The field of Applied Analysis brings together many mathematical topics, such differential equations, dynamical systems, variational calculus, functional analysis, geometry, and approximation theory. The Applied Analysis Group focuses on these mathematical disciplines and their application to the real world around us.
Partial differential equations, especially the nonlinear ones, are as different as animals in the zoo. We study them: do they have solutions, are these solutions unique? How do the solutions behave? How do they depend on parameters? How should we calculate solutions numerically? How do all these properties relate to the real-world situations that generated the equations in the first place? Tools such as functional analysis and variational calculus allow us to create order in the zoo.
EXPLORING THE ZOO OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
EXPLORING THE ZOO OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Connecting PDEs with stochastics
Since the invention of Wasserstein gradient flows, exciting new connections have arisen between optimal transport, (generalized) gradient flows and large deviations. Optimal transport provides tools for analyzing PDEs, large deviations give us the gradient-flow structure underlying many PDEs, and the connection between PDEs and stochastic processes ties all these concepts together.
Dynamical-Variational Transport Costs
We develop a theory of dynamical-variational transport costs (DVTs), a class of large-deviation inspired functionals that provide a variational generalisation of a zoo of existing transport distances. DVTs generate non-homogeneous generalisations of length spaces that will be used to extend metric-space techniques to general length spaces, thereby allowing a variational framework for generalised gradient flows to be rigorously investigated, and the multiscale analysis of such evolutions used in the development of numerical schemes.