Georgia Institute of Technology College of Sciences

If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdős-Rényi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.

We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.

Prompted by the definition for the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of graded rings for which this generating function is rational. We will discuss this class of rings.  This work is joint with Yongwei Yao.

Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work  provides a blueprint for the design and control of novel and robust function in correlated and active systems.