Applications of Homogeneous Dynamics - Lecture 1

Explore the fascinating world of homogeneous dynamics and its wide-ranging applications in this comprehensive lecture. Delve into the fundamental concepts, starting with an introduction to homogeneous dynamics and its role in various mathematical fields. Examine dynamics on homogeneous spaces, harmonic analysis, invariants, and measure rigidity. Investigate the modular group, Euclidean lattices, and the Oppenheim conjecture. Study homogeneous flows, quantitative versions, and the Radness theorem. Discover the connections between homogeneous dynamics and quantum chaos, exploring eigenvalues, hyperbolic billiards, and Hecht operators. Analyze gap distributions, Perry correlation, and randomness in sequences modulo 1. Explore number theoretic sequences and the Alkis McMullan theorem. Learn about applications to the Lawrence gas, quasicrystals, and additive problems. Gain a deep understanding of the interplay between homogeneous dynamics, number theory, and statistical mechanics in this insightful lecture.