Polynomial identity algebras (or PI-algebras) can be seen as generalizations of both commutative and finite dimensional algebras which maintain many ``nice'' properties. In this talk, we give a short introduction to Lie PI-algebras, focusing on the study of multilinear identities. Using these techniques, we demonstrate the existence of an identity for Lie algebras having certain actions on them; such actions arise naturally, for example, when the Lie algebra is graded by a finite group.

I will provide a brief introduction to impartial game theory consisting of examples of games as well as elementary operations on them. This will be followed by a presentation of their syntactic counterparts in Coq, culminating in formal proofs of several basic theorems.

Having studied the equivariant cohomology of smooth toric varieties, we will see to what extent it determines the non-equivariant cohomology. In the equivariantly formal case, one gets the Jurkiewicz-Danilov theorem (with integer coefficients). In general, a complete answer is only known for rational coefficients.