Abstract: Some developments in Algebraic Geometry were influenced by the effort to understand and solve questions of relevance to Number Theory. I will give Weil conjectures as such an example. More recently, Manjul Bhargava came up with a nice example regarding Jacobians of curves. Right now some on-going research work centers round moduli space of vector bundles over curves. I wish to indicate some of these if time permits.

Abstract: Ferdinand Georg Frobenius is credited for pioneering work on Representation Theory. This work seems to have been ignited by a letter from R. Dedekind to Frobenius reporting on his discoveries on to the curious factorization of a determinant that can be associated with each finite group.
We will present an exposition of this problem and its solution which brought to light some key ideas in representation theory.