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# Triangles, Braids, and Hyperbolic Manifolds

### Richard Schwartz

University of Maryland

###
Thursday, October 26, 2000

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** The real hyperbolic plane is a 2-dimensional
space which shares some of the features of the Euclidean plane but is
more exotic. The complex hyperbolic plane is a 4-dimensional space
which contains the real hyperbolic plane as a lower dimensional slice.
The complex hyperbolic plane is a mysterious space, which combines a
bewildering variety of interesting structures--negative curvature,
symplectic geometry, quaternions, the Hopf fibration, etc.--into a
harmonious union. Following a gentle introduction to real and complex
hyperbolic geometry, I will explain what happens when you tile the
real hyperbolic plane by equilateral triangles and then crinkle this
pattern up, into the complex hyperbolic plane. I recently discovered
that this procedure leads to certain 3-dimensional hyperbolic
manifolds, defined in terms of braids. I will try to give the flavor,
if not the substance, of the discovery. My talk will feature some
nice color prints which illustrate the mathematics.

This talk will be accessible to general faculty.