NSF Award
2430026
Computational complexity theory studies the limits of what can be computed and how much time or other resources are required to do so. Randomness plays a key role in this theory because many algorithms that use random choices can solve problems faster and more efficiently than those that do not. In cryptography, which is based on complexity theory, randomness is used to secure data and protect privacy. The study of randomness aims to ultimately improve the performance and security of everyday technology. The objective of this project is to advance the understanding of randomness in computation, with the aim of making progress on long-standing open problems. Specific areas of investigation include pseudorandom generators, which are deterministic procedures that stretch a short random seed into a much longer sequence that "looks random," the complexity of sampling distributions, and the study of mixing of distributions over mathematical structures known as groups. The investigator will foster cross-fertilization between mathematics and computer science. He will also develop publicly-available educational material, such as a book on computational complexity, and lecture notes, surveys, slides, and videos, both at the advanced and introductory levels.<br/><br/>In more detail, the investigator will work on extensions of small-bias generators, any of which can solve central open questions about pseudorandom generators. Recent work first used invariant theory to construct generators for low-degree polynomials over large fields, in fact achieving optimal parameters. The investigator will further develop this technique, with the goal of obtaining comparable pseudorandom generators over small fields, which would solve a long-standing problem in circuit complexity. The study of computational lower bounds for sampling has seen substantial progress in the last fifteen years. The investigator will further develop this area and its applications to data structures and error-correcting codes. The investigator aims to use this angle to make progress on the dictionary problem, a fundamental open problem in data structures. The study of mixing in groups has applications in communication complexity and cryptography. A recent theme has been the study of interleaved sequences of group elements. The investigator will further study interleaved mixing. A concrete aim is to resolve whether computing interleaved products requires large communication even for communication protocols involving many parties, which has been conjectured. Another aim is to use interleaved mixing to provide new separations between deterministic and randomized communication complexity. The proposed research can have an impact on a number of different areas in theoretical computer science and mathematics. Also, the investigator will continue to do research working closely with students at all levels.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
