My research
lies in the field of extremal combinatorics, specifically extremal
set theory questions and their analogues for other discrete
structures.
Given a finite set X, the general problem in extremal set theory
asks how large or small a family of subsets of X can be if it
satisfies certain restrictions. Naturally, this type of question appears
throughout mathematics, and so extremal set theory can be applied in areas
ranging from discrete
geometry to theoretical computer science.
On the other hand, extremal set theory borrows tools from
algebra and probability,
and its connections to other branches of mathematics is one of its most
beautiful features.

Remarkable analogues of extremal set theory results hold for other
objects such as vector
spaces, permutations,
and subsums
of a finite sum. Tantalizingly, while many
results about sets should generalize to different settings, not much is
known about analogues because standard techniques do not always apply.