The 2025 Serge Lang Undergraduate Lecture will be given on Thursday September 18th by Professor Federico Ardila-Mantilla, San Francisco State University.
MUSA and the Department of Mathematics, University of California, Berkeley, present
Title: Inequalities for trees and matroids
Abstract: In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its eigenvalues to bound the size of the addresses in their scheme. We continue their investigation, obtaining more precise spectral information about tree distance matrices. These results, combined with the theory of Lorentzian polynomials, allow us to prove some conjectural inequalities about graphs and matroids that are very easy to state but have taken decades to prove. Along the way we uncover a surprising appearance of Lorentzian polynomials in optimization and economics.
This is joint work with Sergio Cristancho, Graham Denham, Chris Eur, June Huh, and Botong Wang. The talk will assume no previous knowledge of these topics; it will be accessible to anyone with some knowledge of linear algebra.
Bio: Federico Ardila-Mantilla is a Colombian-American mathematician, educator, DJ, and musician. He has received the NSF CAREER Award and the Simons Fellowship for his research, the MAA National Haimo Award for his teaching, and the AMS "Mathematics Programs that Make a Difference" Award for his service.
In his research, Federico investigates objects in algebra, geometry, topology, and applications by understanding their underlying combinatorial structure. His interests include polytopes, matroids, hyperplane arrangements, Lie and Coxeter combinatorics, Hopf algebras, and tropical geometry. He was an ICM Invited Speaker in 2022, a Clay Lecturer in 2024, and an IAS Member in 2024-25.
Federico is committed to fostering an increasingly just, equitable, and welcoming community of mathematicians. He has advised more than 50 thesis students, co-directed the MSRI-UP program for students from underrepresented groups, directs the SFSU-Colombia Combinatorics Initiative, and he hosts over 200 hours of combinatorics lectures online. His efforts are grounded on the following axioms:
- Mathematical potential is equally present among different groups, irrespective of geographic, demographic, and economic boundaries.
- Everyone can have joyful, meaningful, and empowering mathematical experiences.
- Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
- Every student deserves to be treated with dignity and respect.
These statements should not sound revolutionary, but considering the current practices of the mathematical society, they are a pressing call to action.