Zoom link: https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09(link is external)

Let $K\subset S^3$ be a knot and $X$ be a closed smooth four-manifold. Is $K$ smoothly H-slice in $X$? That is, does $K$ bound a smooth null-homologous disk properly embedded in $X\backslash\mathring{B}^4$? The answer to this question can detect exotic smooth structures on $X$. I will describe new tools to compute the (smooth or topological) $\mathbb{CP}^2$ slicing number of a knot $K$, which is the smallest $m$ such that $K$ is (smoothly or topologically) H-slice in $\#^m\mathbb{CP}^2$. This is joint work with Allison N. Miller, Sumeyra Sakalli, and Arunima Ray.