I am currently working on developing a theory of cellular sheaves for chemical reaction networks, I would love to talk about this with anyone that is interested. Here below, I have some of my research projects that I have been involved in:
This Python package offers a suite of tools designed for simulating various random surface growth models. The foundational concepts and methodologies are significantly influenced by the work presented in the book by Barabási and Stanley [BarabasiS95] (1995) on fractal concepts in surface growth.
It is known that the fluctuations in the surface growth models fall into a few universality classes. The most famous one is the Kardar-Parisi-Zhang (KPZ) equation Kardar et al. [KPZ86]
\(\frac{\partial h(t,x)}{\partial t} = \nu \nabla^2 h(t,x) + \frac{\lambda}{2}((\nabla h(t,x))^2 + \eta(t,x), \text{ } t>0, x \in \mathbb{R},\)
Where \(\eta\) is a centered Gaussian noise, which is white in both space and time. The KPZ equation is a stochastic partial differential equation (SPDE)
Despite the variations in the type of surface growth that can occur, the growth of the first Betti number, \(\beta_1\), is roughly linear from computational experiments.
There was a talk given on our work at the AMS Southeastern Sectional Meeting in 2024.