Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

One of the well known advantages of cohomology over homology is the existence of a product on the cohomology groups, called the cup product. Unfortunately, the meaning of this product can be easily obscured by the algebraic formalism of duality. Luckily, on (orientable) manifolds we have a notion of Poincare duality which gives a natural isomorphism between cohomology and homology, and so we get a product on homology, dual to the cup product, called intersection. Geometrically, this is closely related to the usual notion of intersection of curves with multiplicity, and has applications in geometric representation theory. However, if instead of a manifold, our space has singularities (in codim > 1), we lose Poincare duality as an isomorphism, but it turns out we can fix the problem by introducing a new kind of homology called intersection homology where a modified version of Poincare duality lives on. This talk will be split into roughly three main parts. First I will talk about regular homology, intersections, and also attempt to motivate the subject. Next I will give some of the background needed to define intersection homology (pseudomanifolds, stratifications, etc...). Finally I will develop the notion of intersection homology, and state some of the main theorems. The focus will be on examples and concepts rather than proofs. Basic familiarity with manifolds, and algebraic topology will be very useful, but not strictly required.