This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

1 N. Abatangelo and E. Valdinoci, Getting acquained with the Fractional Laplacian.- 2 I. Birindelli et al., Dirichlet problems for fully nonlinear equations with "subquadratic" Hamiltonians.- 3 S. Borghini and L. Mazzieri, Monotonicity formulas for static metrics with non-zero cosmological constant.- 4 U. Boscain and M. Sigalotti, Introduction to controllability of nonlinear systems.- 5 A. Cesaroni and M. Cirant, Introduction to variational methods for viscous ergodic Mean-Field Games with local coupling.- 6 E. Cinti, Flatness Results for Nonlocal Phase Transitions.- 7 M. Cozzi, Fractional De Giorgi classes and applications to nonlocal regularity theory.- 8. F. G. Düzgün et al., Harnack and pointwise estimates for degenerate or singular parabolic equations.- 9 C. Mantegazza et al., Lectures on curvature ow of networks.- 10 L. Mari and L. F. Pessoa, Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview.- 11 C. Mooney, Singularities in the Calculus of Variations.- 12 A. Tellini, Comparison among several planar Fisher-KPP road-field systems.

Serena Dipierro received her PhD from SISSA (Trieste) in 2012, and she is currently an Associate Professor at the University of Western Australia's Department of Mathematics and Statistics. Her research interests include partial differential equations, free boundary problems, nonlinear analysis, nonlocal equations, and calculus of variations.