Complex Mappings From an
Evolutionary Viewpoint
Steve Casey
Department of Mathematics and Statistics,
American University
Tuesday, April 2, 2002 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Tuesday, April 2, 2002 13:00-14:00,
Room 4550
Boulder
Abstract:
The graph of a complex-valued function of a complex variable
lives in four real dimensions. Given that we only live in three
spatial dimensions, visualization is tricky. The talk shows
us how we can use our fourth dimension - time - as a tool to
help us understand complex functions. We watch a given function
evolve in time, starting from the unaltered complex plane and
ending with the range of the function. Mathematically, this
deformation of the function is a homotopy, from the identity
to the target function. The computer provides us with an ideal
tool to see this evolution. We compute a frame-by-frame movie
of the evolution off-line, and show it at relatively high
speed, giving the appearance of a continuous deformation. This
new approach of teaching complex functions appears to lead not
only to a much quicker, but also to a much deeper, understanding
of complex functions.
The talk will present excerpts of the computer projects
used by students in the complex variables class at American
University, and a discussion on how this new learning tool
interfaces with a traditional complex class. We close
by exploring some of the more visually stunning aspects
of complex variables, from views of infinity on the Riemann
sphere, to Escher's tilings of the Poincare disk.
Contact: B. V. SaundersNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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