Super God-Level Top Student
Chapter 976 - 373: Another Spring in the Human World

One of the most counterintuitive aspects of modern mathematics is that everything within it is built upon the precise definitions made by mathematicians. What's even more counterintuitive is that mathematicians can actually revise some definitions within mathematics.

This is because one of the most fundamental and crucial principles of modern mathematics is logical consistency, or rationality.

To put it in layman's terms: no matter how outrageous your definition may be, as long as within a mathematical system all theorems and derivations are based upon a set of accurately defined axioms, and these derivations and conclusions are not mutually contradictory, then it is considered correct.

It can formally exist within mathematical definitions.

For instance, a quintessential example is the definition of the imaginary unit: i^2 = -1.

If one only possesses high school-level mathematical knowledge, upon encountering this equation, the first reaction would likely be, "What the hell is this nonsense?" A real number squared cannot possibly equal a negative number. Based on the basic properties of the Realm of Real Numbers, one can deduce that the square of any real number is always non-negative.

But now the mathematicians insist on defining a number whose square equals -1… And so, they gave it a name: the imaginary number.

Evidently, this imaginary number was purely invented through definition.

The method is quite simple: just throw this "i" into the Realm of Real Numbers. Suppose the Realm of Real Numbers is a set that includes integers, rational numbers, and irrational numbers. Then, add "i" into the mix; at this point, within the set that includes "i," if you perform addition and multiplication, you'll find that real numbers and "i" can no longer be simplified further.

The most you can write it as is in the form a + bi, and this definition thus becomes the complex number.

When the once-princely mathematician Gauss figured out this numerical form, he had to think about how to express it geometrically—and the complex plane emerged. The horizontal axis represents the real part of a complex number, while the vertical axis represents its imaginary part. Any complex number can be represented as a point on the complex plane.

Through Euler's formula, e^iθ = cos(θ) + isin(θ), with a minor adjustment, one soon discovers that any complex number can be represented in polar coordinates =^.

Thus, the rules for multiplication of complex numbers were defined.

In the Realm of Complex Numbers, when two numbers are multiplied together, the result equals the product of their magnitudes and the sum of their arguments: r1·r2·e^i(θ1+θ2).

From this, things become simpler: i × i then equals i^2 = 1·1·e^i(90°+90°), which is essentially equivalent to rotating 1 by 180° on the real number axis, resulting in -1.

See? The great mathematics legends of the past were this audacious—casually defining imaginary numbers, complex planes, and a host of other seemingly bizarre concepts, creating impossible scenarios that later generations of students would struggle with. Through all sorts of then-incomprehensible techniques, they made the impossible possible.

Clearly, Qiao Ze is now doing something very similar to what his predecessors did.

For example, in his paper, Qiao Ze defines the concepts of broad and narrow intertwining properties.

"Broad intertwining refers to the intrinsic connections among all mathematical entities, including but not limited to numbers, polynomials, functions, matrices, groups, rings, etc. These connections manifest through shared mathematical properties or operations and are capable of mutually influencing each other's theoretic outcomes and applications.

Such shared properties include, but are not limited to, arithmetic characteristics, algebraic structures, geometric features, or topological properties. Moreover, there exists at least one operation or mapping method that exhibits similar or interdependent behavior across these different mathematical entities."

"Narrow intertwining pertains to the Intertwining Unification Conjecture: that there exists an algebraic structure and a geometric structure such that within the intertwining framework A⊗G = G⊗A."

To prove this intertwining property, the paper defines a special function I(z) and provides its expression.

I(z) = e^p(z) + e^q(z), and using the zeros and poles of I(z), it investigates the intertwining of the roots of polynomials p(z) and q(z). Throughout the intricate proof process, it establishes a series of lemmas and theorems.

The paper is highly abstract. In fact, what is even more abstract is the timing of its release.

It's New Year's Eve...

The day of family reunions in Huaxia—this is undoubtedly crossing a line.

After all, for mathematicians who still hold ambitions within mathematics, Qiao Ze's paper is certainly unmissable.

Not to mention, this paper bears Edward Witten's co-signature; rumors suggest that Peter Schultz was one of its reviewers. And Schultz came to Xilin largely because of this paper.

Furthermore, considering the paper's extreme difficulty—being in an entirely new field of mathematics, with many of the mathematical symbols newly invented, tackling the issue of mathematical unification—the sheer magnitude of this "buff" can only be described as monumental.

If you don't study it carefully, who will you discuss it with after the New Year?

Thus, for the top mathematicians in Huaxia, the Year of the Snake truly became a uniquely special year.

Celebrate New Year?

Not happening.

Better to research the paper instead.

Fortunately, for mathematicians abroad, there wasn't much of a dilemma—after all, what even is "Spring Festival"?

Moreover, this time Qiao Ze's paper was directly published in an English edition in the "Mathematics Annual"; the official translation reads even more fluently.

It's just that the over two hundred pages certainly demand plenty of time to digest.

As for domestic scholars, although the Chinese edition of the paper is more concise, with just over a hundred pages, the nature of academic papers remains the same: the process of argumentation is consistent. The truly important part—the part that needs time—remains the proofs filled entirely with numbers and symbols.

At the very least, one must be able to understand whether there are flaws or contradictions in the proofs for each lemma and theorem.

It's incredibly challenging, but for those who genuinely love mathematics, it's also incredibly fulfilling.

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