Still Equations
Writing a History of Number Theory with a professional theoretical flavor is a massive undertaking that cannot be accomplished overnight.
Shen Qi would write a few words whenever inspiration struck; he wasn't in a hurry, believing that slow and steady wins the race.
It's Wednesday, coffee time again.
Shen Qi was chatting with several doctoral students in the coffee shop on the third floor of the Mathematics Department.
"Jonas, this is the first time I've seen you at the coffee shop since last September, and you know I've never missed a single Wednesday coffee session," Shen Qi said.
Jonas is also a doctoral student, and this year marks his ninth year at Princeton.
A little over a year ago, when Shen Qi came to Princeton University to study for his master's degree, Jonas was a doctoral student.
It is highly likely that Shen Qi will receive his PhD in a few months, while Jonas will still be a doctoral student.
"I'm qualified to have coffee here, aren't I?" Jonas laughed.
"Of course." Shen Qi nodded, then asked another doctoral student, "Chris, how's your research project going?"
Chris, wearing glasses, said with a very mysterious and serious expression, "I'm about to solve Goldbach Conjecture 1+1."
"Oh, really?" Shen Qi was skeptical. If Chris was telling the truth, it would be explosive news that would shock the Mathematics Community.
"And you, Sebastian, what research topic are you working on?" Shen Qi asked a doctoral student with curly hair.
Sebastian smiled faintly, "I think I have found a general solution that satisfies the Yang- Mills equations in four-dimensional Euclidean space for any compact, simple gauge group."
"You're amazing, Sebastian." Although Shen Qi complimented Sebastian verbally, he harbored doubts in his heart.
Finding this general solution means that one of the Millennium Prize Problem that has puzzled scientists for decades—the Yang- Mills equations—is now fully explained mathematically.
What a lucky day! Chris claims he's about to solve Goldbach Conjecture 1+1, and Sebastian says he's solved the Yang-Micius equations.
Are these two doctoral students truly exceptionally talented, or are they just boasting to the heavens?
Further verification is needed.
Shen Qi was still a little nervous. If the Goldbach Conjecture and the Yang-Micius equations were really solved by Chris and Sebastian, who had been keeping a low profile for several years, then they would become the most dazzling academic stars of today.
The number of times someone can act cool is limited; for every cool person someone else does, you'll have one less to show off.
Shen Qi asked, "Sebastian, could you demonstrate the general solution to the Yang-Mille equations? Of course, you have the right not to, if your research results haven't been published yet."
"I'd be happy to do that." Sebastian stood up, holding his coffee cup, and began writing on the blackboard with chalk.
Unlike other fancy cafes, the Princeton University Mathematics Department's café features several blackboards on the walls, allowing guests to improvise on them when inspiration strikes.
Sebastian sipped his coffee while solving the Yang- Mills equations, completely at ease and confident.
"This..." Shen Qi's heart leaped into his throat. Sebastian had used the symmetry group approach, and that was the right idea. Could it be that he had indeed found the general solution to the Yang-Mille system of equations?
On an ordinary Wednesday, the Yang-Mille equations were solved just like that?
Princeton truly is a place where hidden talents abound!
Sebastian quickly wrote down his answer: Du = IΘu - i8T^aAu^g
"Wow! Sebastian, you're amazing! This year's Fields Medal is yours!" Chris clapped.
"You're just as outstanding, Chris. The Fields Medal is ours." Sebastian smiled at Chris, his eyes filled with tenderness.
"I... Pfft..." Shen Qi almost spat out his coffee. He tapped the blackboard and said with great skepticism, "Sebastian, you're kidding me. I absolutely don't believe that what's written on the blackboard is the general solution to the Yang- Mills equations. It's just the definition of a covariant derivative! However, your earlier handling of symmetry groups was quite interesting. Mathematically speaking, it has some original ideas and academic value..."
"There's not enough blank space on the blackboard, so this is all I can write. In short, all my core ideas are written on the blackboard. You can understand as much as you can," Sebastian said, spreading his hands, and then sat back down next to Chris.
This is an incredible thing. The most precise physical theory in the world is built on a set of equations that no one understands, and to this day, no one has been able to find the general solution to this set of equations.
Shen Qi had never seen what the general solution of the Yang-Mills equations looked like; no one in the world had ever seen it, including Yang Zhenning and Mills, the founders of the Yang-Mills equations.
However, anyone with basic graduate -Mathematics Department knowledge and a certain understanding of Maxwell's equations, Schrödinger Equation, and General Theory of Relativity can immediately determine that Sebastian's answer is unrelated to the general solution of the Yang-Mille equations.
"Jonas, what do you think of Sebastian's solution to the Yang- Mills equations?" Shen Qi asked Jonas.
"Sorry, I don't understand it; it's not my area of expertise. If Chris can write a solution to Goldbach Conjecture 1+1, I think I can offer my opinion." Jonas's expertise is in number theory, and he expressed interest in Chris's claim that Goldbach Conjecture 1+1 was about to be solved.
Shen Qi continued studying the derivation process and conclusions on the blackboard. He felt that Sebastian was just spouting nonsense, but it was also possible that Sebastian was right. His physics level was only level 6, and perhaps he hadn't fully grasped the essence of the Yang-Mille equations?
The Yang-Mille equations are not simply mathematical problems; they are physical theories proposed by physicists who have constructed the Standard Model of particle physics, but they cannot explain them from a mathematical perspective.
For example, a child, with exceptional spatial construction skills, builds an impeccable castle using hundreds of building blocks. He knows how to build, and his work is perfect in practice. However, the child cannot explain from the principles of spatial geometry why he builds the blocks this way. Can he provide a theoretical explanation, suggesting that this construction method is the best in the world?
This little boy is a physicist. He went to ask his dad, a mathematician, "Daddy, I need a mathematical explanation to prove that the castle I built is the best castle in the world."
The mathematician father was also baffled. His expertise was limited; he only knew the result but couldn't provide a fundamental explanation.
The Yang-Mi equations are roughly in the above situation. They can still be used even when the general solution is unknown, and have been used for decades without failing. However, an equation system without a general solution never provides complete peace of mind; what if, under some extremely rare circumstances, it fails?
Similar to the Yang-Mille equations are the Navier-Stokes equations.
Even when humanity cannot achieve breakthroughs in cutting-edge theories, it can still develop and apply technologies at a rapid pace. However, those who engage in theoretical research always want to thoroughly understand the fundamental theories.
Shen Qi was starting to waver after being swayed by Sebastian. Just then, Edward Witten, who had been sitting in the corner observing coldly, spoke up, "Sebastian, you've disappointed me greatly. Is this the result of your four years of doctoral studies?"