# PhD Defence: Algebraic geometry techniques in incidence geometry

**Algebraic geometry techniques in incidence geometry**

In this thesis we are studying: For a given finite set P we define an ordinary circle to be a circle that contains exactly three of the given points. We will find the exact minimum number of ordinary circles, for sufficiently large n, and we will determine which configurations attain or come close to that minimum, our proof being based on a structure theorem for the sets with few ordinary circles. The orchard problem for circles asks for the maximum number of circles passing through exactly four points from a set of n points. We determine the exact maximum and the extremal sets for all sufficiently large n. We also study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points. In particular, we focus on the limits of these probabilities under successive finite field extensions. We are using the Chebotarev density Theorem over finite fields to give another solution to this question, and we generalise this to higher dimensions. We say that an n-variable real function f is expander function if it is true that there is some epsilon > 0 such that for every finite set A in the reals, the image of f on A^n is at least |A|^{1+epsilon}. We will also study a family of four-variable expanders with quadratic growth.