Discrepancy theory studies how well discrete objects can approximate continuous ones. It has numerous applications throughout mathematics and computer science, including numerical analysis, computational geometry, and communication complexity, among others. I will discuss recent examples of the interplay between discrepancy theory and computer science.
I will present positive and negative results on the approximability of classical notions of discrepancy, including the first hardness of approximation results for discrepancy and the first non-trivial approximation algorithms for hereditary discrepancy. The approximation algorithms are intimately related to a new application of discrepancy theory to private data analysis. This connection has led to a characterization of the necessary and sufficient error to answer an arbitrary load of counting queries under differential privacy. The tools developed for these results have also led to progress on a number of long-standing questions in discrepancy theory, such as the discrepancy of arithmetic progressions and of axis-aligned boxes.