# Sasaki-Einstein manifolds and AdS/CFT correspondence

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Sasakian manifolds are odd dimensional analog of Kahler manifolds,

and it is an interesting question to determine when

a Sasakian manifold admits an Einstein metric. Five dimensional

Sasaki-Einstein (SE) manifolds play an important role in AdS/CFT

correspondence, which relates a string theory and a quantum field theory. I

will discuss the existence of SE manifolds and its geometric properties

which will be of great interest to AdS/CFT correspondence.

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# Solving the Twisted Rabbit Problem using trees

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The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?

After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.