In many elliptic variational problems, one usually needs the information on the stability or Morse index to get some good a priori estimates. In this talk I will review a common phenomenon about this estimate and some difficulties for semilinear elliptic equations. I will also discuss my joint work with J. Wei on stable solutions of Allen-Cahn equation, where we get a uniform second order estimate on clustering interfaces.

For a geodesic ball with non-negative Ricci curvature and almost
maximal volume, we give the existence proof of splitting map without
compactness argument. There are two technical new points, the first one is
the way of finding n-directional points by induction and stratified Gou-Gu
Theorem, the second one is the error estimates of projections. The content
of the talk is technical, but we will explain the basic geometric intuition
behind the technical proof. This is a joint work with Jie Zhou.

A useful tool to study a 3-manifold is the space of the
representations of its fundamental group, a.k.a. the 3-manifold group, into
a Lie group. Any 3-manifold can be decomposed as the union of two
handlebodies. Thus, representations of the 3-manifold group into a Lie group
can be obtained by intersecting representation varieties of the two
handlebodies. Casson utilized this observation to define his celebrated
invariant. Later Taubes introduced an alternative approach to define Casson
invariant using more geometric objects. By building on Taubes' work, Floer
refined Casson invariant into a graded vector space whose Euler
characteristic is twice the Casson invariant.  The Atiyah-Floer conjecture
states that Casson's original approach can be also used to define a graded
vector space and the resulting invariant of 3-manifolds is isomorphic to
Floer's theory. In this talk, after giving some background, I will give an
exposition of what is known about the Atiyah-Floer conjecture and discuss
some recent progress, which is based on a joint work with Kenji Fukaya and
Maksim Lipyanskyi. I will only assume a basic background in algebraic
topology and geometry.

A general theme in differential/complex geometry is that curvature positivity conditions imposes strong geometric and topological constraint on the underlying manifold. In this talk, I will discuss a new curvature positivity condition in Hermitian geometry and prove a liouville type theorem for (1, 1)-forms for manifolds satisfying the positivity condition. I will discuss various interactions between this curvature condition and other notions in Hermitian geometry. Lastly, I will discuss some examples and potential applications. 

One can regard a self-map on a space as a dynamical system, and study its
long-term behavior under larger iterations. In this talk, we will introduce
the categorical version of such dynamical systems, and establish invariants
that one can associate to endofunctors of categories from the dynamical
perspective. We will also explain some results on categorical dynamical
systems that are related to holomorphic dynamics, symplectic dynamics,
Teichmuller theory, and Poincare rotation numbers. No background in
dynamical systems is assumed for this talk.