Unless your graduate student is going to do the Langlands project.

1. Analysis, the learning order is as follows:

Mathematical analysis: that is, analysis on the real axis R, calculus

Complex analysis: analysis on complex plane c,

Real analysis: the concept of measure is introduced on the basis of interval, and the definition of integral is abstracted from measure.

Functional analysis: The object of analysis has changed from measurable set (interval) to measurable set (interval).

Measure is introduced into function set, and the properties of function space are studied.

Focus on Banach space and Hilbert space, spectral decomposition.

Harmonic analysis: study the properties of function space and its dual space in a space by means of measure, integral and spectrum.

2.

Algebra and topology

Abstract algebra: It studies the concrete structure of algebra, and the separable normal extensions of groups, rings, fields, modules and fields-Galois extensions.

Topology: Define what types of objects can be measured,

This definition is strictly based on the axiomatization of mathematics.

Differential geometry: Riemannian geometry, which studies the geometry of objects from the smooth differentiable functions on them.

A sufficient object is called a manifold.

This research method abandons the coordinate system, and is similar to algebraic geometry, which is based on axioms in algebra.

The function on an object is studied as an algebraic object.

A prerequisite for this kind of research is "testability", that is, the basic knowledge of real analysis and topology is needed.

Lie group: Study a group with manifold structure, and constantly switch between differential method and algebraic method.

3. Main research branches of number theory

Distribution of prime numbers in natural numbers, integer solutions of integer polynomials, Goldbach conjecture;

Class number in algebraic number field, Galois extension in rational number field and its corresponding L- function;

Integer solutions of curves in algebraic geometry (mainly elliptic curves);

4. Langlands Plan:

The properties of automorphism representation of Adail global number field on reduced groups;

The relationship between automorphic representation and automorphic L- function;

The relationship between automorphic L- functions and number theory L- functions.