Explore the fascinating world of algebraic curves and their isolated points in this hour-long mathematics seminar. Delve into Faltings' celebrated result and its implications for algebraic points on curves defined over rational numbers. Investigate how isolated points behave in families of curves and uncover their consequences for the arithmetic of elliptic curves. Examine the emerging field of using parameterized point geometry to explore curve arithmetic. Learn about collaborative research efforts in this area, including joint work with A. Bourdon, Ö. Ejder, Y. Liu, F. Odumodu, I. Vogt, I. Balçik, and S. Chan.