Applied Math Colloquium Talk - Dr. Greg Reid

Title: Hidden Approximate Symmetry

Abstract: Classical computational algebraic geometry involves exact computational methods for determining properties of solution sets of systems of polynomial equations.  However such methods are unstable when applied to approximate systems.  Recent progress has been made in developing stable methods for the approximate case, including numerical homotopy methods to compute approximate points on every complex solution component (manifold) of such systems.  These methods form the foundation of the new area of numerical algebraic geometry.

I will discuss further developments extending such approximate methods to systems of PDE.  Such methods form the foundation of the new area of numerical geometry of PDE.  Important methods include determining approximate points on PDE systems in their associated geometric (jet) spaces.  Of particular interest will be defining and determining approximate symmetry for such systems, both by perturbation (a symbolic approach) and by a fully numerical approach both developed with collaborators.
  
As the tolerance decreases, the region of maximal symmetry, colored red, decreases.  Classical methods only find the disappointingly low dimensional group (region colored green).  Examples are stationary Schrodinger PDE (2 Figs above left) and Poisson’s PDE for gravitational potential for a gaseous cloud (above right).