Venue: Ramanujan Lecture Hall, ICTS Bangalore
Saturday, June 16 2018 to Wednesday, 20 June 2018
The Gauss-Bonnet theorem in the theory of surfaces is one of the most profound results in mathematics that beautifully connects a certain topological (or shape) invariant with a certain geometric invariant of closed surfaces embedded in Euclidean 3-space with a simple equation. This fascinating interplay between the topology and the geometry of surfaces is best illustrated by looking at polyhedral objects.
Beginning with a discussion of rigid motions of the Euclidean plane, notion of Group actions and quotient spaces (dealing with especial attention to closed surfaces), we move on to a description of Teichmuller space of flat tori. We then introduce the notion of triangulations, Euler characteristic and prove the polyhedral version Gauss-Bonnet theorem, using which we attempt to relate higher genus surfaces and hyperbolic geometry.
Minimal surfaces in 3-d Euclidean space and maximal surfaces in 3-d Lorentz Minkowski space are defined to be zero mean curvature surfaces. The general solutions are given by the Weierstrass-Enneper representations of these surfaces. We will first derive the Weierstrass-Enneper representation of a minimal surface and a maximal surface. We will introduce the Bjorling problem and mention the interpolation problem, namely, given two real analytic curves, is there a minimal surface interpolation them? Next we will introduce maximal surface with singularity and study various types of singularity in maximal surfaces.
We shall discuss some basic properties of fundamental groups and covering spaces and some of their applications.
The Uniformization Theorem implies that except the sphere and torus, all closed and oriented surfaces arise as the quotient of the hyperbolic plane by isometries. This allows an interesting interplay between discrete subgroups of the isometry group, and geometry of the hyperbolic metrics on the surfaces. The first lecture will introduce the various models of the hyperbolic plane and discuss its geometry. The second lecture will introduce Fuchsian groups, and the associated tilings by fundamental domains, with several examples. The third lecture will be an introduction to the moduli space of such Fuchsian groups, focusing on the once-punctured torus. In the fourth lecture, I plan to discuss various ways of tiling the hyperbolic plane with geodesic polygons, and mention some open problems.