Description：

Several problems in number theory concern nonvanishing of determinants of matrices with entries that are (p-adic of complex) logarithm of algebraic numbers. Over the last 150 years there have been several results and conjectures exploring the behaviour of logarithms, and exponentials of algebraic numbers. In this course, I will cover some of these results, in particular, Lindemann—Weierstrass theorem, Baker’s theorem, and Waldschmidt—Masser theorem. We will also see Schanuel’s conjecture, and some equivalent formulations of it (or its weaker version). Further, we will see the Matrix Coefficient conjecture, its implication in Iwasawa theory, and a proposed strategy to tackle it.

Prerequisite：Analysis, complex analysis, some number theory.

Reference：

1.Transcendental Number Theory - Alan Baker, Cambridge University Press 2022

2.Ranks of matrices of logarithms of algebraic numbers I - the theorems of Baker and Waldschmidt—Masser, Samit Dasgupta, Essential Number Theory, Vol 2, No 1, 2023

3.Ranks of Matrices of Logarithms of Algebraic Numbers II: The Matrix Coefficient Conjecture, Samit Dasgupta and Mahesh Kakde, Preprint.

Target Audience：Undergraduate students, Graduate students

Teaching Language: English

Bio: Mahesh Kakde is a professor of mathematics at Indian Institute of Science, Bangalore. He is a visiting faculty at YMSC, Tsinghua University.

Registration: https://v.wjx.cn/vm/h2FDe6c.aspx