Abstract. The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A n−1 on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph M , called the Brauer algebra of type M , and study its structure in the cases where M is a Coxeter graph of simply laced spherical type (so its connected components are of type A n−1 , Dn, E 6 , E 7 , E 8 ). We determine the representations and find the dimension. The algebra is semisimple and contains the group algebra of the Coxeter group of type M as a subalgebra. It is a ring homomorphic image of the Birman-MurakamiWenzl algebra of type M ; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.

Abstract. We introduce tropically unirational varieties, which are subvarieties of tori that admit dominant rational maps whose tropicalisation is surjective. The central (and unresolved) question is whether all unirational varieties are tropically unirational. We present several techniques for proving tropical unirationality, along with various examples.

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