Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions. Donatella Danielli is a Professor of Mathematics at Purdue University. She received a Laurea cum Laude in Mathematics from the University of Bologna, Italy (1989), and a Ph.D. degree in Mathematics from Purdue University (1999) under the direction of Carlos Kenig. Her research areas are partial differential equations, harmonic analysis, and sub-Riemannian geometry. She has been the recipient of an NSF CAREER Award, a Purdue University Teaching for Tomorrow Award, and a Ruth and Joel Spira Award for Graduate Teaching. Before joining the Purdue University faculty in 2001, she held positions at The Johns Hopkins University and at the Institut Mittag-Leffler in Sweden. She is the author of 40 published papers and 2 monographs. She is the creator and organizer of the Symposia on Analysis and PDEs and the Women in Mathematics Days, both at Purdue University.