This award supports theoretical research and education to exploit the power of statistical physics to gain insight into complex systems that lie at the interface of biology, physics, and materials science. The approach applies understanding of materials and materials-related phenomena across disciplinary boundaries into biologically inspired problems with potential implications for applications. The project contains three major aims to investigate:1.)  How spatial obstacles change the distribution of genes that appear at a particular place on a chromosome populations of invading organisms. Of particular interest is gaining insight into how environmental inhomogeneities can shape not only the boundaries at front of a population of invading bacteria but also the genetic structure of the bacteria. This research has potential to contribute to understanding the evolution of resistance of bacteria to antibiotics, as when opportunistic pathogens such as Pseudomonas aeruginosa invade catheters connected to hospitalized patients.2.) How mechanical properties of very thin shells are affected by temperature. This builds on the observation that fluctuations that arise with increasing temperature lead to unusual mechanical properties that describe distortions of a sheet over long distances. The PI aims to understand what happens when the sheet is rolled in to a thin spherical shell. This research has potential to contribute to developing strategies for the delivery of drugs to affected areas of the body, as well as having implications for mechanical systems assembled on scales 1000 times or so smaller than the diameter of a human hair.3.) Networks that describe neural development in animals and humans, and in particular how loss of connections among neurons and the strengthening and weakening of neural connections lead to neural circuits learning various functions.The project will contribute to the training of students and postodocs in modern theoretical methods on problems with impact across disciplinary boundaries.TECHNICAL SUMMARYThis award will support theoretical research and education in the development and application of statistical physics methods to diverse problems in materials science and biophysics. The PI will tackle problems that challenge theory and lead to intriguing confrontations with experiments. The issues addressed include spatial population genetics near obstacles and constrictions, the soft condensed matter physics of thin thermalized shells and the theory of the eigenfunctions and eigenvalues that control the nonlinear dynamics of directed localization in sparse networks with spatial randomness. Nonequilibrium statistical dynamics and population genetics that incorporate genetic drift, mutations, migrations, and competition and cooperation have played a crucial role in the evolutionary history of many species, in particular on solid surfaces. Examples include the migrations of invasive species, or bacterial invasions of animal tissue. Using the tools of nonequilibrium statistical dynamics and population genetics the PI will examine how these phenomena affect genetic lineages in spatial media containing obstacles. Because biological organisms do not typically grow up in well-mixed test tubes or featureless Petri dishes, it is important to understand how they behave in the presence of environmental inhomogeneities. The PI's research on thermalized shells builds on the "extreme mechanics" of thin plates and shells, characterized by the highly nonlinear Foeppl - von Karman equations. The background curvature of thermally excited spherical shells presents new challenges relative to flat plates, which will be addressed by generalizing graphical summation and renormalization group methods that have proven useful for sheet polymers. Finally, the project will investigate directed localization and the associated nonequilibrium dynamics in strongly non-Hermitian matrices (involving both excitatory and inhibitory connections) that arise naturally in simple models of interacting ecosystems and in sparse neural networks. The PI's theoretical research will determine how the intricate fractal eigenvalue spectrum that controls the spontaneous activity and induced response changes with an increasing ratio of inhibitory to excitatory connections and with a variable bias for the transfer of information in a particular direction.    Strongly interdisciplinary by nature, the research could provide insights into controlling human pathogen invasions, the development of drug delivery strategies, and human and animal neural development. In addition, the project will contribute to the training of students and postodocs in modern theoretical methods with a wide area of applicability.