教師教學研究介紹

 My research area is number theory. Especially, I have been studying problems related to the “Langlands program”. To convey the flavor of the Langlands program, let us consider the following phenomena.
Let p be a prime number and N(p) be the number of integers n=0,…,p-1 such that p divides n²-2. Then it is known that N(p) is governed by the residue of p (modulo 8):

• When p=2, N(p)=1.
• When the residue of p (modulo 8) is 1 or 7, N(p)=2.
• When the residue of p (modulo 8) is 3 or 5, N(p)=0.
  This theorem is called (part of) the quadratic reciprocity law.

Next, let us think about replacing the square with the cubic. We let M(p) be the number of integers n=0,…,p-1 such that p divides n³-2. Then, the behavior of the number M(p) cannot be described simply by the residue of p (modulo any integer). Its description is given in the following magical way. We consider an infinite product (and expand it as an infinite series in q^r) as follows:

Then we have M(p)=a_p+1 for any prime number p different from 2,3.
This surprising fact is one of the consequences of the Langlands program. Roughly speaking, the Langlands program suggests that two totally different mathematical objects can be related in a mysterious and miraculous way; the one is modular forms (or more generally, automorphic representations), and the other one is Galois representations. In the above example, the infinite product is a typical example of a modular form while the number M(p) can also be understood in terms of Galois representations. As we can see from this example, the Langlands program itself is quite interesting. However, more importantly, it provides numerous applications in number theory. Because automorphic representations and Galois representations a priori have totally different natures, it could be possible to study the one through the other one once the Langlands program is established.

I moved to the department of mathematics in August 2024. Since then, I have taught 4 topic courses in total: 李型有限群表示論 (Fall 2024), 局部朗蘭茲對應導論 (Spring 2025), 代數群理論 (Fall 2025), p進簡約群結構論 (Spring 2026). All the themes of these courses are related to the Langlands program. The Langlands program is a huge framework that drives much of modern number theory and engages many researchers. However, entering the subject is not easy at all since it requires a wide range of tools. As for myself, I still have much to learn, including many foundational things. I strongly feel that NTU students are all highly motivated and exceptionally talented (to be honest, I am always a bit frightened!). I hope to share at least some part of this subject with wonderful students, so that we may grow together through it.