Georgia Institute of Technology College of Sciences

We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example, we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise because of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap’s long side. This modeling approach and analysis can be extended to directionally growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior. As an example, we discuss how our model reveals how changes to a cell-shape describing parameter may manifest at the population level of the microbial collective. Specifically, we discuss mechanisms revealed by our model on how we may be able to control spatiotemporal patterning by modifying cell shape of a given strain in a multi-strain microbial consortium.

Recording Link: https://bluejeans.com/s/0g6lBzbf0XT

In modern data analysis, the datasets are often represented by large-scale matrices or tensors (the generalization of matrices to higher dimensions). To have a better understanding or extract   values effectively from these data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss the applications of CUR decompositions on developing efficient algorithms or models to robust decompositions or data completion problems. Additionally, some simulation results will be provided on real and synthetic datasets. 

In this talk, we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.

We consider a subadditive potential $\Phi$. We obtain that for $t \to \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\to \infty$  to the maximal Lyapunov exponent and entropy of maximizing measures. Moreover, we show that if a $SL(2, \mathbb{R})$ one-step cocycle satisfies pinching and twisting conditions and there exist strictly invariant cones whose images do not overlap on the Mather set then the Lyapunov-maximizing measures have zero entropy.

Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations.  One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.