I will discuss the Green-Tao proof for existence of arbitrarily long arithmetic progressions in the primes. The focus will primarily be on the parts of the proof which are related to notions in complexity theory. In particular, I will try to describe in detail how the proof can be seen as applying Szemeredi's theorem to primes, by arguing that they are indistinguishable from dense subsets of integers, for a suitable family of distinguishers.
Renowned composer and pianist William Bolcom discusses his career as a composer and performer with Artist-in-Residence Derek Bermel.
We give new results on the number-on-the-forhead (NOF) communication complexity of the multiparty pointer jumping problem.
The origional motivation for this problem comes from circuit complexity. Specifically, there is no explicit function known to lie outside the complexity class ACC0. However, a long line of research in the early 90's showed that a sufficiently strong NOF communication lower bound for a function would place it outside ACC0. Pointer jumping is widely considered to be a strong candidate for such a lower bound.
Green and Tao used the existence of a dense subset indistinguishable from the primes under certain tests from a certain class to prove the existence of arbitrarily long prime arithmetic progressions. Reingold, Trevisan, Tulsiani and Vadhan, and independently, Gowers, give a quantitatively improved characterization of when such dense models exist. An equivalent formulation was obtained earlier by Barak, Shaltiel and Wigderson.
We give an algorithmic proof of Forster's Theorem, a fundamental result in communication complexity. Our proof is based on a geometric notion we call radial isotropic position which is related to the well-known isotropic position of a set of vectors. We point out an efficient algorithm to compute the radial isotropic position of a given set of vectors when it exists.
Algorithms spend time on performing arithmetic computations, but often more on moving data, between the levels of a memory hierarchy and between parallel computing entities. Judging by the hardware evolution of the last few decades, the fraction of running time spent on communication is expected to increase, and with it - the demand for communication-avoiding algorithms. We use geometric, combinatorial, and algebraic ideas and techniques, some of which are known in the context of expander graphs, to construct provably communication-optimal algorithms.