Krishnendu Bhowmick

Title:
Convexity, sumsets and discrete geometry

Abstract: Additive combinatorics is the study of the combinatorial properties of sets of numbers, particularly with respect to the operations of addition and multiplication. This thesis is primarily concerned with the number of arcs (which are colinear triple free sets in finite fields) and properties of convex sets in difference sets. The key tools throughout are from discrete geometry, extremal combinatorics and probability theory.

The main results are, finding a tight bound for total number of Arcs and number of arcs of fixed size in \(\mathbb{F}_q^2\) in chapter \ref{ch.1}, a tight bound for size of largest possible convex sets in difference sets in chapter \ref{ch.2}, and some results about local differences determined by convex sets in chapter \ref{ch.3}. Chapter \ref{ch.erdos} provides an elegant, elementary solution to an old problem of Erd\H{o}s. Chapter \ref{ch.ext} consists of couple of geometric problems,  including a simple proof of a conjecture of Brass, Moser and Pach.

The thesis includes joint work with Oliver Roche-Newton, Ben Lund, and Miriam Patry, published in Discrete \& Computational Geometry, Mathematika, and Integers.

Location: RICAM, SP2 416-2