This section contains general information about the course.

## Weekly outline

### 31 July - 6 August

Week 1

Modeling

"A theory is an idealized approximation to the reality."### 7 August - 13 August

Week 2

Sets and Relations

Books to look through:1. Theory of Justice by John Rawls 2. Social Choice and Individual Values by Kenneth Arrow

Theme : Geometry, Space and Time

### 14 August - 20 August

Week 3

Structures and Evolution of Number Systems

- Cannot see the wood for the trees.

### 21 August - 27 August

Week 4

Are you aware of ABACUS? Look out for Chinese remainder theorem. Do you see the role of primes in system of congruences?

- Do you agree that, "There is no compulsive relationship between linearity and time invariance." ? What do we mean by domain and range operations while talking about functions? In connection with Digital Signal Processing, is delay a domain or a range operation? Can you qualify linearity as a behavior of the range?

- Explore : "Integral domain can be embedded in a field of fractions." in context of the Z-transform.

### 28 August - 3 September

Week 5

- Symmetry is something we do with an object and still some relationships between parts of object do not change.

Two interesting software packages for ALGEBRA

- "It is like, different concrete objects having the same inner souls."

### 4 September - 10 September

Week 6

### 11 September - 17 September

Week 7

Social Contract

The term**social contract**describes a broad class of philosophical theories whose subject is the implied agreements by which people form nations and maintain a social order. In laymen's terms this means that the people give up some rights to a government in order to receive social order. - Wikipedia

### 18 September - 24 September

Week 8

Theories are there because of limitations, primarily to overcome limitations.

### 25 September - 1 October

Week 9

"You could not have predicted the invention of a wheel."

Stone's Representation Theorem : (Existential)

Every boolean algebra is isomorphic to the algebra of subsets of some set.

### 2 October - 8 October

Week 10

- A representation of a group G is a homomorphism from G to GL(V)...

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For a finite group G, there is a finite number of 'atomic' smallest possible matrix representations from which all other matrix representations are 'built'.

### 9 October - 15 October

Week 11

- Theorem: Let G be a `finite' group. P:G->GL(V) be a representation, where V is a vector space then, If V
_{1}is an invariant subspace of the group then we can always find V_{2 }which is also an invariant subspace of the group, and V_{1}+V_{2}(direct sum) is the whole vector space.

• Joseph Fourier 1768-1830

Elena Prestine: Applied Harmonic Analysis : Models of the Real World

• F.G. Frobenius 1849-1917

Charles Curtis: Pioneers of Representation Theory

### 16 October - 22 October

Week 12

EXAM WEEK

### 30 October - 5 November

Week 14

SEMESTER BREAK### 6 November - 12 November

Week 15