Research directions

My background, from my late undergraduate studies and my master's studies, lies in combinatorial optimization, namely polyhedral combinatorics. My research, from my PhD studies on, focuses on applying optimization techniques — in particular those of combinatorial optimization and semidefinite programming — to problems in other branches of mathematics, like geometry and coding theory.

Problems in extremal geometry, like the sphere packing problem, have provided the background for my PhD thesis and much of my research since then. I have been able to successfully extend ideas from optimization in order to obtain new results for such problems, and in the process have acquired other interests, such as group representation theory, harmonic analysis, and functional analysis.

One of my goals is to develop new optimization methods and better understand the ones that exist. This is partly done through applications since, as an eminently applied subject, optimization is mainly developed through its applications. But some of my research has also been directly concerned with the development and understanding of optimization methods, for instance with a better understanding of how semidefinite programming formulations of combinatorial problems can be devised. Of particular interest is my work on Grothendieck inequalities. Grothendieck inequalities can be seen as an early occurrence in mathematics of a paradigm now well-established in the realm of approximation algorithms: the determination of the integrality gap of an easy-to-compute relaxation of a difficult combinatorial problem. So this is a nice example of how ideas from optimization naturally occur in mathematics.

In the near future, I plan to include on this page a much more complete account of my past and current research. Come back soon to see more.