It's an amazing feeling.

Pang Xuelin never imagined that Ponzi geometry, which was originally used to solve number theory problems, could be associated with nonlinear partial differential equations.

The sudden inspiration suddenly diverged, and in an instant, all kinds of wonderful ideas began to emerge in Pang Xuelin's mind.

……

"In a system of partial differential equations related to surfaces, the first thing that needs to be solved is the existence of complex structures! At this point, we can start with a classic old question! That is, given a near-complex structure J on a 2n-dimensional differential manifold M, when is this near-complex structure induced by the complex structure?"

……

"A given near-complex structure J is induced by a complex structure if and only if there are local real coordinates {x^1,x^2,x^3 in a neighborhood at each point...... x^2n-1,x^2n}, such that J∂xj=∂x^j+n,J∂x^j+n=-∂x^j, because if such a local coordinate card set exists, the conversion function between the complex coordinate card set {x+ix^n+1,...,x^n+ix^2n} fits into the Cauchy-Riemann equations, and thus is a pure function; The inverse proposition is clearly true. Next, the problem can be boiled down to finding such a good coordinate system, or solving some system of first-order linear differential equations. ”

……

"High-dimensional cases: Newlander-Nirenberg theorem. The near-complex structure M is a (1,1) tensor field, so it can act on the cotangent bundle. At each point p∈M, the complex tangent space TpMc can be decomposed into the straight sum of the two subspaces corresponding to the eigenvalue ±i. According to the continuity, the straight-sum decomposition ...... of the complex cut plexus can be obtained."

……

"Lemma: Let M be a tight Riemann manifold. Consider the differential equation δu=f(x,u), f:M*R→R is a smooth function. If u-,u+∈C^2(M) is such that u-≤u+, δu-+f(x,u-)≥0 ,δu++f(x,u+)≤0, then there is a solution x∈C^∞(M) satisfying u-≤u≤u+......"

……

Time passed minute by minute, and lines of symbols like heavenly books flowed out quickly under Pang Xuelin's pen, filling one manuscript paper after another.

Pang Xuelin wandered in the ocean of mathematics, improving the theoretical framework of Ponzi geometry step by step, and enriching its flesh and blood.

The more he studied, the more Pang Xuelin felt that the Ponzi geometry theory he created was a vast space behind it.

It's like Galois, who pioneered group theory, and elevated algebra to a whole new field.

Pang Xuelin even faintly realized why the old man Grothendieck studied far-aberish geometry back then.

Ponzi geometry was created on the basis of far-abelian geometry, and on the basis of Ponzi geometry, Pang Xuelin faintly felt that algebra and geometry were merging with each other.

From the Cartesian era, which organically combined algebra and geometry through coordinate axes to form analytic geometry, and then to Riemann's creation of the theory of algebraic geometry, algebra and geometry, two important tributaries of mathematics, are not only very different, but also have a deep internal connection with each other.

However, in an era when the branches of various disciplines are becoming more and more fragmented, it is almost impossible to unify the two propositions of algebra and geometry.

However, the Ponzi geometry theory proposed by Pang Xuelin has made algebra and geometry faintly converge, and there is really a bridge of communication between the two.

Perhaps the old man Grothendieck had a similar idea, but unfortunately the old man left early and only proposed a theoretical framework of far-abelian geometry.

Today, Pang Xuelin's Ponzi geometry proposed on the basis of far-abell geometry is fulfilling the unfulfilled wish of Mr. Grothendieck.

This theory can not only solve the relevant problems in the field of number theory, but also play an important role in the field of nonlinear partial differential equations.

It is important to know that the main body of differential equation research at present is the nonlinear partial differential equation (NLPDE).

Many important natural science and engineering problems can be attributed to the study of nonlinear partial differential equations.

Mathematical models in many fields of real life can be described by NLPDE, and the basic equations of many important physics, mechanics and other disciplines are themselves NLPDE.

In addition, with the deepening of research, some problems that can be approximated by linear differential equations must also consider the influence of nonlinearity, so the research on NLPDE, especially the research work on NLPDE to solve the exact solution, shows very important theoretical and application value.

However, both in reality and in the mathematical community on the wandering earth, there has been no way to provide a universally effective method for finding the exact solution of nonlinear partial differential equations.

The advent of Ponzi geometry has made it possible for people to effectively solve the exact solution of nonlinear partial differential equations.

This means that Ponzi geometry plays an important role not only in the field of mathematics, but also in the fields of natural sciences and engineering.

It is a new mathematical tool that can fundamentally advance humanity in the natural sciences and engineering.

What Pang Xuelin needs to do now is to further refine this theory.

One day, two days, three days......

One week, two weeks, three weeks......

Before I knew it, a month and a half had passed.

During this period, Pang Xuelin slept when he was sleepy, ate when he was hungry, and spent the rest of his time on research.

As the days passed, his hair began to grow longer, his face was covered with stubble, his clothes were getting dirtier, and his whole body began to smell strongly.

But he didn't realize it, except to think and keep resting, eating

……

On this day, Liu Xin came to Pang Xuelin's office again and asked the technical secretary Li Yiyi: "Xiao Li, Professor Xiao Pang hasn't come out yet?" ”

Li Yiyi: "No, Secretary-General, Professor Pang has been in it for more than a month, and now the distance between the Earth and Jupiter is getting closer and closer, if Professor Pang doesn't come out again, the last orbit change window time is gone." ”

Liu Xin said: "I believe him, Xiao Pang is not a big talker, by the way, what about Xiao Xia?" ”

Li Yiyi: "My sister-in-law went back to cook for Professor Pang, and during this time, in addition to cooking for the professor and going home to sleep, my sister-in-law waited here every day, and also tried her best to cook all kinds of delicious food for Professor Pang." ”

Liu Xin sighed and said, "It's really embarrassing for her, when Xiaopang comes out, remember to notify me immediately!" ”

As he spoke, he turned around and was just about to leave when he saw Yao Bingxia coming over with a food box.

"Xiao Xia, you girl, you still cook for Xiao Pang every day, are you afraid that we will starve him?"

Yao Bingxia reluctantly showed a smile on his face and said: "Grandpa Liu Xin, Brother Xiaolin is used to the taste I make, I'm afraid he won't be used to it......"

"You guys!"

Liu Xin shook her head and was about to speak, when she suddenly noticed that Yao Bingxia's face was a little wrong, the food box in her hand fell to the ground with a snap, and then Yao Bingxia ran past him like a wind......

Liu Xin turned around and saw Pang Xuelin, who had completely changed his appearance, coming out of the office at some point and hugging Yao Bingxia, who was in tears, into his arms.

Liu Xin waited for a moment, and when the two of them had enough warmth, he stepped forward and said, "Xiaopang, have you solved it?" ”

Liu Xin's eyes fell on Pang Xuelin's face.

After not seeing him for a month and a half, Pang Xuelin became unkempt, pale, and full of stubble, and his whole person seemed to have lost a lot of weight, but his eyes became clearer and brighter than ever.

Pang Xuelin turned his head and smiled: "Old man, fortunately, I am not dishonored!" ”