Our group has been working in proof complexity for more than two decades and has played an important role in the development of the field. The novelty of this project is its focus on the role of the following two concepts in proof complexity: incompleteness and pseudorandomness. The incompleteness phenomenon is well understood in the context of stronger arithmetical theories, but little is known in bounded arithmetic. As it may be extremely difficult to solve the problems about incompleteness in the polynomial time domain, one of the approaches we will try is proposing axioms that will decide these questions. Similarly, pseudorandomness has been intensively researched in computational complexity, but its role in proof complexity still needs more research.