Solving Doubts

Princeton has more than one Fields Medal winner.

It is not uncommon for students to receive instruction from Fields Medal winners in the classroom.

Classes taught by Fields Medal are usually packed, and Shen Qi's Functional Analysis class was no exception.

Graduate students in the Princeton Mathematics Department choose their own advisors and then delve into one or two main research areas under their guidance.

At the same time, graduate students must also complete the necessary credits.

For example, Yu Lei chose Shen Qi as his supervisor, and his postgraduate research focused on Function Theory, including Real Analysis and Complex Analysis.

In addition to Real Analysis and Complex Analysis, Yu Lei also needs to study general graduate courses such as Functional Analysis, Lie groups, Lie Algebra, and Partial Differential Equations.

You should allocate more energy to courses that are your main focus, and for other courses, allocate study time and effort appropriately according to your individual circumstances.

There is overlap between Functional Analysis and Function Theory and algebra theory, and Function Theory and algebra theory are the main research areas of Yu Lei and Ralph.

Not only Yu Lei and Ralph came, but even Zhou Yu'an, who is about to start his doctoral studies, came.

Zhou Yu'a Yu Lei traveled together; both of them lived in the graduate student dormitory.

The two arrived just in time, reaching the classroom one minute before class started.

To his surprise, all the seats were already taken. Ralph arrived 10 minutes early, and some Ralph students arrived even earlier than him.

"Professor Shen only teaches one class a month, he's really popular." Zhou Yu'an could only sit on the ground. His main focus was Differential Geometry and Partial Differential Equations. He came mainly because of Shen Qi, and he had already earned the credits for Functional Analysis last semester.

"Who says otherwise? Little Brother Qi is a newly crowned winner of the Fields Medal, he's at the peak of his career." Yu Lei was also sitting on the ground, felt proud. A strong mentor makes a strong student, and a mentor who is a big name in the Fields Medal brings benefits to the student.

At that moment, a curly-haired boy in the next seat gave up his seat and said to Zhou Yu'an, "Zhou, you can sit here."

"Thank you, La'shir, but I can sit on the floor." Zhou Yu'an politely declined the curly-haired boy's offer to sit down. He was quite well-known among the graduate students in the Mathematics Department. He was Shen Qi's undergraduate classmate and was doing research with Professor Faltings, and was about to enter the doctoral program.

This is Zhou Yu'an's third semester in the graduate program of the Department of the Mathematics Department at Princeton University. He has proven himself with his grades and gained the recognition of his classmates, some of whom even offered him their seats.

"Little Brother Yu, you're amazing! You've already made a name for yourself. You're the next Little Brother Qi." Yu Lei saw this clearly; his target for catching up was right in front of him— Zhou Yu'an.

Just then, a girl with mixed-race features entered. She hurried along, carrying a math book in her hand.

"Not a second off, just right." The girl timed it even more precisely, and she smiled at Shen Qi.

Shen Qi gestured for her to sit down. He recognized the girl; she was the most beautiful girl in the Mathematics Department at Princeton University. She, Yu Lei, and Ralph enrolled in the graduate program of the Mathematics Department at Princeton University in January of this year.

Seating is only allowed on the floor; all available chairs are occupied.

But beautiful women are treated differently; some men offered this mixed-race woman their seats.

The mixed-race girl refused several boys' offers to sit down. Instead, she sat on the floor next to Yu Lei and asked, "Hey, Yu, do you have any extra pens?"

"Yes, Heidi, it's my pleasure to help." Yu Lei handed the mixed-race girl a pen and three sheets of paper. He was used to using his laptop to take notes on the teacher's key points, but he also brought a pen and paper in case he needed them.

"Thank you," the mixed-race girl said with a smile; she had a good impression of the handsome Yu Lei.

Shen Qi, who was secretly observing everything that happened in the classroom, paid special attention to Zhou Yu'an, Yu Lei, and Ralph.

Zhou Yu'an earned the respect of his classmates with the highest academic level among his peers.

Yu Lei had just enrolled, had a handsome face and was quite popular with girls. He was good at interacting with girls of all types.

Ralph, an American native, is somewhat introverted. He is very hardworking and diligent, but doesn't have many friends.

Shen Qi began lecturing, and in this edition of the Functional Analysis textbook, he added the Muller -Shen theorem for Banach space, as well as Shen Proximity Theorem.

Including Wash's theorem and Riemann Theorem, Shen Qi's name appears in two Princeton mathematics textbooks: Functional Analysis and number theory.

The classroom atmosphere was lively, and the students asked a number of questions, including some that were imaginative and quirky.

Getting the Fields Medal bigwigs to hang his name on the blackboard would be a special kind of psychological satisfaction.

"Hanging the blackboard" refers to a teacher being unable to answer a student's question, or being unable to provide an effective solution or proof.

Shen Qi once hung up a blackboard. It was when he was a doctoral student and was teaching a tutoring class. An Arabia undergraduate student asked Shen Qi how to prove the Riemann Hypothesis. At that time, Shen Qi was speechless.

Shen Qi eventually proved the Riemann Hypothesis, and from then on he never hung up a blackboard again.

"So, are you satisfied with my answer regarding the Banach space, Yu Lei?" Shen Qi asked in English.

Yu Lei actively participated in class discussions and demonstrated his abilities. He just raised a question about Banach space.

Shen Qi answered Yu Lei's question perfectly, Yu Lei was very satisfied and benefited greatly.

"Yu Lei, your question is very common and not technically difficult. I can answer it." Zhou Yu'an muttered softly in Chinese.

"Little Brother Yu, don't discourage me like that, my fragile heart can't take it." Yu Lei didn't seem upset at all, after all, he had only been at Princeton University for a few months and was still in the learning and growth stage.

Shen Qi looked around the audience and asked, "Does anyone have any questions? If not, let's move on to the next part of the lesson."

The mixed-race girl raised her hand. "Professor Shen, I have a question?"

"Speak your doubts, Heidi." Shen Qi fulfilled his duty, imparting knowledge, teaching skills, and resolving doubts.

Heidi has delicate features and fair skin with fine pores, suggesting she may have some Asian ancestry. She said, "We all know that if either the linear dimension or the orthogonal dimension of Hilbert space H is infinite, then the Hilbert space H must be a finite n-dimensional Euclidean space Rn. Conversely, how can we explain that a maximal linearly independent set of vectors and a maximal orthogonal basis can be the same in a finite n-dimensional Euclidean space? How can we explain that the linear dimension and orthogonal dimension of the Hilbert space are equal, and both are finite n-dimensional? Professor Shen, can you answer my question?"

Heidi's tongue-twisting questions piqued the curiosity of the male students in the classroom. They entered the same thought process as Heidi, waiting for Shen Qi's answer.

Yu Lei was taken aback. This girl, who looked like a goddess, was incredibly capable! Princeton truly is a place where outstanding talents emerge!

On stage, Shen Qi clapped his hands three times in quick succession, and said in a high-pitched, affected voice, "Heidi, our Heidi, Princeton's Heidi! Hey, Heidi, you should audition for Hollywood. I guarantee Heidi Lamarr's math scores are definitely not as good as yours."

Heidi smiled sweetly and said, "Why not?"

Wow!

The boys in the classroom were excitedly shouting "Ooh ooh!" Most of them had heard of Heidi Lamarr's legend, and Heidi Lamarr was even the dream goddess of some of them during their nightly masturbation sessions.

Heidi Lamarr was once a top Hollywood actress, renowned for her stunning beauty and considered on par with contemporaries like Vivien Leigh and Ingrid Bergman.

Physical beauty alone is not enough to make someone the object of Princeton boy's fantasy.

Heidi Lamarr was a super-achiever; her major in college was applied mathematics and communications engineering. She was an inventor who invented frequency hopping technology, which is considered the origin of CDMA.

Heidi Lamarr, known as the "Mother of CDMA" and the "Mother of Spread Spectrum," had acting as a side job, which unexpectedly made her a Hollywood star.

In Shen Qi's Functional Analysis class, a mixed-race girl with the same First-Name as Heidi Lamarr raised a very insightful question about Hilbert space.

Faced with a female student who possessed both beauty and intelligence, Shen Qi certainly couldn't hang her on the blackboard. He smiled and began to explain.