By the Numb3rs Fall 2010 - Research

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NEW GRANTS

Juan Manfredi’s award from the National Science Foundation is on Analysis of the p-Laplacian

In many problems in physics and engineering the relevant energies are proportional to the square of the velocity. The resulting equations are linear and model very well small variations from equilibrium. But more substantial variations are better modeled by considering non-quadratic energies. In this proposal we study some aspects of the mathematical theory of the equations that result from the minimization of energies (or other quantities in physics and engineering) that are given by power laws.

Manfredi explores the connections between p-harmonic functions and stochastic games. It may shine new light into some optimization problems that can be formulated in spaces quite more general than Euclidean space (graphs, trees, length spaces). In addition, Manfredi will use the formulation of Tug-of-War games in simple graphs to mentor several freshman students who used computer simulation to run these games, and are exposed to mathematical thinking early in their undergraduate career.

Anna Vainchtein receives an award from the National Science Foundation to study “Kinetics of Lattice Phase Transitions”

Anna Vainchtein received an award from the National Science Foundation to study “Kinetics of Lattice Phase Transitions”. This proposal is concerned with physically-motivated modeling of materials undergoing martensitic phase transition, a diffusionless deformation of crystal lattice from the high-symmetry parent austenite phase to the low-symmetry martensite phase which can exist in several symmetry-related twin variants. This project has been funded for 3 years.

Jon Rubin receives an award from the Nation Science Foundation to study the use of mathematical modeling and analysis to investigate how the brain generates rhythms that drive respiration, locomotion, and other rhythmic processes

The topic of this project is the use of mathematical modeling and analysis to investigate how the brain generates rhythms that drive respiration, locomotion, and other rhythmic processes. The work will involve various techniques from dynamical systems theory, which allows us to study solutions to complicated systems of differential equations, and will break new ground in the analysis of how intrinsic neuronal properties interact with the architecture of connections among neurons to generate particular activity patterns. The results will lead to biological predictions about how the respiratory system can adapt to extreme conditions, such as oxygen deprivation, and how it may be possible to restore locomotor rhythms after spinal cord injury.