Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise ....Singular phenomena for nonlinear partial differential equations arising in applications. The development of nonlinear Partial Differential Equations (PDEs) in Australia is recognized worldwide through the outstanding contributions of mathematicians from the ANU, University of Sydney and other top Australian Universities. This project undertakes research in the PDEs field and follows directions of very current interest at an international level. Beyond the ANU, the project will enhance expertise in Australia in very active areas of mathematics research related to applications in physics, biology and other applied disciplines. Moreover, it will foster collaboration with mathematicians of international standing from Australia and abroad. Read moreRead less
Optimal Control of Stochastic Partial Differential Equations. The problem to control a stochastic process so as to minimize a certain cost functional arises in many areas of Applied Sciences, Engineering and Mathematical Finance. An important practical question is to find, for a given cost functional, the optimizing control in a feedback form. We propose new tools to construct such optimal controls for a class of stochastic processes which are solutions to stochastic partial differential equati ....Optimal Control of Stochastic Partial Differential Equations. The problem to control a stochastic process so as to minimize a certain cost functional arises in many areas of Applied Sciences, Engineering and Mathematical Finance. An important practical question is to find, for a given cost functional, the optimizing control in a feedback form. We propose new tools to construct such optimal controls for a class of stochastic processes which are solutions to stochastic partial differential equations. As an outcome of this project we will obtain methods to determine the optimal control policies for a large variety of cost functionals and degenerated stochastic partial differential equations, in particular those arising in modelling of volatility in Finance.Read moreRead less
Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contri ....Geometric variational problems and nonlinear partial differential systems. We will investigate several important problems on non-linear partial differential systems, bridging analysis, differential geometry and mathematical physics. Harmonic maps are the prototype of maps minimizing the Dirichlet energy. The liquid crystal configuration generalizes the harmonic map with values into two dimensional spheres. The Yang-Mills equations originated from particle physics. We will make fundamental contributions to these topics: Regularity problem and energy minimality of weakly harmonic maps, Weak solutions of the liquid crystal equilibrium system, Yang-Mills heat flow and singular Yang-Mills connections.
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