Biographical:
    Archimedes
    Rene Descartes -- The Arithmetization of Geometry
    Pierre Fermat -- Prelude to Calculus
    M. C. Escher -- Symmetry and Beauty
    Kurt Godel -- The Impact of Incompleteness
    David Hilbert and the Formalist Program
    Felix Klein and the Erlannger Programm
    Bolyai, Lobachevsky and Gauss -- The Discovery of Non-Euclidean Geometry
    Benoit Mandelbrot -- The "Father" of Fractals
    G.F.B. Riemann and the "Curvature" Revolution of the Conception of Space
   
Topical:
    Archimedean and Platonic Solids
    Graph Theory and the Four-Color Theorem
    Origami Geometry -- The Axioms of Paper-Folding
    Logical Paradoxes and Fallacies
    Topology (or rubber-sheet Geometry)
    The Geometry of Kaleidocycles
    Projective Geometry and Art
    Penrose Tilings and Non-Periodic Tilings
    Knots and the Study of DNA
    The Geometry of Space-Time, Relativity
    Spherical Geometry
    The Geometry of Linkages
    Solid (3-dimensional) Geometry
    Computer Graphics and Virtual Reality

Teaching and Education:
    Visualization in Mathematics Education
         (What is appropriate role?)
    Proofs and Abstract Reasoning in Secondary Geometry Courses
    The use of Technology in the Teaching of Geometry (or Mathematics generally)
    Discovery-Learning in the Teaching of Geometry (or Mathematics generally)
    Psychology and Geometry: Perception and Learning
         (How do we "best" learn geometrical ideas?)
    Turtle Geometry and Logo

Advanced:
    Differential Geometry of Surfaces
    Minimal Surfaces
    The Geometry of Splines and NURBS
    Sphere Packing and the Kepler Conjecture
    The Poincare Conjecture
    The Geometry of CAT scans and other medical imaging systems