# Analysis

### Directors of the specialization

Hans-Olav Tylli and Xiao Zhong

### Persons responsible for discussing the study plans

Ilkka Holopainen, Hans-Olav Tylli and Xiao Zhong

### Model study plans

#### Example 1

This example concerns a student interested in a solid background in analysis, possibly with interests in applications, who is interested in moving go work in the private sector after completing his/her master degree (containing 55 cr. in mathematics + 30 cr. pro gradu + 5 cr. seminar + 30 cr. computer science or some other subject).

The student considered in the example did not do Vector analysis II or Differential equations II in the candidate studies so these courses need to included in the master studies. It is recommended that the core courses (Real analysis I and Functional analysis) are taken in the Autumn of Year 1. The optional courses can be chosen freely, and the recommended total amount of credits in mathematics is at most 30 credit points per semester.

**Year 1, Autumn**

- Differential equations II, 5 cr. (if not taken earlier)

- Vector analysis II, 5 cr. (if not taken earlier)

- Real analysis I, 5 cr. (core)

- Functional analysis, 10 cr. (core)

- Harmonic analysis, 10 cr. (optional)

- Minor subject, 10 cr.

**Year 1, Spring**

- Fourier analysis I, II, 10 cr. (optional)

- Complex analysis I, 10 cr. (optional)

- Partial differential equations I, 10 cr. (optional)

- Real analysis II, 10 cr. (optional)

- Introduction to differential geometry, 10 cr. (optional)

- Spectral theory, 10 cr. (optional)

- Minor subject 5 cr.

**Year 2, Autumn**

Minor subject 15 cr.

Optional analysis course, 10 cr. (depending on total credit points)

Pro gradu work seminar starts

Pro gradu work starts

**Year 2, Spring**

pro gradu completed, 30 cr., and pro gradu seminar 5 cr.

Optional analysis course, 10 cr. (depending on total credit points)

Minor subject 10 cr.

#### Example 2

This example concerns a student who is interested in analysis and who is interested in applying to graduate school after completing the master degree. The example is based on assumption that the student has chosen Vector analysis II and Differential equations II courses in his/her candidate studies. Also, the student wants to study only mathematics but no other subjects in his/her master studies (studies then contain 55+30 cr. math. + 30 cr. pro gradu + 5 cr. seminar). It is recommended that the core courses (Real analysis I and Functional analysis) are taken in the Autumn of Year 1. The optional courses can be chosen freely, and the recommended total amount of credits in mathematics is at most 30 credit points per semester.

**Year 1, Autumn**

- Real analysis I, 5 cr. (core)

- Functional analysis, 10 cr. (core)

- Harmonic analysis, 10 cr. (optional)

- optional advanced course, 5-10 cr.

**Year 1, Spring**

- Fourier analysis I, II, 10 cr. (optional)

- Complex analysis I, 10 cr. (optional)

- Partial differential equations I, 10 cr. (optional)

- Real analysis II, 10 cr. (optional)

- Introduction to differential geometry, 10 cr. (optional)

- Spectral theory, 10 cr. (optional)

**Year 2, Autumn**

Optional analysis course, 10 cr. (depending on total credit points)

Optional analysis course, 10 cr. (depending on total credit points)

Pro gradu work seminar starts

Pro gradu work starts

**Year 2, Spring**

pro gradu completed, 30 cr., and pro gradu seminar 5 cr.

Optional analysis course, 10 cr. (depending on total credit points)

Optional analysis course, 10 cr. (depending on total credit points)