Each year, distinguished geometers visit Austin to give special short courses that cover recent developments in geometry. These Perspectives in Geometry lectures are open to all geometers; students and postdocs are especially encouraged to attend. These lectures are designed to be a pathway to enter the field. We plan to record video of the lectures and post them, along with written text, on this website.
Perspectives in Geometry
- Dates: March - April 2013
- Subject: Proof of the Caratheodory Conjecture
-  Abstract: In this talk we give a brief history of the conjecture and an overview of the proof. This involves the geometric reformulation of the proof in terms of surfaces in the space of oriented lines – a 4-manifold endowed with a neutral Kaehler metric (of signature (2,2)). By arguments of global analysis, we show how the conjecture follows from the existence of certain holomorphic discs with boundary. The existence of these discs is proven by mean curvature flow with respect to the neutral metric.
-  Abstract: In this talk we introduce the geometry of neutral Kaehler 4-manifolds. These naturally arise on the space of oriented geodesics of 3-manifolds of constant curvature. In the case of Euclidean 3-space, the neutral metric captures the geometry of 2-dimensional submanifolds perfectly and allows for a complete reformulation of the conjecture. Geometric aspects of the proof will be explained, including the notion of the angle matrix of the intersection of two positive definite surfaces, which gives the boundary conditions for mean curvature flow.
- Abstract: In this talk, the analytic aspects of the proof of the conjecture will be described. This includes the global argument which relates the classical Riemann-Hilbert problem to umbilic points, as well as mean curvature flow with boundary which establishes the existence of holomorphic discs.
- Abstract: In this talk we describe how the proof of the conjecture can be extended to provide a local index bound for umbilic points on smooth surfaces in Euclidean 3-space. The bound (index less than or equal to 3/2) is weaker than that established by Hans Hamburger for real analytic surfaces in the 1940’s (less than or equal to 1). This leads to the prediction of the existence of smooth, non-real analytic surfaces with “exotic” umbilic points of index 3/2.