# Capstone course arrangements 2016-17

**To: ALL NEW-curriculum MATH students admitted in 2013-14 or before**

- In order to graduate by summer 2017, you must have completed, as your capstone course, either MATH4900 (Seminar) or MATH4400 (Project), but not both.

Please note the following points:- MATH4900, MATH4400 are MUTUALLY EXCLUSIVE: having taken either, you will be barred from taking the other.
- A few students may want to take MATH4400. Such a student will need to check whether he/she is qualified for taking this course. Refer to CUSIS for the specific conditions.
- MATH4900 will be offered in the First Semester only. It is expected most of you will take this course. Delaying the course without appropriate justification may result in delay in graduation.

- For students who have already passed both of MATH2050, 2060 and intend to take MATH4900:

There are seven sections of MATH4900 in the First Semester 2016-17. Each has its own theme. Registration for MATH4900 will be done in CUSIS, on a first-come-first-serve basis. - For students who have not yet completed both of MATH2050, 2060:

In principle, you are not eligible to take either of the capstone courses. However, if you intend to graduate by summer 2017, you may make an application to the department for a place in MATH4900 in the First Semester 2016-17. Please submit your application at the following web page, from 1000hrs, 10th August, 2016:

http://www.math.cuhk.edu.hk/capstone-seminar

Applications will be closed at Monday 1400hrs of the second week of the semester (12th September, 2016). Late applications will NOT be entertained.

If your application is successful and you are taking no more than four other MATH courses, you will be registered into MATH4900 by the department during the add-drop period. - For students who intend to take MATH4400 (Project) instead of MATH4900:

You need not register for MATH4900 in CUSIS. It is your responsibility to look for a potential project supervisor yourself. You are advised to do so as soon as possible.

#### Themes of MATH4900

MATH4900A Elementary perspectives of knot theory

MATH4900B Continued fractions

- The representations of numbers by continued fractions
- Convergences as best approximations, the approximation of algebraic irrational numbers and transcendental numbers.
- Measure theory of continued fractions.

MATH4900C Studies in Operations Research

In this section, small projects and problems in the area of OR will be explored. The aim is to lead students to a thorough understanding of the art and science of practical applications in OR. Students may realize the successful way of undertaking projects and what obstacles are present to thwart success.

MATH4900D Classical Analysis (infinite products, Tauberian theorems, the Gamma function, Bernoulli numbers, special functions, and elliptic integrals)

Topics will be selected from the book “Invitation to Classical Analysis” by Peter Duren.

MATH4900E p-adic numbers

We are all familiar with the fact that the system of real numbers arises as the completion of the system of rational numbers, and its importance in mathematics. But there are other ways of doing

completions of the system of rational numbers, and the system of p-adic numbers then arises, for each prime number p. In mathematics (in particular in number theory), p-adic numbers are regarded as equally important as the real numbers. In this capstone course, we will encounter many aspect of p-adic numbers: calculus, algebra, geometry, and applications.

MATH4900F Harmonic analysis and applications

In this section, students will learn some basic harmonic analysis, and investigate some interesting applications of harmonic analysis in areas as diverse as geometry, combinatorics, and number theory. The unity across different fields of mathematics will be emphasized, as material will often be drawn from more than one area of mathematics. Students are expected to supplement the text by materials they gather or develop on their own.

MATH4900G

- Inverse problems
- Importance and applications of inverse problems
- Formulations of mathematical models of inverse problems
- Mathematical properties and analyses of inverse problems
- Mathematical and numerical methods for solving inverse problems

- Numerical methods for partial differential equations (PDEs)
- Background and applications of PDEs
- Numerical methods for solving PDEs
- Stability and convergence of numerical methods