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Penn Undergraduate Mathematics Colloquium

Wednesday, October 13, 2004 - 4:30pm

Shelly Harvey

MIT

Location

University of Pennsylvania

DRL A6

Tea will be served at 4:00 PM in DRL 4E17.<BR> This program is made possible by a grant from the Fund to Encourage Women (FEW) of the Trustees' Council of Penn Women.

A knot in three-dimensional space can be obtained by taking a string, tying it up and then gluing the ends together. One could imagine that this knot could be very complicated. It may be surprising for you to know that knots are very important in mathematics and outside of mathematics. In knot theory, we try to distingusish knots that cannot be continuously transformed one to the other inside of three dimensional space. One would like to know when a knot is the same as the "unknot" (the knot that is not knotted). A more difficult question is whether a knot is the same as the unknot when we allow our transformations to "take place" in four-dimensional space! In this talk, we will define a knot and talk about some ways we distinguish different knots. To this end we will explain how we can associate a surface to a knot, how we can associate a matrix to a knot, how we can associate a group to a knot and how we can associate a "braid" to a knot. We also discuss how one can build "twisted" 3-dimensional spaces out of knots and how some knots can be "undone" in 4-dimensions. This talk will be accessible to undergraduate math majors.