## Departmental PhD Thesis Exam – Travis Squires

Wednesday, September 7, 2011, 10:10 a.m., in BA 6183

PhD Candidate:  Travis Squires

PhD Advisor:  Sergey Arkhipov

Thesis Title:  Lie 2-Algebras as Homotopy Algebras Over a Quadratic Operad
(http://www.pispace.org/thesis/Thesis-Squires.pdf)

We begin our discussion by introducing some notions from operad
theory; in particular we discuss homotopy algebras over a quadratic
operad. We then proceed to describe Lie 2-algebras as homotopy
algebras over a given quadratic operad using a theorem of Ginzburg
and Kapranov. Next we briefly discuss the structure of a braided
monoidal category. Following this, motivated by our discussion of
braided monoidal categories, a new structure is introduced which
we call a commutative 2-algebra. As with the lie 2-algebra
case we show how a commutative 2-algebra can be seen as a
homotopy algebra over a particular quadratic operad. Finally
some technical results used in previous theorems are mentioned.
In discussing these technical results we apply some ideas about
distributive laws and Koszul operads.