Explore homotopy continuation and its applications in science and engineering through this 47-minute seminar by Jonathan Hauenstein from the University of Notre Dame. Gain insights into tracking solutions for parameterized nonlinear equation systems and understand how changing parameters like temperature, pressure, and leg lengths affect solution behavior. Delve into an overview of homotopy continuation followed by practical applications in chemical equilibria, kinematic synthesis for mechanical engineering, and variational inference for biological model parameter estimation. Learn about key concepts such as the Implicit Function Theorem, phase equilibrium, Gaussian curvature, and adaptive annealing. Enhance your understanding of applied geometry and algebra in this Society for Industrial and Applied Mathematics (SIAM) presentation.