Vertex algebras are a class of noncommutative, nonassociative algebras that arose out of conformal field theory in the 1980s, and were axiomatized by Borcherds in his proof of the Moonshine Conjecture. They have applications in many areas of mathematics including representation theory, combinatorics, finite group theory, number theory, and algebraic geometry. I will give an introduction to the subject and will discuss some recent progress on a certain classification problem.
Classical Schur-Weyl duality relates the irreducible characters of the symmetric group to the irreducible characters of the general linear group via their commuting actions on tensor space. We investigate the analog of this result for the group of unipotent upper triangular matrices . In this case the character theory of is unattainable, so we must employ supercharacter theory, creating a striking variation.
The normal subgroups of a finite group can be realized as intersections of stabilizers of the irreducible -modules. A supercharacter theory of is an analogue to the representation theory of where the theory is built from nearly irreducible modules. Via stabilizers, each such theory gives a sublattice of the lattice of normal subgroups, along with similar theories on each subquotient of the lattice. Now given a sublattice of the lattice of normal subgroups of a finite group and supercharacter theories on the covering relations, it is natural to ask under what conditions a supercharacter theory of can be built that respects the imposed covering relations. Recently, Aliniaeifard gave a construction which allows one to build from a sublattice a supercharacter theory with as its associated sublattice. However, this construction does not allow for the choice of supercharacter theories on each subquotient. In this talk, we will discuss Aliniaeifard's construction as well as some methods being developed to refine it to account for more general covering relations.
The Brill-Noether varieties of Riemann surface C are objects of classical interest. They parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension, i.e. equations for the curve. When C is general, these varieties are well understood: they are smooth, irreducible, and have the "expected" dimension. As one ventures deeper into the moduli space, past the general locus, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill--Noether theorem, which determines the dimensions of the Brill-Noether varieties on a general curve of fixed gonality, i.e. "general inside a chosen special locus”.
In the first hour, I will give an introduction to the tropical linear series techniques that allow combinatorial tools to be used to attack such problems. In the second hour, I will explain some of the new ideas used, which introduce logarithmic geometry as a new tool in Brill-Noether theory. This is joint work with Dave Jensen.