  This thesis establishes new quantitative records in several problems of
incidence geometry and growth. After the necessary background in Chapters 1, 2
and 3, the following results are proven.
  Chapter 4 gives new results in the incidence geometry of a plane determined
by a finite field of prime order. These comprise a new upper bound on the total
number of incidences determined by finitely many points and lines, and a new
estimate for the number of distinct lines determined by a finite set of
non-collinear points.
  Chapter 5 gives new results on expander functions. First, a new bound is
established for the two-variable expander a+ab over a finite field of prime
order. Second, new expanders in three and four variables are demonstrated over
the real and complex numbers with stronger growth properties than any functions
previously considered.
  Finally, Chapter 6 gives the first bespoke sum-product estimate over function
fields, a setting that has so far been largely unexplored for these kinds of
problems. This last chapter is joint work with Thomas Bloom.
