Benoit Charbonneau & Sprio Karigiannis, Pure Mathematics Department, University of Waterloo.
Talk
1:
"Flow
of
connections
within
a
complex
gauge
equivalence
class"
(Benoit
Charbonneau)
Given a vector bundle on a Kähler manifold, and a connection on this vector bundle, one can hope to minimize the part of the curvature parallel to the Kähler form following a heat flow. We will explore the details of this construction.
Talk
2:
"Which
exponential
map
is
that?"
(Spiro
Karigiannis)
In Riemannian geometry there are two notions of "exponential maps" that, although sharing similar properties, are very different. One such map is the flow (or integral curves) of a vector field, obtained by solving a first order ODE, with no metric required. Another such map is the "geodesic flow" which flows the manifold along geodesics of the metric given an initial point and initial velocity. This "exponential map" is obtained by solving the geodesic equations, which are second order nonlinear ODE's. Nevertheless, by passing to the total space of the tangent bundle, one can view this second map as a special case of the first, in a suitable sense. If time permits, we will also discuss the "homotopy formula" for the Lie derivative of forms and its analogue for the geodesic exponential map.