This collection of open problems has been circulated since 2018 when, encouraged by
Sean Prendiville, I prepared a draft for the Arithmetic Ramsey Theory workshop in …

This book explains some recent applications of the theory of polynomials and algebraic
geometry to combinatorics and other areas of mathematics. One of the first results in this …

Arithmetic combinatorics is often concerned with the problem of controlling the possible
range of behaviours of arbitrary finite sets in a group or ring with respect to arithmetic …

We prove new exponents for the energy version of the Erdős–Szemerédi sum-product
conjecture, raised by Balog and Wooley. They match the previously established milestone …

In this paper we characterize real bivariate polynomials which have a small range over large
Cartesian products. We show that for every constant-degree bivariate real polynomial f …

The past decade has seen numerous major mathematical breakthroughs for topics such as
the finite field Kakeya conjecture, the cap set conjecture, Erdős's distinct distances problem …

This book surveys the mathematical and computational properties of finite sets of points in
the plane, covering recent breakthroughs on important problems in discrete geometry, and …

In a recent paper, Cook et al.(Compos Math 154: 2150–2194, 2018) used the splitting type
of a line arrangement in the projective plane to study the number of conditions imposed by a …

We study some natural operators acting on configurations of points and lines in the plane
and remark that many interesting configurations are fixed points for these operators. We …

We study configurations of n points and n lines that form Θ (n 4/3) incidences, when the point
set is a Cartesian product. We prove structural properties of such configurations, such that …