Jordan Larson | Searcharxiv

arXiv2025

Representation varieties of RAAGs

We investigate the $G$-representation varieties of right-angled Artin groups (RAAGs) for various Lie groups $G$. We show these varieties are connected for a large class of such $G$, including $\mathrm{SU}(n), \mathrm{Sp}(n)$ and $\mathrm{U}(n)$, while they are generally not connected for other large classes, such as $\mathrm{SO}(n)$ and $\mathrm{Spin}(n)$ for $n \geq 3$. When $G = \mathrm{SO}(3)$ we determine the number of connected components of the representation variety associated to any RAAG that is also a 3-manifold group.

math.GT↗

arXiv2025

Representation varieties and genus-three Torelli maps

We consider the family of Torelli homeomorphisms on a genus-three surface given by powers of a fixed bounding pair map. For each such homeomorphism $\phi$ we determine the number of connected components of the fixed point set of the induced map on the representation variety of the surface, as well as the number of connected components of the representation variety of the mapping torus of $\phi$.

math.GT↗

arXiv2025

A Method for Analytical Solutions in the Lattice Boltzmann Method

Analytical solutions to the lattice Boltzmann Equation make it possible to study the method itself, explore the properties of its collision operator, and identify implementations of boundary conditions. In this paper, we propose a method to find analytical solutions where the macroscopic flow profile is known. We test this method on bulk Couette flow aligned and inclined to the simulation lattice with the quadratic and entropic equilibrium distributions. Our method indeed provides an analytical solution to these flows when using the quadratic distribution. When the flow is aligned to the lattice, our method provides an analytical solution using the entropic distribution for practical relaxation times and shear rates. We show that a small even order truncation of the formal solution is optimal for accuracy-compute-time trade-off. In the inclined case, our method does not conserve momentum, by a small relative error, when using the entropic distribution. We also discover that entropic lattice Boltzmann method is not compatible with the angled Couette flow. We discuss the application of our method to more complicated flows.

physics.flu-dyn↗