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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.

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