I was a top student in America.

Professor Philip's mathematical analysis workshop is primarily for university students preparing for the Putnam Mathematics Competition.

Li Ao went upstairs and entered the small lecture hall, where the first few rows were already mostly occupied.

He had just found an empty seat when Benjamin followed him in.

Their eyes met briefly before they looked away, neither of them speaking.

Professor Jensen, an expert in real analysis, sat in the back row. He had come specifically to sit in on the meeting today. When he saw that Li Ao and Benjamin were both there, he nodded in satisfaction.

Not long after, Professor Philip walked onto the podium, carrying a paper cup of coffee in one hand and lecture notes in the other.

The classroom immediately fell silent.

"There are quite a lot of people today."

Professor Philip put down his coffee cup, glanced at the audience, and casually began before getting straight to the point: "In this class, we'll talk about uniform convergence, Fatou's lemma, and Jensen's inequality chains, and also see how they're used in estimating the mean of convex functions."

……

This workshop is a high-intensity, advanced course.

For more than two hours straight, Li Ao listened attentively.

Although he has already completed AP Calculus and half of "Principles of Mathematical Analysis," and has the added benefit of the term "abstract thinking," this level of coursework still cannot be taken lightly.

After the class, he felt he had gained a lot.

He also noticed that a professor from the coaching staff sitting not far away would occasionally glance in his direction.

Li Ao felt a little uneasy. It seemed that a perfect score on the initial assessment test had indeed caught the attention of the coaching staff.

get out of class was about to end.

"The essence of analysis is not computational techniques, but the use of the language of limits and the structure of inequalities to clearly explain the dynamic process of 'infinity'."

After Professor Philip finished his final summary, the classroom was filled with the rustling sound of people packing up their things.

The students got up and left one after another.

[Under intense study conditions... Intelligence +0.1]

As Li Ao was looking at the pop-up panel notification, Benjamin's slightly excited voice suddenly came from beside him:

"Professor Philip, hello. Your lecture on the mean estimation of convex functions was particularly enlightening for me. I had previously read your paper on weak form estimation and Hardy-Littlewood maxima carefully, and I'm so glad to have the opportunity to listen to your lecture in person today."

Li Ao looked up and saw that Benjamin had somehow gotten to the front of the podium.

Professor Jensen, who was standing nearby, also walked over, carrying a stack of lecture notes.

Seeing this, Benjamin quickly turned to Professor Jensen and struck up a conversation with him familiarly:

"Professor Jensen, could you please take another look at the draft of the project on the optimal constant of Hardy's inequality that I sent you last week? I've replaced one of the lemmas with a more concise proof."

Li Ao had thought the two professors would take the opportunity to chat with Benjamin for a while, but Professor Jensen dismissed him with just a few words: "I've looked at the draft, the ideas are alright, we'll talk about it later. I heard you got a perfect score on the placement test? Well done, keep up the good work."

Upon hearing this praise, Benjamin wore a humble expression, but couldn't suppress the slight smirk on his face.

But his joy didn't last long before he saw Professor Jensen pick up his lecture notes and walk straight toward Li Ao.

Benjamin's expression froze on his face.

Did you understand today's seminar?

Professor Jensen stopped beside Li Ao's desk and asked, "Today's lecture is completely different from the high school competition; it's much more abstract. Old Philip skipped a lot of the basics when he taught the undergraduates, and he went a bit fast, and it's all symbolic real number theory."

Li Ao quickly stood up: "I can basically keep up, but there are a few parts that I need to go back and re-analyze. I have indeed learned a lot."

Li Ao had met this black belt coach at the opening ceremony, so it wasn't surprising that the coach could call him by name.

After all, the coaching staff had arranged for him to attend this class, and besides, the coaches had probably already circulated that perfect score paper from yesterday.

Professor Jensen nodded, then asked:

"You're also studying advanced mathematics on your own now, right? What books have you been reading lately?"

"Rudin's *Principles of Mathematical Analysis*, and Strang's *Introduction to Linear Algebra*."

Li Ao truthfully replied, "I just got to the vector space section of the book that Strang is working on."

"Oh? You've picked a good book, and you read it pretty quickly."

Professor Jensen was slightly surprised, but quickly got to the point, "The solution to the last question on your diagnostic test was to first prove a two-sided approximation estimate of a weighted Hermite-Hadamard inequality, right? Come on, let's go to the conference room next door and talk about it."

"Okay, Professor." Li Ao immediately agreed, grabbed the things on the table, and followed Jason out of the classroom.

Benjamin stood there, staring at the two figures as they walked away, his eyes filled with confusion.

He didn't understand. Everyone else got full marks, he had an IMO gold medal, and he was in the professor's research group. Why would Jason care so much about a newcomer?

However, he simply took a deep breath, quickly adjusted his expression, turned around, put on a smile, and started talking to Professor Philip again.

……

Li Ao followed Jason into the conference room and saw David sitting next to him, stuffing lecture notes into a folder.

He greeted him politely, "Hello, Mr. David."

David looked up at this and nodded with a smile, and the two didn't talk any more.

Professor Jensen handed over several blank sheets of draft paper.

Li Ao pulled out a chair, sat down, and immediately began writing down the proof process.

Due to limited space on the exam paper, he wrote many boundary conditions rather roughly, so now he's simply filling in the steps as well.

After writing the last line of formula, he pushed the paper over:

"Professor, this is my complete proof of the two-sided approximation of the weighted Hermite-Hadamard inequality. The last question on the diagnostic test was just a discretized special case of it."

Professor Jensen was already very interested in the short answer section of the exam paper, but now that he had the full version, he immediately lay down on the table and started working on it carefully, wanting to see if Li Ao had really proven the chain of inequalities.

David also came over, grabbed a pen, and added a few weighting conditions.

At first, the two appeared normal, but as they continued calculating, David involuntarily raised an eyebrow, and Professor Jensen's pen stopped moving.

He put down his pen, looked up at Li Ao, and showered him with praise:

"The proof is very rigorous. You have indeed derived a two-sided extension of the weighted Hermite-Hadamard inequality, which is much more accurate for estimating the mean of weighted convex functions. It is especially useful for estimating integral errors and solving inequality extrema problems."

Professor Jensen patted the draft paper. "I suggest you turn it into a short note, polish it a bit, and submit it for publication."

Although Li Ao was very confident in his proof, he was still somewhat surprised to hear that he could publish a paper.

At this stage, he is focused on how to get into the national team during the summer camp and has no intention of doing scientific research immediately.

Unexpectedly, the ideas I casually organized while trying to improve my "intelligence" attribute actually formed the prototype of a short mathematical analysis article, which was quite an unexpected bonus.

Just as I was pondering this, Professor Jensen spoke up again:

"You'll all have to write your own papers sooner or later, so this is a good opportunity to practice. Feel free to ask me anything if you don't understand anything, and ask David for help with formatting issues. If this is actually published, it will be very helpful for your college applications later."

He paused for a moment, then continued, "Since you got a perfect score on the placement test and were able to keep up with today's seminar, you can come and listen to undergraduate classes more often. You can choose whichever of the regular problem-solving classes you attend during the summer camp."

"Also, if you think you can handle it, you can come to my research group to do some small projects, such as helping to organize some data on the estimation of the mean value of convex functions. By the way, Benjamin from earlier is also in our group."

Having said all that, Li Ao had no reason to refuse.

Anyway, in his previous life he had helped many students revise their English papers, so he was very familiar with the format and layout.

He thought for a moment and then nodded in agreement: "Yes, Professor."

David pretended to organize his notes, but his ears were practically perked up.

Upon hearing that his advisor not only encouraged Li Ao to write his thesis but also directly brought him into the project team, Li Ao's hands involuntarily slowed down.

It would be normal if it were a senior undergraduate student joining the group, but this kid has only been at summer camp for a few days.

No sooner had Li Ao stepped out of the conference room than David couldn't help but ask:

"Professor, isn't it a bit too early to have Leo try writing a paper now?"

Professor Jensen tidied up the draft papers on the table, and without looking up, asked in return:

"Have you forgotten where this is? The National Math Competition Summer Camp!"

"There has never been a shortage of geniuses here."

……