We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when $2k<n+1$. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen’s example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

In this talk, I will discuss a direction of study in topology: Section problems. There are many variations of the problem: Nielsen realization problems, sections of a surface bundle, sections of a bundle with special property (e.g. nowhere zero vector eld). I will discuss some techniques including homology, Thurston-Nielsen classication and dynamics. Also I will share many open problems. Some of the result are joint work with Nir Gadish, Justin Lanier and Nick Salter.

For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial.   A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. Based on them, an efficient method is given for computing symmetric tensor decompositions.