Dr. Manuel Rivera

Assistant Professor
Purdue University


Manuel was born in San Juan, Puerto Rico. He started discovering mathematics during his teenage years at the University of Puerto Rico. He was an undergraduate at MIT and finished his PhD in algebraic topology at CUNY Graduate Center under the supervision of Dennis Sullivan. He has held postdoc positions in Paris 6, University of Miami, and Cinvestav in Mexico City. He is currently an Assistant Professor at Purdue University. 

Manuel works in the intersection of topology, geometry, and algebra. He is interested in using tools from homotopy theory to understand the algebraic nature of geometric space. In classical homotopy theory, two topological spaces are considered to be equivalent if they have the same “shape”  in a sense broader than homeomorphism. Then one develops a framework to study natural constructions that involve spaces and continuous maps and are invariant under this new notion of equivalence. This framework can be abstracted, axiomatized, and exported to other settings to study mathematical objects considered up to a specified notion of equivalence weaker than isomorphism. There are many ways to attach natural algebraic invariants to spaces capturing different geometric structures and properties. The goal is then to use the abstract framework to handle these algebraic invariants and unlock effective analyses of quantitative and qualitative aspects of geometric objects. Manuel is interested in applying this philosophy to study the geometry and topology of manifolds.

“During Hispanic Heritage Month the US celebrates the cultures and contributions of citizens with origins in Spain and Latin America. The date September 15 turns out to be the anniversary of independence for Costa Rica, El Salvador, Guatemala, Honduras and Nicaragua. Mexico and Chile celebrate their independence in September as well. Puerto Rico, where Manuel is from, is a Latin American country that has not achieved its independence being under US rule since 1898. It is one of the world’s oldest colonies. Many of the island’s inequities in access to quality education and academic research are intricately tied to its political status. This is a complicated issue with different angles and subtleties. Manuel hopes this is resolved in the near future so that Puerto Rico is free to develop a system compatible with their culture and needs under which mathematics and science can flourish, and that this will eventually be part of the Hispanic Heritage Month celebrations.“