A theoretical research and training program is proposed in the fundamental area of growth of unstable interfaces and related problems, a fast developing area in contemporary research. The growth of an interface is said to be unstable if a smooth initial interface eventually develops a fractal structure. Under this umbrella one studies important and distinct phenomena such as growth of bacterial colonies and cancer cells, dynamics of chemical reaction, propagation of crystallization fronts, dynamics of the Quantum Hall droplets, transport of 1D fermions, Diffusion Limited Aggregation (DLA), surface instabilities in hydrodynamical flows, Integrable Models, and 2D Quantum Gravity and Matrix Models. Laplacian Growth (LG) is the simplest nontrivial model of such unstable growth. In LG the velocity of the interface is proportional to the gradient of some scalar field with an additional condition of incompressibility. This scalar field can represent very different physical quantities such as electrostatic field, food concentration, or a pressure field. It is an excellent context in which to study the general phenomena of unstable growth. The proposed project looks beyond current LG formalism to the possibility of using symmetries and the introduction of stochastic noise to the problem, making it more realistic and relevant. It will allow one to: i) consider the selection problem in any geometry of Laplacian Growth as a problem of stability against stochastic noise; ii) introduce new formalisms for the growth more closely connected to experiment; iii) approach the problem of finding the (multi)fractal properties of the interface; iv) formulate the Laplacian Growth as a Hamiltonian problem. Broader impact: 1. LG and DLA are good research area for graduate and undergraduate students to be introduced into nonlinear dynamics, Integrable Models, and other contemporary areas of mathematics and physics. It will allow a student to see the effects of nonlinear couplings, see the emergence of new fundamental dynamical properties which arise due to stochasticity. The student will gain an experience in interdisciplinary research. 2. The PI and the students will publish in peer reviewed journals and give presentations at conferences. In addition the PI will use every opportunity for media appearances and public lectures to communicate the results to a broader public. 3. The proposed research will contribute to the development of a better understanding of the dynamics of growth of unstable interfaces in all the applications mentioned above. These areas are of the great importance for the technology, medicine, and environmental studies.
The growth of unstable interfaces is a fast developing area in contemporary research. The growth of an interface is said to be unstable if a smooth initial interface branches in the process of growing and eventually develops a fractal structure. Such processes are abundant in nature. Almost every growth in complicated biological systems is of this class. Under the umbrella of ``growth of unstable interfaces'' one studies important and distinct phenomena such as growth of bacterial colonies and cancer cells, dynamics of chemical reaction, propagation of crystallization fronts, dynamics of the Quantum Hall droplets, transport of 1D fermions, Diffusion Limited Aggregation (DLA), surface instabilities in hydrodynamical flows, Integrable Models, and 2D Quantum Gravity and Matrix Models. Laplacian Growth (LG) is the simplest nontrivial model of such unstable growth. LG process is actively studied by both mathematicians and physicists, as well as engineers . This project looked beyond current LG formalism to the possibility of introduction of noise in the process. This noise always present in nature and in unstable processes is expected to lead to significant corrections to the pure mathematical formulation of the problem. The problem to correctly introduce nose in an unstable process is mathematically very complicated. In the project a specific way of doing so was proposed. The resulting formulation did not lead to the clear answers of the physical questions. However, it did lead to a much dipper understanding of the role different regularization schemes play in the LG. In my current research I am trying to classify and their universality classes. In the meantime the research on LG problem branched into the researched in forced magnetization dynamics, thermoelectricity of topological insulators, and dynamics of Cold Fermionic gases. This research in turn resulted in many publications in prominent peer reviewed journals. The Laplacian Growth and DLA were great research area for graduate and undergraduate students. They were introduced into nonlinear dynamics, Integrable Models, and other contemporary areas of mathematics and physics. It allowed them to develop a wide range of skills, from analytical calculations to computer simulations of nonlinear dynamics. The experience students have got from the combination of nonlinear dynamics and instability regularization can be applied to many different fields. It will allow them to formulating correct questions for unstable systems and their proper regularizations, as well as to see the emergence of new fundamental dynamical properties. In addition, due to the vast applicability of this research, the students gained an experience in interdisciplinary research and an ability to connect research in different areas.
