Combinatorics, Algebra, & Topology Seminar
All talks are from 1200-1300 in the designated room unless otherwise specified.
Highest--weight representations and global Weyl modules: from classical Lie algebras to YangiansPrasad SenesiCatholic University of AmericaTime: 12:00 PMHighest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Particular examples of highest-weight representations of certain infinite--dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding Loop algebras and quantum groups. Along the way, the utility of highest-weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago).
Quiver representations, theta functions and tropical curves countingMan-Wai "Mandy" CheungHarvard UniversityTime: 12:00 PMScattering diagrams, theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will describe those objects in terms of quiver representations. After that, we will give a expression of tropical curves counting by quiver representations. Last one is a joint work in progress with Travis Mandel.