The Combinatorics of CAT(0) Cube Complexes

Abstract:

There are numerous contexts where a discrete system moves according to local, reversible moves. The configuration space, which contains all possible states of the system, is often a CAT(0) cube complex. When this is the case, we can use techniques from geometric group theory and poset theory to understand, measure, and navigate these spaces. I will present a self-contained introduction to these ideas, and discuss some applications: finding shortest paths, planning robotic motion, and sampling random lattice paths.

The talk will include joint work with many people, including Tia Baker, Naya Banerjee, Hanner Bastidas, César Ceballos, John Guo, Megan Owen, Seth Sullivant, Coleson Weir, and Rika Yatchak.

Hosted by Math Department: https://www.math.princeton.edu/events/combinatorics-cat0-cube-complexes-2024-12-02t213000

Prof. Federico Ardila - Bio

Professor Federico Ardila-Mantilla of San Francisco State University and the IAS will provide the Fall 2024 Horizons Lectures

Federico Ardila-Mantilla is a Colombian-American mathematician, educator, and musician. He has received the NSF CAREER Award and Simons Fellowship for his research, the MAA National Haimo Award for his teaching, and the AMS National "Mathematics Programs that Make a Difference" Award for his service work with MSRI-UP. 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:

1. Mathematical potential is equally present among different groups, irrespective of geographic, demographic, and economic boundaries.

2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

4. 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.