The field of integer programming and combinatorial optimization (IPCO) concerns the search for optimal solutions to problems where there are many possible combinations or configurations of smaller choices which build up a solution. This ranges from widespread applications in logistics and industrial engineering, to novel programs that increase the rate of successful kidney transplants by matching donors and compatible patients.
Research on IPCO deals both with fundamental questions between pure mathematics and theoretical computer science, as well as experimental work on algorithms for different problems arising from applications. By fundamental, we refer to basic research on the mathematical structures (especially polyhedra) defined by the set of possible solutions to a given problem. I work on this particular discipline of IPCO, which is called polyhedral combinatorics. Greater knowledge on the underlying polyhedron can have a great impact on our ability to solve the corresponding problems.
My doctoral project is to study such mathematical structures arising from an interesting class of models; specifically, three models defined over the structure of trees in graph theory.
The interest in this project is that the selected problems are both general enough, as each of the three models can apply to several real world applications in areas like communication networks or utilities distribution, while somehow interconnected, which allows for a systematic approach to the study of the corresponding polyhedra. It is therefore rather appealing to investigate those problem simultaneously during the course of a PhD degree.
- 2021. The unsuitable neighbourhood inequalities for the fixed cardinality stable set polytope. Paper 87, pages 115-124. In:
- 2021. Graphs and Combinatorial Optimization: from Theory to Applications - CTW2020 Proceedings. Springer. 240 pages. ISBN: 978-3-030-63071-3.
- 2018. The matching relaxation for a class of generalized set partitioning problems. Discrete Applied Mathematics. 253: 153-166. doi: 10.1016/j.dam.2018.05.033
- 2016. Combinatorial relaxation bounds and preprocessing for berth allocation problems.
I deeply admire Federico Ardila, a mathematician and professor at both Universidad de Los Andes and San Francisco State University. His essay "Todos Cuentan" (Notices of the AMS, 2016) is an inspiring expression of his pedagogical and transformative efforts.
I humbly believe in his approach to tackle the underrepresentation problems in mathematics and science. His ideas build on the following foundational axioms.
Axiom 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.