NSF Award Supports New Research Opportunities in Mathematics
Author: vsiefken
Iowa State University Professor Bernard Lidický has received a new award from the National Science Foundation (NSF), recognizing the significance and potential impact of his research in mathematics. The NSF is the nation’s leading federal agency supporting fundamental scientific discovery and innovation, providing competitive funding for projects that advance knowledge and strengthen the research enterprise.
Lidický, a professor in Iowa State University’s Department of Mathematics and an internationally recognized researcher in graph theory and combinatorics, has built a distinguished record of scholarship, collaboration, and mentorship.
The Department of Mathematics celebrates this accomplishment and looks forward to the discoveries and collaborations that will emerge from this NSF-supported research initiative.
ABSTRACT
Extremal combinatorics seeks to identify and characterize optimal objects within a given class, revealing rich mathematical structure with connections across many areas. Over the past two decades, computer-assisted methods have transformed the field, most notably Razborov’s flag algebra framework (2007), which translates extremal problems into semidefinite programs that can be solved using modern optimization software. This versatile approach has led to breakthroughs on longstanding open problems in graphs, hypergraphs, permutations, oriented graphs, point sets, embedded graphs, and phylogenetic trees. Despite being introduced nearly twenty years ago, the community is still working on understanding and extending the method.
The PIs will advance the flag algebra method by extending it to new models and developing new techniques to tackle longstanding open problems in extremal combinatorics. Building on their previous work, they have introduced a stability method for extremal constructions with iterative structure, a blow-up technique connecting local graph structure with graph limits, and adaptations that make flag algebra methods effective even for small graphs, leading to results such as the solution of Erdős’ pentagon conjecture and the computation of small Ramsey numbers. They plan to further develop these techniques and make them more applicable to other problems. The PIs expect that the proposed objectives will draw on tools from linear and nonlinear programming to obtain exact results. Additionally, the PIs plan to work with graduate students at their schools as well as graduate students from other schools during annual workshops co-organized by the PIs.