Fall 2019

** Organizers: ****Anna Marie Bohmann, Rares
Rasdeaconu and Ioana Suvaina**

** Fridays, 3:10-4:00pm in SC 1310 (unless
otherwise noted) **

Friday, October 18th

__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.

Friday, Oct 25th -- no meeting (Fall Break)

__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.

** **