# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### On some fully non-linear PDEs and their applications in Kaehler geometry

In this talk, we will discuss the so-called inverse $\sigma_k$ equation on Kahler manifolds. We will concentrate on examples of singular solutions and their application in Kahler geometry. Parts of the work are joint with Mijia Lai, Jian Song and Ben Weinkove.

##### Deforming $G_2$ conifolds.

A theorem of Dominic Joyce says that the moduli space of compact $G_2$ manifolds is smooth of dimension equal to the third Betti number of the manifold. We study the moduli space question for noncompact $G_2$ manifolds with one end, asymptotic to a metric cone of $G_2$ holonomy. This includes the explicit Bryant--Salamon manifolds as examples. We prove that this moduli space is smooth and unobstructed when the rate of convergence to the cone at infinity lies within a certain range. The dimension of this moduli space includes a component which is topological and a component which is analytic, arising from the existence of certain solutions to an eigenvalue equation on the link of the asymptotic cone. We also consider the moduli space question for compact $G_2$ manifolds with isolated conical singularities.