Current Topics Seminar
| Speaker | Luke Gutzwiller |
| Title | Homology and Holes |
| Date | April 23 4:00pm PDL C-36 |
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A topologist is a mathematician who can't tell a doughnut from a coffee cup. More precisely, from a topological viewpoint a doughnut and a coffee cup seem essentially the same because one can be continuously deformed into the other, without acts of unnecessary violence like tearing a new hole. A doughnut and a pretzel, however, are very different topologically, because they contain different numbers of holes. We would like to be able to tell when two spaces are essentially the same and when they are not. Algebraic topology attempts to put such questions on a firmer basis by defining what exactly we mean by "the same", and characterizing properties like "having n holes" in terms of algebraic objects like abelian groups. There are myriad often-terrifying ways of associating algebraic information to topological spaces. They go by exotic names like singular homology, de Rham cohomology, complex cobordism, or Morava K-Theory, and their definitions can seem Byzantine and forbidding. The basic features of homology and cohomology theories can, however, be illustrated with little pain for smooth compact manifolds. By the end of this talk, anyone with a passing familiarity with manifolds should be ready to throw around terms like "representable homology classes", "tubular neighborhoods" and "the Pontrjagin-Thom construction" with wild abandon at their next party, and hopefully gain some intution as to how and why homology counts holes.
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