Algebraic methods in pseudorandomness and circuit complexity

Abstract

In this thesis, we apply tools from algebra and algebraic geometry to prove new results concerning extractors for algebraic sets, AC⁰-pseudorandomness, the recursive Fourier sampling problem, and VC dimension. We present a new construction of an extractor which works for algebraic sets defined by polynomials over F₂ of substantially higher degree than the previous state-of-the-art construction. We exhibit a collection of natural functions that behave pseudorandomly with regards to AC⁰ tests. We also exactly determine the F₂-polynomial degree of the recursive Fourier sampling problem and use this to provide new partial results towards a circuit lower bound for this problem. Finally, we answer a question posed in [MR15] concerning VC dimension, interpolation degree and the Hilbert function.