Dr. Matthew Badger, Assistant Professor of Mathematics, has been awarded a five-year, $410,000 grant (2017–2022) from the National Science Foundation’s Faculty Early Career Development (CAREER) Program for his project “Analysis and Geometry of Measures.” A CAREER award is considered the highest distinction the NSF provides to a junior researcher in the mathematical sciences. Dr. Badger earned his PhD from University of Washington in 2011, and prior to joining the faculty at UConn, he was a James H. Simons Instructor and NSF postdoctoral fellow at Stony Brook University from 2011–2014.
The area of mathematics in which Dr. Badger specializes is called geometric measure theory. Measures are abstract generalizations of length, area, or volume, which assign a size to every mathematical set. They were first developed by Lebesgue in 1902 to define the integral of certain discontinuous functions that could not be computed using the Riemann integral. Lebesgue gave a rigorous definition of the length of a subset of the real line. Geometric measure theory was born out of attempts to define the length of subsets of Euclidean spaces.
Imagine, for example, that you wanted to measure the length of a piece of wire or string. This is easy: unwind the string into a straight line, place it next to a ruler, and read off its length. What if the string was cut into two pieces? This is easy, too: use the ruler to measure the length of the first piece of string, repeat with the second piece of string, and then add the two lengths together. What if the string is cut into infinitely many pieces, some short, some long, some as short as a fleck of dust? The pieces are scattered and cannot possibly be lined up next to a ruler without misplacing some of the pieces. How can you measure the length of the string now?
An answer to the question of how to assign length to subsets of Euclidean space was given by Carathéodory in 1914 and was studied in greater depth by Besicovitch in the 1920s and 1930s. In a recent series of three papers, Dr. Badger and his collaborator, Dr. Raanan Schul of Stony Brook University, solved a 70-year-old problem in geometric measure theory from the 1940s of how to identify the “length-like” part of an arbitrary measure in Euclidean space. The research component of Dr. Badger’s CAREER award is directed at related problems, whose resolution would significantly increase our understanding of the internal structure of general measures.
Dr. Badger’s CAREER project also has a significant educational component, with activities aimed at training and professional development for graduate students and postdocs in analysis inside and outside of UConn. One highlight is a pair of linked conferences Dr. Badger will organize for young researchers working in geometric measure theory and related areas. The first conference (for postdocs) will take place at UConn in Fall 2017; the second conference (for graduate students) will take place in Spring 2019. Several postdocs from the first conference will be invited back to UConn to give mini-courses for graduate students in the second conference. For more information about these upcoming events, please contact Dr. Badger at matthew.badger@uconn.edu.