In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.

Since the symmetric group is a finite group it’s representation theory is not too complex, however in this special case we can realise these representations in a particular nice combinatorial way using young tableaux and young symmetrizers. I will introduce these ideas and use them to describe the representation theory of Sn over the complex numbers.

in this talk I will try to introduce some key ideas and concepts about random walks on discrete spaces, with special interest on random walks on Cayley graphs.

This will be a little potpourri containing some of the recent developments on the model theory of F_p((t)) and of algebraic extensions of Q_p.