INTELLECTUAL MERIT.

This work was driven by the philosophy that many objects, relationships, and procedures in pure and applied mathematics are best understood by studying the rich discrete structures underlying them. A few of the main contributions are the following:

1. a proof of da Silva's 1987 conjecture, which established that her "positively oriented matroids" in combinatorics are the same objects as the "positroids" discovered in 2006 and recently used by physicists as a much more efficient alternative to Feynman diagrams.

2. a study of "CAT(0) cube complexes", the high-dimensional spaces of non-positive curvature that can be built by gluing cubes together; a purely combinatorial description, and an algorithm to navigate CAT(0) cube complexes efficiently.

3. the development of efficient algorithms to move robots optimally by using the combinatorial structure of CAT(0) cube complexes.

4. the solution to intricate problems about the enumeration of curves on surfaces with certain properties, by "tropically" reducing them to the enumeration of integer points in polyhedra.

5. a counterexample of a 2007 conjecture by Holtz and Ron on box splines, motivated by numerical analysis.

6. a combinatorial explanation of the close relationship between the Gelfand-Tsetlin polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes: two families of geometric objects arising in the representations of Lie algebras.

7. a proof of Devadoss and Forcey's 2008 conjecture that the "nestomultiplihedron", a geometric object born out of topological constructions, can be realized as a polyhedron.

8. a proof of Stanley's 2001 positivity conjecture on a family of polynomials in combinatorics called "staircase Schur functions". 

BROADER IMPACTS.

This research program served as the mathematical backbone of the SFSU-Colombia Combinatorics Initiative, which seeks to help build and support an increasingly diverse, engaged, and active community of mathematicians that serves the needs of all sectors of society. Through joint courses and research projects, students in the US and Colombia participated in their first international academic experience, while making serious scientific contributions. The results of this work have been highlighted at the Committee on Education of the American Mathematical Society and in the Notices of the AMS.

The project directly involved about 200 undergraduate and Master's students, approximately half of whom plan to continue on to Ph.D. programs. About 75 of the students are in Colombia, a country lacking a strong mathematical tradition. Of the 100 students in the US, more than half are women, and more than half come from underrepresented groups.

About 150 additional participants were involved in the project through their participation in the biannual Encuentros Colombianos de Combinatoria. Hundreds of others around the world who are involved indirectly, through an online archive of videos in graduate combinatorics developed under this grant.

Last Modified: 10/06/2016
Modified by: Federico Ardila