I am an explorer/creator. I explore/create ideas that do not exist yet.
I believe every human being is creative, so I want to create a creativity generator where everyone can efficiently/optimally maximize micro/macro creativity.
When I was 17, I got exposed to Riemann Hypothesis by some popular math book, so I read his original paper, but I almost understood nothing that he was saying. Later, I realized I needed to study complex analysis to understand his paper, so naturally, I taught myself complex analysis. Riemann's paper was an inspiration to me that he only published one landmark paper on number theory, but it became a new field in mathematics. Let me briefly mention the paper, "On the Number of Primes Less Than a Given Magnitude." [1] From one perspective, prime numbers seem to be sporadic but follow a simple statistical law(Prime Number Theorem). Riemann extends PNT to the positions of the non-trivial zeros of a single analytic function, namely the zeta function. So claiming that all non-trivial zeroes are lying in zeta function is one of the most profound problems in mathematics remained a deep mystery after 162 years. Riemann also studied the analytic functions of the zeta function, but the most notable one is a functional equation.
Later, I have been exposed to another Riemann's landmark paper, in fact, his dissertation, "On the Hypotheses that Underlie Geometry." [2] Again, even though this paper did not have many math equations, I understood little at first. This paper also inspired me to realize that we can do deep mathematics without equations, and we can study objects without seeing them. Roughly speaking of his paper, he proposed a new perspective to geometry that a) topological invariants: the genus of a surface and b) generalize into a high number of dimensions it was radical that later gave birth to Einstein's general relativity.
Riemann possibly noticed or did not notice the connections between his two papers. But in the twentieth century, some of the most important results are unraveled between the two connections.
André Weil noticed a formal analogy between the structure of the rational numbers and that of the meromorphic functions on a Reimann surface.
Weil's own words:
"The mathematician who studies these problems has the impression of deciphering a trilingual inscription. In the first column, one finds the classical Riemannian theory of algebraic functions. The third column is the arithmetic theory of algebraic numbers. The column in the middle is the most recently discovered one; it consists of the theory of algebraic functions over finite fields. These texts are the only source of knowledge about the languages in which they are written; in each column, we understand only fragments." [3]
Then he conjectured this version RH is not limited to curves but should be confirmed in every dimension. I was inspired by his prophetic vision that foresees the existence of a theory of homology for algebraic varieties defined over finite fields. In other words, Weil's intuition—topological invariants from the world of geometric shape should propagate to algebraic varieties over finite fields.
Alexander Grothendieck was inspired by the construction of Weil's theory of homology and developed EGA and FGA. These development lead to Pierre Deligne's proof of the RH for varieties over finite fields. [5]
Robert Langlands conjecture a higher level of Weil's conjecture that a) an entire family of analytic functions, automorphic L-functions satisfy an analytic continuation and a functional equation, b) a) + the linear representations of Lie groups and c) automorphic L-functions contain the most essential information of the Weil cohomologies. In a special case, Andrew Wiles made the breakthrough by proving the Taniyama-Shimura-Weil conjecture, which led to the proof of Fermat's Last Theorem. On the other hand, Drinfeld and Gérard Laumon initiated geometric Langlands. In geometric Langlands, there are identities between orbital integrals that also appear in the harmonic analysis of Lie groups are called the Fundamental Lemma. Ngô Bảo Châu proved the Fundamental Lemma [6] by looking at identities derived from comparing specific geometric shapes. Technically called Hitchin's integrable systems, roughly speaking the motion of certain spinning tops.
Currently, my research is rooted in the trinitarian analogy. My particular interests lie in the Langlands program, which unifies objects from arithmetic geometry with analytic/topological objects, such as p-adic Hodge theory. I am at developing AI to construct stochastic representations for a number-theoretic framework, in the form of arithmetic quantum chaos and stochastic cohomology invariants.
B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, November 1859, 671-680.
Riemann, & Weyl, H. (1923). Über die Hypothesen, welche der Geometrie zu Grunde liegen (3. Aufl.). Springer.
A 1940 Letter of André Weil on Analogy in Mathematics Translated by Martin H. Krieger
On the Riemann Hypothesis in Function-Fields André Weil Proceedings of the National Academy of Sciences Jul 1941, 27 (7) 345-347; DOI: 10.1073/pnas.27.7.345
Deligne, Pierre. The Weil conjecture: I. Mathematical Publications of IHÉS, Tome 43 (1974), pp. 273-307. http://www.numdam.org/item/PMIHES_1974__43__273_0/
Ngô, Baochâu. The fundamental lemma for Lie algebras. IHÉS, Tome 111 (2010), pp. 1-169. doi: 10.1007 / s10240-010-0026-7. http://www.numdam.org/articles/10.1007/s10240-010-0026-7/
The email address consists of "classfieldtheory" concatenated with "@outlook.com".