COURSE OVERVIEW:

Standard Complexity Theory and Beyond

The theory of computation studies the abstract properties of computation which arise in concrete applications, for instance, in the design and analysis of algorithms. The logical properties of computability are well understood, thanks to the efforts of mathematical logicians (Turing, Goedel, Church, etc) going back to the 1920's. But in computer science, we are deeply interested in finer points of computation related to computational complexity. These issues are much harder to resolve and include such famous questions as the ``P versus NP'' problem. ``Standard'' complexity theory has been developed to investigate such phenomenon and is a currently thriving field. In this course, we want to explore two frontier areas of complexity theory:

The first one is very old -- almost as old as beginning of the theory of computation. How can we give a proper foundation for real computation? This is sometimes viewed as the foundation for numerical analysis. While the standard theory addresses computation over a countable set (like finite strings or natural numbers), we now ask about computation over an uncountable set. After over 50 years of work, there is still debate about the proper foundation for this theory. Two dominant schools here are the analytic school and the algebraci school.
The other frontier is very young: looming in the horizon, we have the possibility of computers built upon quantum principles. What is the proper computational model here? Does quantum complexity have radically different properties than the standard theory?

We will divide the course into roughly three equal parts:

(1) Introduction to Standard Complexity Theory
(2) Real Computation
(3) Quantum Computation