Seminar meets Wednesday at 1:30 PM in Room 617 on the sixth floor of Wachman Hall.
Shiferaw Berhanu Temple University
Consider a function f which is a holomorphic function or a harmonic function on a domain D ⊆ C. If f is flat at an interior point p ∈ D, then f ≡ 0 on D. This is referred to as the unique continuation property of holomorphic (harmonic) functions. However, if say f is smooth up to the boundary, and flat at a boundary point, f may not vanish on D. In fact, f may not even have a single zero in D. We will describe conditions that guarantee unique continuation for harmonic and holomorphic functions. In 1993, M. S. Baouendi and L. P. Rothschild conjectured a generalization of their boundary uniqueness result to solutions of general second order elliptic equations with real analytic coefficients. We will briefly discuss the positive solution of this conjecture in our recent work.
Gerardo Mendoza, Temple University, Microlocal analysis is an approach to (a tool for, a point of view for) the analysis of linear partial differential equations. It allows for an exquisitely refined analysis of problems about existence and regularity of solutions of linear partial differential equations.
One of its most powerful tools comes for the interplay with the symplectic structure of the cotangent bundle. Symplectic transformations, which in some sense are the allowed changes of variables in the cotangent bundle, are taken advantage of as "quantized symplectic transformations.'' They allow us to transform classes of differential operators into simpler models that can in many cases be studied in detail with much profit.
In this talk I plan to illustrate what microlocal analysis is by explaining some concepts and interesting theorems. I'll discuss
- Regularity, wave front set
- Propagation of singularities
Solvability already illustrates the role of the symplectic structure of the cotangent bundle.
David Futer, Temple University,
This talk is about the weird world of hyperbolic geometry and the role it plays in topology. In the Euclidean geometry that we studied in high school, you can adjust the scale of a triangle without changing its angles. Once you introduce curvature, this is no longer true: the angles of a triangle completely determine its area. On a surface with many handles, measured appropriately, the number of handles also determines the area.
These ideas become even more powerful in dimension 3, where the topology of a 3-dimensional manifold — the way in which it connects to itself — determines not only its volume but all sorts of other geometric properties. We will visualize such manifolds as universe that you fly through on a spaceship, with geometric features flying by. Indeed, this interplay between geometric measurements and topological features can help cosmologists figure out what sort of topology our physical universe has.