Abstract: Nonlocality has become increasingly noticeable in nature. The modeling and simulation of its presence and impact motivate new development of mathematical theory. In this lecture, we focus on nonlocal models with a finite horizon of interactions, and illustrate their roles in the understanding of various phenomena involving anomalies, singularities and other effects due to nonlocal interactions. We also present some recent analytical studies concerning nonlocal operators and nonlocal function spaces. The theoretical advances are making nonlocal modeling and simulations more reliable, effective and robust for applications ranging from classical mechanics to traffic flows of autonomous and connected vehicles.

Bio: Qiang Du is the Fu Foundation Professor of Applied Mathematics of Columbia University where he is also a faculty of Data Science Institute and co-chairs the Center of Computing Systems for Data-driven Science. He received his B.S. from the University of Science and Technology of China (USTC) and Ph.D from Carnegie Mellon University. A Fellow of SIAM and AMS, he now serves as the EIC of SIAM Journal of Applied Mathematics (SIAP) and the founding co-EIC of Communications of the American Mathematical Society (CAMS). Some of the recognitions for his work include Frame Faculty Teaching Award (1992), Feng Kang Prize in Scientific Computing (2005), SIAM Outstanding Paper Prize (2016), USACM T.J.R. Hughes Medal (2021). He was also an invited speaker of International Congress of Mathematicians (ICM 2018).