**Project leaders**: Professor Ilyas Kahn and Professor Alec Payne

**Project manager**: George Daccache

**Team members**: Matthew Chen, Jack Qian, Paul Rosu

Our project explores a differential equation related to a special type of seven-dimensional space known as a $G_2$ manifold, which plays a significant role in both Riemannian geometry and theoretical physics. This equation describes a mathematical process called the Laplacian flow, which transforms a space that is "close" to a $G_2$ manifold to be "more like" an actual $G_2$ manifold. Imagine the manifold as an object made of clay that gradually changes shape when gently tugged. The Laplacian flow is like the tugging, guiding the transformation of the manifold into a desired form. During this transformation, we sometimes encounter singularities, which are points where the process stops making sense, like a sudden crack in the clay. These singularities can reveal important details about the behavior of the flow and understanding them is necessary for any geometric or topological applications of the flow. In our research, we have successfully numerically simulated new flows that form singularities in finite time, and studied how different changes in their initial conditions can lead to different behavior. These results help us better understand the fundamental nature of this complicated process.

Our project uses numerical simulation to study the finite-time singularities of the Laplacian flow of closed $G_2$-structures defined on seven-dimensional manifolds. A $G_2$ structure on a manifold is defined by a "non-degenerate" 3-form $\varphi$ that encodes the manifold's geometric information (including a Riemannian metric). For this underlying metric to have exceptional holonomy group $G_2$, the 3-form $\varphi$ must be both closed ($d\varphi = 0$) and co-closed ($d^* \varphi = 0$). The Laplacian flow, governed by the equation $\partial_t \varphi_t = \Delta_{\varphi_t} \varphi_t$, starts with $G_2$-structures that are closed but not necessarily co-closed, aiming to evolve them into torsion-free $G_2$-structures. We focus on flows of $G_2$ structures defined on the manifold $\Lambda^2_- S^4$ which are invariant under an action of $Sp(2)$. The $Sp(2)$ invariance reduces the Laplacian flow equation to a (1+1)-variable system of two parabolic equations. We numerically simulated several solutions of this system by using a backward Euler scheme and finite difference methods and observed the formation of finite time singularities, as indicated by the blow-up of geometric quantities. To investigate these singularities, we used rescaling methods to determine the microscopic behavior of the flow around the singularity. As the flow approaches a singularity, we build a sequence of rescaled solutions by "zooming in" by increasingly large factors. Often, these sequences converge to a limit solution, called a singularity model. By starting our flow with a perturbation of a well-understood $G_2$-structure (a recently discovered shrinking soliton), we observed the formation of finite time singularities. Some flows beginning with significant perturbations of the shrinker still formed singularities modeled by the shrinker. However, some perturbations led to singularities not modeled by the shrinker, showcasing interesting new behavior. These findings are the first numerical demonstrations of such singularities, offering new insights into the dynamics of $G_2$-structures under the Laplacian flow.

Shrinker initial condition showing the two functions in the reduced PDE and their derivatives. This plot represents the initial setup for the $G_2$ structure before perturbation.

Blow-up sequence of a perturbation of the shrinker showing the functions and their derivatives evolving and increasingly resembling the shrinker with higher k-values, indicating that the singularity is modeled by the shrinker.

Blow-up sequence of a perturbation of the shrinker showing divergence from the shrinker model with higher k-values, indicating that the singularity is not modeled by the shrinker.