Dive into an extensive 3-hour 47-minute lecture on advanced mathematical concepts for game development. Explore a wide range of topics including smoothing and easing functions, Hermite interpolation, splines, complex numbers, Euler's formula, dual numbers, quaternions, vector algebra, geometric algebra, and their applications in game development. Learn about the imaginary unit, complex plane, and Euler's identity. Discover the reasons behind using quaternions in games, alternative perspectives on vector algebra, and the intricacies of geometric product and multivectors. Gain insights into 3D geometric algebra, full 3D multivectors, and the generality of geometric algebra. Conclude with discussions on quaternions from a geometric algebra perspective, 4D games, and practical animation techniques in Unity.