# Welcome!

I'm Taylor J. Klotz, a mathematician who likes to go by TK, and this is my website dedicated to research, teaching, and a smidge of unrelated shenanigans. I can be contacted via my gmail: Taylor.Klotz.23

The graphic on the left is something fun I made in Mathematica. It's a surface in $$\mathbb{R}^3$$ which is swept out in time by a solution to the vortex filament equation (VFE), given by $$\frac{\partial \gamma}{\partial t}=\gamma'\times\gamma'',$$ where $$\gamma(s,t)$$ is a time-dependent curve in $$\mathbb{R}^3$$ and the prime notation denotes differentiation with respect to the curve parameter $$s$$. The VFE is a model for the centerline of a vortex in an incompressible ideal fluid.
The specific solution used in the image to the left arises from taking the 1-soliton solution to the 1-dimensional focusing nonlinear Schrödinger equation $$i\frac{\partial \psi}{\partial t}+\frac{\partial^2\psi}{\partial s^2}+\frac{1}{2}\psi|\psi|^2=0$$ and using the Hasimoto transformation $$\psi(s,t)=\kappa(s,t)e^{i\int_{s_0}^s\tau(\sigma,t)\,d\sigma}$$ to find the curvature $$\kappa(s,t)$$ and torsion $$\tau(s,t)$$ of a solution to the VFE. We can then use those invariants to write down examples of solutions, and each cyan-colored curve on the surface to the left is precisely such a solution of the VFE with a fixed $$t$$-value.