I will discuss the adiabatic theorem in quantum mechanics and the appearance of Berry's phase in the adiabatic evolution on periodic trajectories in parameter space. I will describe how one can observe the Berry phase and some experiments that demonstrate it. I will also discuss similar phenomena in classical mechanical systems and try to explain how they are related. If time permits, I will discuss the relation between the geometric phases and holonomy, which is the change of a vector parallel transported around a closed loop on a curved surface.
As references, I can recommend the original paper by Berry:
Berry Michael Victor 1984Quantal phase factors accompanying adiabatic changesProc. R. Soc. Lond. A39245–57
https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry120.pdf
And the paper describing how the geometric phase can be interpreted as the holonomy of parallel transport:
Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase
Barry Simon, Phys. Rev. Lett. 51, 2167
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.2167
If you want to read a lot and like maths, there is a good book
Dariusz Chruściński , Andrzej Jamiołkowski, Geometric Phases in Classical and Quantum Mechanics
https://link.springer.com/book/10.1007/978-0-8176-8176-0
There is also an older book, which is a collection of papers with some introduction, and which has more physics examples:
Geometric phases in physics, Frank Wilczek; Alfred Shapere, 1989
This event is part of the master lecture series organized by the Section for Theoretical Physics.