"Distinguished guests and dear students, today is an extremely glorious day for the Department of Mathematics of Princeton University, and the special thesis defense ceremony of Junxin, a student from the Department of Mathematics of Princeton University, was held in the auditorium of the Department of Mathematics of Princeton University. Let's invite our old friend Junxin to the podium! ”

With a round of warm applause, Junxin got up from the stage and came to the podium. He bowed respectfully to the audience.

"I would like to invite Junxin to sit on the podium." The host said to Junxin.

This was said in advance, Junxin nodded, didn't say anything, raised his feet and walked like a podium.

"Please sit at the designated seats for the members of the dissertation defense committee, the chairmen and vice-chairmen of the dissertation defense committee, and the other professors and students who came to attend the dissertation defense ceremony are invited to sit in turn."

"Now, let's hand over this stage to Junxin!" Speaking of this, the host walked off the stage with interest.

Junxin turned on the microphone in front of him, tried it gently, and then began his speech today:

"Dear supervisors, professors, guests and students, good afternoon, thank you for taking the time to listen to my dissertation defense, and now we officially start today's content."

The Irish Roots Program is an influential research program published by Felix Klein in 1872 entitled Comparative Perspectives on New Geometric Studies, named after Klein at the time of Ireagles. The programme proposed a new solution to the geometric problems of the time. This was the first influential mathematical program in the field of mathematics in modern times, and led the study of mathematical geometry problems almost throughout the late nineteenth century. ”

"Similarly, from 1917 to 1922, Mr. Hilbert creatively put forward conjectures about the relevant problems of mathematical proofs in order to save traditional mathematics, which we call the Hilbert Program, although it was declared bankrupt because of the formulation of Gödel's incompleteness theorem, but he successfully led the study of mathematics into the era of mathematical foundations."

"And in 1967, our esteemed Professor Robert Langlands, in a letter to Professor Andre Weiy." Speaking of this, Junxin's eyes looked at Professor Lang Lanz in the audience, and Professor Lang Lanz slightly bowed forward to express his gratitude.

It is a set of far-reaching conjectures that accurately predict possible connections between certain seemingly unrelated areas of mathematics. In the future study of mathematics, the Langlands program will certainly be a significant issue. ”

"I think last year I published a paper in the Annals of Mathematics on the relationship between the Taniyama-Shimura conjecture and Fermat's theorem. To be more accurate, we would have to say the Taniyama-Shimura-Weiyi conjecture, the latter being a geometric object with a profound arithmetic nature, but the former being a highly periodic function derived from a very different field of mathematical analysis. In my opinion, if Fermat's theorem is proved, it is also an important support for Langlands' program. From this point, it can be seen that the Langlands program proposes a network of relations between the Galois representation in number theory and the self-defending type in analysis."

"Let's systematically sort out the relevant content of the Langlands Program. The roots of Langlands' program can be traced back to one of the most profound results of number theory, the law of quadratic reciprocity. The law of quadratic reciprocity first arose in the 17th century in Fermat's time, and Gauss gave its first proof in 1801. A question that is often asked in number theory is: When two prime numbers are divided, is the remainder perfectly squared?"

"The quadratic reciprocal law reveals a curious connection between two seemingly unrelated questions about the prime numbers p and q, which are: "Is the remainder of p divided by q perfectly squared?" With the question "Is the remainder of q divided by p perfectly squared?" Although there have been many proofs of this law (Gauss himself gave six different proofs), the law of quadratic mutual reciprocity remains one of the most amazing facts in number theory. In the twenties of the twentieth century, Sadaharu Takagi and Amy Atin discovered other, more general laws of mutual reversal. When we look at Langlands' program in reverse, we find that one of the original motives of Langlands's program was to provide a complete understanding of the law of reciprocity in the more general case. ”

"Please open the proceedings and turn to page 12, where I give two main foreshadowings of the Langlands Program. That is, the definition starting from Atin's reciprocal law: given a field of numbers on Q with a Galois group as an commutative group, Artin's reciprocal law assigns an L-function to any one-dimensional representation of the Galois group, and asserts that these L-functions are equal to some Dirichlet L-functions (an analogy of the Riemann ζ function, expressed by the Dirichlet feature). The exact connection between these two L-functions constitutes Artin's reciprocal law. ”

"On the basis of Artin's law of reciprocity, as long as a proper generalization of the Dirichlet L-function is found, and the person who does this is Professor Heck. Professor Heck has linked the all-pure automorphic form (a fully pure function defined in the upper semi-complex plane that satisfies certain functional equations) with the Dirichlet L-function. On this basis, Professor Langlands generalized Heck's theory to apply to self-defending cusp representations (self-defended cusp representations are some kind of infinite-dimensional irreducible representations of general linear groups GLn on Q-Adair rings). Each finite-dimensional representation of the Gharois group from a given number field is equivalent to an L-function from a self-defending cusp representation. ”

From this result, no, it should be from this conjecture that Professor Langlands proposed a series of contents on the group theory aspects of number theory, thus linking pure mathematics and analytical mathematics to form a large-scale Langlands program. This is true both the Funnage principle and the generalized Ramanujan conjecture. ”

"We went back to this place for the history of the book, and then we started to get into my thesis. The central part and central idea of the paper is that for any given functional domain, a precise connection between the Galois group representation and the self-supporting type accompanying the domain is established. ”

That is, my proof corresponds to the overall Langlands program, which provides such a complete understanding of the more abstract so-called functional domains rather than the usual case of number fields. We can think of a domain of functions as a collection of quotients of polynomials that can be added, subtracted, multiplied, and divided like rational numbers. ”

"Please open the paper to page twenty-eight, and here is the work of a man named Vladimir Drifeld."