Seminar meets Wednesday at 1:30 PM in Room 617 on the sixth floor of Wachman Hall.
Ed Letzter, Temple University
Shiferaw Berhanu, Temple University
The classical theorem of Frobenius says that if a system of k smooth vector fields on a manifold M are linearly independent at each point and are closed under Lie brackets, then M is a union of immersed submanifolds whose tangent spaces at each point are spanned by the given vector fields. We will discuss a significant generalization of this theorem for any collection of vector fields that may not be closed under brackets. The vector fields will not be assumed to be linearly independent, in fact, any number of them may vanish on any set. If time permits, applications will be presented. These orbits have applications in Control Theory, Partial Differential Equations, Several Complex Variables, Complex Geometry, and Foliation Theory.
Samuel Taylor, Temple University
We’ll explore the following two questions:
1) What geometric structures can be associated to a random 3-dimensional manifold?
2) How likely is it that two random elements of a group G commute?
Although these questions appear to be quite different, we’ll see how they fit into the common framework of counting problems in geometry group theory. Our talk will be a friendly overview of this framework and stress many problems — some of which have been recently answered and some of which are still open.