**Speaker:** Zhifu Xie

**Title:** On the Uniqueness of Convex Central Configurations in the Planar 4-Body Problem

**Abstract:** A central configuration is a specific arrangement of masses, and a planar central configuration can lead to a homographic periodic solution. It is crucial for understanding the dynamic behavior of the N-body problem, and the question of its finiteness has been a challenge for mathematicians in the 21st century. For the planar four-body problem, its finiteness has been proven by computer-assisted proof in 2006 by Hampton and Moeckel, but there is still much to understand. One conjecture is that there exists a unique convex central configuration for any four positive masses in a given order. Many research paper has attempted this question by assuming either having some equal masses or having restrictions of the geometric shape such as a trapezoid or co-circular shape. In this talk, we provide a rigorous computer-assisted proof (CAP) of the conjecture for four masses belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem. Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its surrounding neighborhood.

**Bio:** Dr. Zhifu Xie holds the Wright W. and Annie Rea Cross Endowed Chair in Mathematics and Undergraduate Research at the University of Southern Mississippi, where he is a Professor of Mathematics. Zhifu earned his B.S. in Mathematics Education from Chongqing Normal University in 1998, his M.S. in Mathematics from Chongqing University in 2001, and his Ph.D. in Mathematics from Brigham Young University in 2006. He began his academic career as an assistant professor at Virginia State University in 2007 and was promoted to full professor in 2015. Zhifu’s outstanding research, teaching, service, and integration of knowledge earned him recognition as a 2012 SCHEV Outstanding Faculty Award Finalist (SCHEV refers to the State Council of Higher Education for Virginia). He joined the University of Southern Mississippi in 2016 as the Cross Endowed Chair. His research interests range from classical celestial mechanics to differential equations, including their applications in fields such as reaction-diffusion equations and infectious disease models. Zhifu is renowned for his innovative approaches to classroom teaching, which incorporate new techniques and materials. He actively engages undergraduate students in his cutting-edge mathematical research, overseeing their internships and individual projects, and finds great satisfaction in watching their progress.