A theorem of Liouville asserts that the simplest conformal transformations on Euclidean space---translations, dilations, reflections, and inversions---generate all conformal transformations when the dimension is at least 3.  I will describe a new proof of this theorem which is shorter and more elementary than the argument, due to Nevanlinna, found in most modern textbooks.

We discuss a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in a space form. The estimate in Euclidean space was first conjectured by Alexander, Hoffman, and Osserman in 1974 and proven in full generality by Brendle and Hung in 2017. We will show the sharp area estimate also holds…

In traditional Euclidean geometry, points serve as the foundational elements for constructing and analyzing space. In contrast, Laguerre geometry, a non-Euclidean geometry, uses oriented circles (or hyperspheres, in the context of higher dimensions) and oriented lines (or hyperplanes), as fundamental objects. Here, a “point” is simply a circle with radius zero, i.e. having no special…

Khovanov homology is a link invariant which categorifies Jones polynomial. In this talk we present several results concerning Khovanov homology of fibered positive links; in particular, we extend the result by Stosic stating that braid positive links have vanishing Khovanov homology in homological grading 1. We also explore Khovanov homology of certain cable links and…