Based on the theory learned in the course Applied Information Theory we will continue to explore some fundamental topics concerning reliable transmission of information and applications of the information theory in another scientific fields.

In the first part of this course we will follow Shannon's ingenious approach for representing any communication system geometrically. Signals and noise are represented by points in high-dimensional spaces. Using advanced geometric reasoning a number of basic results in information and communication theory are deduced with deep understanding of some very complex subjects.

In the second part of this course we will discus some successful and controversial applications of information theory in cryptography, physics, biology, as well as at estimations of ultimate limits in computation and communication.

The course will cover approximately the following topics:

• Information as a universal interpretation of Kolmogorov's mathematical expectation
• Claude Elwood Shannon, the founding father of Information theory
• Sampling and quantization - basic steps from the real to the digital world
• Geometry of signals and codes
• Optimal error control codes
• Simplex conjecture - strong and week interpretation
• Asymptotic bounds on communications
• The Channel coding theorem - from geometrical point of view
• Error exponent and channel capacity
• Generalized error exponent and channel capacity
• Channel coding theorem and ultimate limits of computation and communication
• Information theory and security
• Information theory and physics
• Information theory and genetics
• Future topics of information theory: Compressed Sensing and the Hilbert-Kolmogorov super-compression of information in neural networks.