The graduate student seminar meets from 12:30 to 2:00 pm Tuesdays in Goldsmith 300. Want to give a talk? Email Matt Cordes at mcordes@etc... for details.
A quiver is simply a finite graph with arrows for edges, and a representation of a quiver is just a collection of vector spaces and linear maps corresponding to the vertices and arrows respectively. Quiver representations have become ubiquitous in modern mathematics having connections to such diverse areas as representation theory, mathematical physics, Lie theory, Coxeter groups, hyperbolic geometry, algebraic geometry, etc...
A central problem in representation theory is to classify all indecomposable representations of a given quiver. This problem has a really nice solution if there are only finitely many (or "almost finitely many'') such indecomposables, but in general is essentially an impossible problem. In the past 20 years a fair amount of progress has been made in understanding this impossible problem by introducing so called "moduli spaces'' which describe how representations vary in continuous families.
This talk is supposed to serve as a gentle introduction to this circle of ideas and to provide a little context for Kiyoshi's series of NOSY seminars on semi-invariants starting next week. In particular, I will assume absolutely no background in quivers or even algebraic geometry (though, things will probably make a lot more sense with some geometric background).
This is a joint paper with Jonah Ostroff (for his minor exam). It starts with an elementary concept of a special set of planar tree which we used to develop the very interesting theory of cluster categories, semi-invariants and representations of quivers using trees to illustrate a special case of general formula which have recently been developed by many authors. More precisely, a cobinary tree is a planar tree in which every node has either one parent and two children or one child and two parents. The internal edges of the tree give the formulas for the semi-invariants for the corresponding cluster in the cluster category of type \( A_n \). With a few examples, you should see what this means.
The second lecture will be a continuation of the first lecture with the aim of applications to Morse Theory at the end of the lecture. I will use the combinatorics of trees to obtain formulas for the incidence matrices which arise in a family of Morse functions. The stability theorem for semi-invariants translates into stability conditions for representations which gives a formula for the entries of the incidence matrix. There will not be any proofs. I will just do some simple examples to illustrate the concepts. You should be able to do the computation for any MCT at the end of the lecture.
In this [pizza] talk, I will introduce the history and basic results of Diophantine approximation, concerning about the question: how well can real numbers be approximated by rational numbers? Then, we will play a [never-ending] game, which is similar to putting pepperoni on pizzas, to answer a deep result about the abundance of [badly approximable] numbers that cannot be approximated well by rationals.