computational theory serves as the backbone of computer science. it has developed many important methods and fundamental results, which have significant impacts to many other branches of computer science. randomization plays a crucial role in the development of theoretical computer science. it has become a basic tool in the fields of computational complexity, algorithm design, cryptography, and machine learning. another tool, algebrization, provides methods that transform local combinatorial properties into algebraic problems, and bring a global point of view for computational problems. algebraic methods were seen as core technologies in some recent breakthroughs of theoretical computer science.. the pis of this proposal have systemic publication records in the two methods, and feel their importance to the fields of computational complexity, algorithm design, and machine learning. we plan to have a unified approach to study both randomization and algebrization methods, and apply them to problems in the borders of computational complexity, algorithm, and machine learning. we will explore how problems in the fields of complexity theory and algorithm design can be effectively solved utilizing randomization and algebrization, and will show how the two tools are crucial in several areas of theoretical computer science.
