Super God-Level Top Student

Now that he had pointed out the direction, with ample research time, there might be unexpected gains.

"Princeton Research Institute just released the four fundamental theorems and twelve example problems of Super Helical Space Algebra?"

"Yes, master. All the information is publicly available under the new algebra research direction section on the official website. Would you like me to pull up the data for you now? Cutie..."

If netizens saw the Dou Dou in Qiao Ze's computer acting like this, their feelings would probably be quite complicated.

After all, this thing is aggressive and belligerent on Weibo and other apps, and even if it occasionally speaks in a friendly tone, it is most likely being sarcastic.

However, when Qiao Ze used it, it became as fawning as a bootlicker.

This really vividly explained what "an orange grown in the south is an orange, but grown in the north, it's a bitter orange" meant.

"Hmm," Qiao Ze responded.

Soon, the research content published on the official website of the Institute for Advanced Study in Princeton appeared before Qiao Ze in the form of images.

In addition to the two theorems that Robert sent to Edward, the image also presented two more theorems.

They were about the topological properties of superspiral space algebra and descriptions of the Mott insulating phase in quantum phase transitions and strongly correlated systems—quite interesting.

Apart from the second one, each summarized theorem was followed by two to three names.

Then there were the twelve related example problems.

From Qiao Ze's perspective, all twelve problems were very simple.

They were essentially based on the three theorems that had already been published.

But for beginners, they were indeed useful.

This also inspired Qiao Ze. Your next chapter is on My Virtual Library Empire

Although he did not intend to waste too much time on popularizing superspiral space algebra, he could still give a little help to mathematicians and physicists who were devotedly studying this field.

After all, setting problems was an easy task for him, hardly taking much time.

He could also incidentally extend it to the corresponding transcendental geometry.

Thinking this, he took action.

Very soon, Qiao Ze directly designed two problems.

The first problem was an advanced topic on super helical space algebra: Consider a high-dimensional superspiral space algebra model with the Hamiltonian [ H =-t\\sum_{j=1}^{N}(c_{j\\uparrow}^{\\dagger}c_{j+1\\uparrow}+ c_{j\\downarrow}^{\\dagger}c_{j+1\\downarrow}+ext{h.c.})

Prove that under certain conditions the system's ground state may undergo a spin-density wave (SDW) phase transition, that is, the formation of a periodic arrangement with spin order in the system. Analyze the conditions for the spin-density wave phase transition at zero temperature in this model and provide the corresponding physical explanations.

The second question was about the transcendental geometry he was studying.

Qiao Ze dubbed it "The Door Through the Dimensions." The problem wasn't difficult, but it was very special.

The problem description was as follows:

Suppose there exists a mysterious Door of Dimensions in the universe, which connects four-dimensional space and six-dimensional space. The mathematical description is: [ V =\\int d^4x \\sqrt{g}\\left(\\frac{1}{2}\\mathbf{R}+\\frac{1}{2}\\nabla\\phi \\cdot \\nabla\\phi - V(\\phi)\\right)]

Where, ( V ) represents the action of the Door of Dimensions, (\\sqrt{g}) is the square root of the metric in four-dimensional spacetime, (\\mathbf{R}) is the scalar curvature of four-dimensional spacetime, (\\nabla\\phi) is the gradient of the scalar field in six-dimensional space, and ( V(\\phi)) is the potential energy term associated with the scalar field interaction.

In this six-dimensional space, a curve ( C ) is defined as a path that connects the two sides of the Door of Dimensions and meets the following condition. This path ( C ) has a length ( L ), and its action is minimized, considering that in the four-dimensional space, the metric is (\\sqrt{g}= 1 ) and the scalar field is (\\phi =\\phi_0 ).

Find the solution: the curve ( C ) in the six-dimensional space with minimal action.

Hint: The relevant theories of superspiral space can be used to solve this, and the least action should correspond to the movement equation satisfied by the path (\\mathbf{x}(t)).

After formulating the questions, Qiao Ze had Dou Dou post them directly.

To ensure everyone could understand, he specifically used both Chinese and English for the problem statements.

Especially for some unique terms in new mathematics, Qiao Ze also provided explanations, which were considerate and did not require any thanks from the other party.

One might say that everyone is contributing to the advancement of academia.