Alfonso Sorrentino minicourse "Actionminimizing methods in dynamics and geometry" (February 17–March 4, 2020, Steklov Mathematical Institute of RAS, Moscow)
In these lectures we discuss John Mather's variational approach to the study of convex and superlinear Hamiltonian systems, what is generally called AubryMather theory. Starting from the observation that invariant Lagrangian graphs can be characterised in terms of their "actionminimizing" properties, we shall describe how analogue features can be traced in a more general setting, namely the socalled Tonelli Hamiltonian systems. This approach brings to light a plethora of compact invariant subsets for the system, which, under many points of view, can be seen as a generalisation of invariant Lagrangian graphs, despite not being in general either submanifolds or regular.
Besides being very significant from a dynamical systems point of view, these objects also appear in the study of weak solutions of the HamiltonJacobi equation (weak KAM theory) and play, as well, an important role in other different contexts: such as analysis, geometry, mathematical physics, billiard dynamics, etc. We shall also see how similar results can be also extended to some nonconservative setting, namely the case of socalled conformally symplectic systems.
Tentative course content:
 From KAM theory to AubryMather theory: actionminimizing properties of invariant Lagrangian graphs.
 Tonelli Lagrangian and Hamiltonian on compact manifolds.
 Mather theory: Actionminimizing invariant measures, Mather sets and minimal average actions.
 Weak KAM theory: HamiltonJacobi equation, weak (sub)solutions, actionminimizing curves, Aubry sets and Mane sets.
 AubryMather theory for conformally symplectic systems.
Some References:
 S. Maro', A. Sorrentino: "AubryMather theory for conformally symplectic systems" Comm. Math. Phys., 354 (2): 775808, 2017.
 A. Sorrentino: "ActionMinimizing Methods in Hamiltonian Dynamics. An Introduction to AubryMather Theory". Mathematical Notes Series Vol. 50 (Princeton University Press), 2015.
Financial support:
The visit of Alfonso Sorrentino is supported by the Simons Foundation (grant No. 615793). The event is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 0751520191614).
RSS:
Organizer
Sorrentino Alfonso, Università degli Studi di Roma — Tor Vergata
Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow Steklov International Mathematical Center 

Alfonso Sorrentino minicourse "Actionminimizing methods in dynamics and geometry", Steklov Mathematical Institute of RAS, Moscow, February 17–March 4, 2020 


February 17, 2020 (Mon) 

1. 
Actionminimizing methods in dynamics and geometry. Lecture 1 A. Sorrentino February 17, 2020 17:00–18:30, Steklov Mathematical Institute of RAS, Moscow, Steklov Mathematical Institute, Room 430 (8 Gubkina)






February 19, 2020 (Wed) 

2. 
Actionminimizing methods in dynamics and geometry. Lecture 2 A. Sorrentino February 19, 2020 09:30–11:00, Steklov Mathematical Institute of RAS, Moscow, Steklov Mathematical Institute, Room 430 (8 Gubkina)






March 4, 2020 (Wed) 

3. 
Actionminimizing methods in dynamics and geometry. Lecture 3 A. Sorrentino March 4, 2020 11:30–13:10, Steklov Mathematical Institute of RAS, Moscow, Steklov Mathematical Institute, Room 313 (8 Gubkina)





4. 
Actionminimizing methods in dynamics and geometry. Lecture 4 A. Sorrentino March 4, 2020 14:00–15:30, Steklov Mathematical Institute of RAS, Moscow, Steklov Mathematical Institute, Room 313 (8 Gubkina)





