# Voula Collins

### First Position

Postdoctoral Associate, University of Connecticut### Dissertation

*Crystal branching for non-Levi subgroups and a puzzle formula for the equivariant cohomology of the cotangent bundle on projective space*

### Advisor

### Abstract

My research lies at the intersection of combinatorics with both representation theory and algebraic geometry. I have two different research projects, both of which use combinatorial tools to tackle different problems in these two other fields. My first project revolves around expanding the use of crystal graphs in branching problems. In the second I compute the structure constants of the regular and equivariant cohomology rings of the cotangent bundle to projective space, using Maulik-Okounkov classes as a basis. First I do so directly in Theorem 4.1.1 and then I put forth a conjectural positive formula which uses a variant of Knutson-Tao puzzles in Conjecture 4.3.2. The proof of the puzzle formula relies on an explicit rational function identity that I have checked through dimension 9.