On the fields of algebraic and differential rational invariants: calculation, application, and interaction

At the beginning of the talk, we will recall Rosenlicht's theorem, which describes rational invariants and quotient spaces of actions of algebraic groups on algebraic varieties.
Next, we will discuss the actions of algebraic Lie pseudogroups on algebraic differential equations and their rational differential invariants, as well as the Lie–Tresse theorem, describing the fields of rational differential invariants.
To put it simply, Rosenlicht's theorem says that the quotient space of an algebraic group after removing singular orbits is an algebraic variety where the field of rational functions is isomorphic to the field of rational invariants.
Similarly, the Lie–Tresse theorem states that regular orbits of the action of algebraic pseudogroups on algebraic differential equations also form a differential equation.
Depending on the time available, we will consider examples of calculating and applying algebraic and differential invariants.