## Graduate

## Curriculums

**Curriculum Schedule for Graduates of the Department of Mathematics, PKU ((trial implementation)**

**Oct. 25 ^{th}, 2018**

**Intermediate Courses:**

**1. Analytics and Partial Differential Equations**

*Real analysis. Harmonic analysis.* (offered respectively in the first and second semester)

*Complex analysis.*

*Functional analysis II.*

*Second order linear partial differential equations. Hyperbolic partial differential equation. *(offered respectively in the first and second semester)

__Courses offered every two years:__

*The basis of linear analysis. Calculus of variations. The theory of functions of several complex variables.*

__Courses for qualification tests:__

*Functional analysis II. Harmonic analysis. Complex analysis. Second order linear partial differential equations. Hyperbolic partial differential equation.*

**2. Ordinary Differential Equations and Dynamical Systems**

*The Qualitative Theory of Ordinary Differential Equations. Ergodic Theory. Dynamical System. *

__Courses for qualification tests:__

*The Qualitative Theory of Ordinary Differential Equations.*

**3. Algebra and Number Theory**

__Courses offered every year:__

*Abstract Algebra. Commutative Algebra. *(offered respectively in the first and second semester)

*Number Theory I. Number Theory II.* (offered respectively in the first and second semester)

*Algebraic Geometry I. Algebraic Geometry II.* (offered respectively in the first and second semester)

*Group theory. Theory of Group Representations.* (offered respectively in the first and second semester)

*Homological algebra.*

*Lie Groups, Lie Algebras and Representations.*

*Theory of Geometric Representation.*

*Modular Form.*

Courses of Basic Applied Algebra: *Cryptography.* etc.

__Courses offered every two years:__

*Finite Field.*

*Homogeneous Flows, Moduli Spaces and Arithmetic.*

__Courses for qualification tests:__

*Abstract Algebra II. Representation Theory. Algebraic Geometry. Number Theory.*

**4. Geometry and Topology**

__Courses offered every year:__

*Homology Theory. Homotopy Theory. *(offered respectively in the first and second semester)

*Differential Manifold. Differential Topology.* (offered respectively in the first and second semester)

*Introduction to Riemannian Geometry.*

*Complex Geometry.*

*Symplectic Geometry.*

__Courses offered every two years:__

*Low-Dimensional Manifolds.*

*Introduction to Hyperbolic Geometry. Geometric Group Theory.* (offered in turn)

__Courses for qualification tests:__

*Differential Geometry. Algebraic Topology. Differential Topology.*

**5. Mathematical Physics**

__Courses offered every year:__

*Mathematical Methods of Classical Mechanics.*

**Mathematical Methods of Quantum Mechanics.*

**Gromov-Witten Theory.*

__Courses offered every two year:__

**Introduction to Quantum Field Theory.*

**Introduction to Condensed Matter Physics.*

**Biomathematics.*

__Courses for qualification tests:__

To be determined.

**6. Mathematical Logic, Combination and Discrete Mathematics**

*Combinatorial Mathematics.*

**Mathematical Logic.*

*Probability Theory.*

**Discrete Mathematics.*

**Mathematics in Information and Big Data.*

__Courses for qualification tests:__

To be determined.

**7. Other Courses**

__Courses offered every year:__

*Mathematical Skills Training.*

__Courses offered every two year:__

*History of Mathematics.*

**Courses of Special Topics:**

__Topics in Analysis: __Complex Analysis, Harmonic Analysis, e.g. *Function Theory of Several Complex Variables.* etc.;

__Topics in Partial Differential Equation: __courses offered in the first and second semester;

__Topics in Ordinary Differential Equation & Dynamical System: __courses offered in the first and second semester, e.g. *Smooth Ergodic Theory.* etc.;

__Topics in Algebra:__ courses offered in the first and second semester;

__Topics in Geometry and Topology:__

Algebraic Topology: *Category Theory.*

Geometric Topology: *Introduction to Contact Topology. Foliation*. etc.

Differential Topology: courses offered in the first and second semester, e.g. *Morse Theory. Floer Homology Groups.* etc.

Riemannian Geometry: courses offered in the first and second semester, e.g. *Gromov Geometry.* *Geometric Analysis.* etc.

Global Differential Geometry: *Fiber Bundle Geometry. *etc.

*Moduli Space & Gauge Field Theory: courses offered in the first and second semester.

__Topics in Mathematical Physics:__

courses offered in the first and second semester for two years, including Gromov–Witten Theory, Fukaya Category Theory, Mirror Symmetry, etc.

**Courses of Hotly Discussed Topics:**

Every year at least one course will be offered in accordance with the latest developments in mathematics.