Speaker:  Mitchell Faulk, Vanderbilt University

Title:  An extension of Yau's theorem to asymptotically conical manifolds (and orbifolds)

Abstract: Yau's original solution to Calabi's conjecture states that, given a prescribed form representing
the first Chern class on a compact  Kahler manifold, it is possible to find a Kahler metric whose Ricci form
is the given one, and moreover, this metric is unique. Yau's solution involves solving a Monge-Ampere
equation via a continuity method, and uses certain a priori estimates to obtain regularity on the candidate
solution. I'll discuss a variant of this theorem in the setting where the Kahler manifold is asymptotic
to a cone. The precise existence results in this setting depend on the linearized equation, and in particular
on the Fredholm index of the linearized operator. In the compact case, the index is always zero, but in
this asymptotic setting, the index varies as a step function of the decay rate of the prescribed Ricci form.
I'll discuss some small improvements on an existence statement from Conlon-Hein in the case where
Fredholm index is the first negative value. The techniques involved are linear.

Speaker:  Jocelyne Ishak, Vanderbilt University

Title:  New case of rigidity in stable homotopy theory

Abstract: In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C,
but looking at the homotopy level alone does not provide us with higher order structure information. Therefore,
we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the
underlying model structure can we recover? This question has been investigated during the last decade, and some
examples have been studied, but there are still a lot of open questions regarding this subject. In this talk, we will take
a tour through the world of stable homotopy theory. This tour will start at the spectra level and Bousfield localization,
then we will pass to the rigidity question and what it means in this world.  We will end our tour by talking about
a new case of rigidity, which is the localization of spectra with respect to the Morava K-theory K(1), at p=2.