  The restriction and Kakeya problems in Euclidean space have received much
attention in the last few decades, and are related to many problems in harmonic
analysis, PDE, and number theory. In this paper we initiate the study of these
problems on finite fields. The restriction problem then becomes a question
about general bounds for certain types of exponential sums, while the Kakeya
problem has an algebraic geometry flavor, asking for the extent to which lines
in different directions can overlap. In many cases the Euclidean arguments
carry over easily to the finite setting (and are in fact somewhat cleaner), but
there are some new phenomena in the finite case which deserve closer study.
