# Books written by Alessio Figalli.

Alessio Figalli is an Italian mathematician working primarily on calculus of variations and partial differential equations. In 2018 he won the Fields Medal "for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability".

## The Monge-Ampere Equation and Its Applications

The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation.

## Optimal Transportation and Action-Minimizing Measures

In this book we describe recent developments in the theory of optimal transportation, and some of its applications to fluid dynamics. Moreover we explore new variants of the original problem, and we try to figure out some common (and sometimes unexpected) features in this emerging variety of problems.

## Optimal Control With Applications in Space and Quantum Dynamics

Several complete textbooks of mathematics on geometric optimal control theory exist in the literature, but little has been done with relevant applications in control engineering. This monograph is intended to fill this gap.

## Partial Differential Equations and Geometric Measure Theory

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods.

## Nonlocal and Nonlinear Diffusions and Interactions

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems.