and warm.

In his ordinary and peaceful life, Shen Qi was very concerned about the physics paper he submitted to PRL, which was his passion.

Shen Qi's passion triggers are the same as Ou Ye's passion triggers, and Ou Ye's passion triggers will also keep Shen Qi excited.

An old married couple needs a bit of passion to spice things up, and for Ou Ye, that passion came from discovering the mystery of integer points on a type of Elliptic Curve.

Elliptic Curve have always been a fascinating topic in number theory and related fields. When Wiles proved Fermat's Last Theorem, he actually proved it indirectly by proving a class of results concerning Elliptic Curve.

"You're amazing, Ye! You're following in Wiles' footsteps." Shen Qi was delighted to see that Ou Ye had created something new on her own, without relying on him at all.

It was only while having soup tonight that Shen Qi learned that Ou Ye had made research progress on integer points of Elliptic Curve.

Ou Ye shook her head. "It's not even a result. I only thought of it yesterday; it's just an idea."

Shen Qi: "Then let's make this idea a correct proposition. I won't offer you too much professional advice; I feel like throwing up whenever I see number theory. All I can do is cheer you on. If you have any questions, ask your advisor; Professor Lyndon Strauss will definitely give you more help."

"Well, what about you? Has PRL accepted your physics paper?" Ou Ye asked.

"No news yet, I'm waiting for news too."

Shen Qi waited and waited for several days, and finally received good news: PRL accepted his paper "Research on Defect Topology Based on Spherical Stable Homotopy Group".