About the Program
The goal of this thematic program is to bring together leading experts in Analysis and Partial Differential Equations for a trimester in IMPA to present a perspective on the current trends on selected research topics and to foster the scientific collaboration with researchers in the Brazilian and South American communities. The focus of the program will be on the analysis of elliptic and kinetic PDEs including the following topics:
Special focus will be given to young researchers and graduate students through a number of activities programmed for the period. These include:
Research Topics
Fully nonlinear equations
The research in nonlinear elliptic equations is one of the most developed in mathematics and of great importance because of its applications in broader scientific disciplines such as differential geometry, fluid dynamics, phase transitions and mathematical finance. In the past few years, we have witnessed many new developments in the theory of homogenization and nonlocal diffusive processes related to the study of fully nonlinear equations. This has fostered a better understanding of problems in material science, optimal design and valuation of financial options, just to mention a few examples. They also arise naturally as the equations governing any probabilistic model whose values may take long jumps. This program will present the latest developments in the subject by exploring recent and modern techniques, and addressing further directions and important open problems.
Free boundary problems
Free boundary problems are the central subject in the study of phenomena where phase transitions occur. They arise when one attempts to describe a discontinuous change of behavior in a physical or biological quantity. Applications appear in stopping time for optimal control, ground water hydrology, plasticity theory, optimal design and problems in superconductivity. Typical examples are the evolution of an ice-water mixture, an elastic membrane constrained to stay within a given region and in the description of laminar flames as an asymptotic limit for high energy activation. In recent years, we have seen a substantial progress in the understanding of free boundary problems involving nonlocal operators. These operators appear in the study of reaction-diffusion problems in plasma physics, semi-conductor theory and flame propagation in the presence of turbulence or long-range interactions. This program will present the research in the subject by examining the latest achievements and results obtained in the field.
Particle systems and kinetic equations
Kinetic theory was originally devised to explain macroscopic properties of gases. Such initial work was culminated with the introduction of the Boltzmann transport equation by L. Boltzmann in 1872. Today, modern kinetic theory has emerged as an indispensable tool to give a quantitative description of diverse phenomena in many sciences such as physics, chemistry, biology, social sciences and economics. Applications ranging from semi-conductors, plasmas, cell dynamics, flocking, traffic, granular materials, to neuron networks, random wave propagation, multi-scale modeling and inverse problems are among the scope of kinetic theory. Interestingly, the underlying equation is always the Boltzmann equation, that is, a transport term equated to an integral (usually non-linear) operator. The mathematics of such equation, even after the progress of the last 30 years, is at the early stages; basic questions such as existence and uniqueness of solutions for general initial configurations and the rigorous validation of the equation are formidable (and open) tasks. This program will address some of the important questions and the mathematics developed in recent years to answer them, and give perspective of the new trends, open problems and relevant questions in the theory for the years to come.