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**1 - 2**of**2**### RESEARCH STATEMENT

"... My reseach interests lie in algebraic topology and homotopy theory, and my doctoral work is focused on the realization of Π-algebras. More precisely, my goal is to apply the obstruction theory of Blanc-Dwyer-Goerss [9] to the case of truncated Π-algebras. Since the obstructions to existence and uniq ..."

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My reseach interests lie in algebraic topology and homotopy theory, and my doctoral work is focused on the realization of Π-algebras. More precisely, my goal is to apply the obstruction theory of Blanc-Dwyer-Goerss [9] to the case of truncated Π-algebras. Since the obstructions to existence and uniqueness live in certain Quillen cohomology groups of Π-algebras, my main focus has been the algebraic problem of understanding these groups better and trying to compute them. 1. Background 1.1. Π-algebras and the realization problem. One of the main ideas of algebraic topology is to describe spaces by associating algebraic invariants to them. The homotopy groups πi(X) of a space X are an important example and have been studied extensively. They are a collection of groups (abelian if i> 1), but they also carry additional structure: a π1-action on higher groups, Whitehead products πi × π j → πi+ j−1, and primary homotopy operations corresponding to precomposition by maps between spheres, i.e. πn(S j) × π j → πn. This algebraic structure is known as a Π-algebra and was formally introduced in [20]. The prototypical Π-algebra is the homotopy Π-algebra π∗(X) of a

### THREE CROSSED MODULES

, 812

"... We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché). ..."

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We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché).