Lecture Series: May 2-3 2025, UPR-RP, Anfiteatro CN-142.
Colloquium: May 5, 2025, 3:30- 4:30pm, UPR Mayagüez, Monzon 201
This event is part of the Algebraic structures in topology conference series sponsored by the National Science Foundation.
Speaker
Laurentiu Maxim (Professor, University of Wisconsin, Madison)
Lecture Series
Title: A guided introduction to intersection homology and applications
Intersection homology was introduced by Goresky and MacPherson in order to recover some of the classical results and properties of manifolds (like Poincare duality, Lefschetz type theorems and Hodge theory for complex manifolds) in the context of singular spaces. The guiding principle of intersection homology is that these results hold once one considers only chains that meet the singular locus with a controlled lack of transversality.
Lecture 1: (May 2, 2025, 4:00 – 5:00 pm, UPR-RP, CN-142) I will give a motivated introduction to intersection homology, by first recalling basic homological properties for manifolds, and then illustrating by examples how these properties can be restored in the singular context.
Lecture 2: (May 3, 2025, 10:30-11:30 am, UPR-RP, CN-142) I will give a second definition of intersection homology using sheaves, and discuss the Kaehler package for the intersection homology groups of complex projective varieties.
Lecture 3: (May 3, 2025, 2:30 – 3:30 pm, UPR-RP, CN-142) I will briefly indicate several applications of intersection homology: (i) for studying the topology of Hilbert schemes of points on smooth complex surfaces; (ii) Stanley’s proof of McMullen’s conjecture, describing the existence of a simplicial polytope with a prescribed face vector; and (iii) Huh-Wang’s proof of Dowling-Wilson’s conjecture for realizable matroids.
Colloquium
(May 5, 2025, 3:30- 4:30pm, UPR Mayagüez, Monzon 201)
Title: Curvature, asphericity, and positivity in complex geometry
Abstract: As conjectured by Hopf in the 1930s, curvature conditions are expected to restrict the topology of a smooth manifold. I will overview recent progress on various open problems involving aspherical manifolds, curvature, and positivity of (co)tangent bundles in the Kaehler context, including conjectures of Hopf and Singer and some of their Hodge-theoretic variants.
Organizers
Manuel Rivera (Purdue), Heralal Janwa (UPR-RP), Iván Cardona (UPR-RP), Gabriel Montoya (UPR-RP), Luis Cáceres (UPRM), Mona Merling (UPenn)
Contact
We plan the stream the lectures online via Zoom. For more information, including the Zoom coordinates, contact the organizers at: manuelr at purdue dot edu
Sponsors