Applications of Frobenius beyond prime characteristic.
Abstract: Recall that the Frobenius morphism is simply the map sending an element in a ring of prime characteristic $p>0$ -- say, a polynomial with coefficients in a finite field -- to its $p$-th power. Though simple to define, Frobenius has proven to be a useful and effective tool in algebraic geometry, representation theory, number theory, and commutative algebra. Furthermore, and remarkably, some of the most interesting applications of Frobenius are to the study of objects defined over the complex numbers, and more generally, over a field of characteristic zero! In this talk, we will discuss some of these applications, with an eye towards classical singularity theory and birational algebraic geometry, both over the complex numbers.