AF: Large: Theory of Computation - Pushing the State-of-the-Art

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This project is aimed at understanding a variety of fundamental questions in the theory of computation. It will be carried out via the postdoctoral mentoring program at the Institute for Advanced Study, and as such the specific topics of focus will evolve with the postdocs present each year. Current foci include, among others, the following:- The power of formulas. Formulas is the most basic mathematical and computational descriptive mechanisms. Understanding their minimal length for natural problems captures at once limitation on the space requirements, as well as the potential parallelism inherent in the problem. The award will focus on the most challenging direction, which has so far resisted attack - proving limitation of formulas. More generally, the researchers will pursue proving limitations of other natural computational models, especially arithmetic computation.- The power of relaxationsThe "meta-algorithms": Linear and semi-definite relaxations of integer programs, are among the most fruitful and powerful techniques for solving (or finding approximate solutions) to optimization problems. The award will focus on understanding the limits of these techniques. This work ties in naturally to understanding the limitations of natural proof systems and of natural strategies for search algorithms.- Peeking inside the "black-box"One of the most useful paradigms in programming and learning is the encapsulation of objects as black-boxes, to which only input-output access is allowed. With this utility come limitations which can hopefully overcome if we are allowed some access into the internal workings of the black box. Modeling such access, in scientific experiments, machine learning and computational complexity is a challenge the researchers plan to pursue.Computational complexity, a foundational core of computer science, has proved itself a remarkably deep and fruitful fountain of problems, ideas and techniques over the past decades. The research agenda is expected to be a driver of innovation in Theoretical Computer Science and related disciplines. Some of the areas of study have potential implications outside theory, especially machine learning, coding theory, scientific discovery and more. On the educational side, the mentoring program furthers the quality of IAS postdocs to serve as outstanding teachers, graduate advisors and academic leaders and innovators in one of the most exciting branches of science today. Whether IAS alumni pursue a career in academia or industry their impact on technology and on the training of new generations undergraduate and graduate education is immense.