Let $K$ be an algebraically closed field. Given three elements of some Lie algebra over $K$, we say that these elements form an $sl_2$-triple if they generate a subalgebra which is a homomorphic image of $sl_2(K).$ In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of $sl_2$-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic $p$. In particular, I will focus on the results in classical Lie algebras, which can be found as subsets of $gl_n(K)$.

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.