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.

Introduction
Outline
What is homogeneous dynamics
Dynamics on a homogeneous space
Examples
Harmonic analysis
Invariants
Measure rigidity
Modular group
Euclidean lattice
Oppenheim conjecture
Homogeneous flows
Quantitative versions
Radness theorem
Quantum chaos
Eigenvalues
Hyperbolic billiard
Hecht operators
Generic eigenvalues
Gap distribution
Perry correlation
Randomness Sequence Modulo 1
Exponential Limit Distribution
Number theoretic sequences
Alkis McMullan
The gap distribution
Distribution theorem
Outline of lectures
Applications to the Lawrence gas
Applications to quasicrystals
Applications to additive problems
Conclusion