Explore the fascinating world of branching random walks in this comprehensive lecture by Parthanil Roy from the Indian Statistical Institute. Delve into the behavior of growing particle systems, their applications in physics, biology, and ecology, and discover how they connect to various scientific models. Learn about two important conjectures formulated by renowned physicists Éric Brunet and Bernard Derrida, and understand how Roy's theorem verifies these conjectures in a significant special case. Gain insights into the long-term behavior of branching random walks and their snapshot appearance after extended periods. The talk also covers related topics such as simple random walks, Euclidean geometry, and random walks in higher dimensions. Engage with the speaker's expertise in probability theory, ergodic theory, and hyperbolic dynamics while exploring the interdisciplinary nature of this mathematical concept.