Series: Mathematics Colloquium

Date: Thursday, September 16, 2004

Time: 4:00 - 5:00 PM

Place: 201 Thomas Building

Host: Augustin Banyaga

Refreshments: 3:15 - 3:45 PM, 321 Whitmore

Speaker: Margaret Symington, Georgia Tech

Title: 
  
  What Does a Quartic Hypersurface Have in Common With a Spherical
  Pendulum?

Abstract:

  Symplectic geometry has its origins in classical mechanics but most
  modern techniques are heavily influenced by complex algebraic
  geometry.  With both the origins and the modern techniques in mind,
  I will use the spherical pendulum to illustrate physically some of
  the topology involved in the mirror symmetry of K3 surfaces,
  examples of which are hypersurfaces given by an equation of degree
  four.  Specifically, the set of positions and velocities (or linear
  momenta) of a spherical pendulum (its phase space) breaks up into
  subsets with fixed energy and fixed angular momentum.  For most
  values of the energy and angular momentum these sets are tori, but
  for one special value it is a pinched torus.  We will identify this
  pinched torus physically and see how it sits among the neighboring
  tori.  Meanwhile, certain projections of a K3 surface to S^2 have
  level sets that are either tori or pinched tori.  Such projections
  always have 24 pinched tori.  Why 24?  The answer lies in the
  induced "integral affine geometry" on S^2.