The generalized Ricci flow is first studied by Streets and Tian.
It can be viewed as the Ricci flow of connections with torsion which has
many applications in non-Kähler geometry. In this talk, we will study the
fixed points of the generalized Ricci flow which are called the steady
generalized Ricci solitons. Similar to Ricci flow, generalized Ricci flow is
also a gradient flow so we can further compute the variation formulas and
see some linear stable examples. Our main result is to show that the concept
of dynamical stability and linear stability are equivalent on the steady
gradient generalized Ricci solitons.

The Monge-Ampere equation det(D^2 u) = 1 arises in prescribed
curvature problems and in optimal transport. An interesting feature of the
equation is that it admits singular solutions. We will discuss new examples
of convex functions on R^n that solve the Monge-Ampere equation away from
finitely many points, but contain polyhedral and Y-shaped singular
structures. Along the way we will discuss geometric and applied motivations
for constructing such examples, as well as their connection to a certain
obstacle problem.

I will introduce the harmonic level set method developed by Stern in 2019.
This technique has been used to prove the positive mass theorems in various
settings, for example, the Riemannian case, the spacetime case, the
hyperbolic case, and the positive mass theorem with charge. I will focus on
the positive mass theorem for asymptotically hyperbolic manifolds. We give a
lower bound for the mass in the asymptotically hyperbolic setting. In this
setting, we solve the spacetime harmonic equation and give an explicit
expansion for the solution. We also prove some rigidity results as
corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri.

The deformed Hermitian-Yang-Mills equation, which will be
abbreviated as dHYM equation, was discovered around the same time in the
year 2000 by Mariño-Minasian-Moore-Strominger and Leung-Yau-Zaslow using
different points of view.

In this talk, first, I will skim through Leung-Yau-Zaslow’s approach in a
simple way. Then I will introduce the C-subsolution which is introduced by
Székelyhidi and Guan, I will go over some known results of the dHYM
equation, and I will bring up my previous results. Last, I will show some of
my recent works which will appear soon.