Abstract: This series of lectures provides an introduction to the basic calculus of pseudodifferential operators defined on Euclidean spaces. We will start by reviewing the space of Schwartz functions, the convolution, the Fourier transform, and their basic properties. Then we prove two important results for studying pseudodifferential operators: the Fourier inversion formula and the Plancherel theorem. We will proceed by finding an asymptotic expansion for the symbol of formal adjoint and composition of pseudodifferential operators. We will end the lectures by introducing a notion of ellipticity and constructing parametrices for elliptic pseudodifferential operators.

Abstract: A higher order automorphic form is a generalization of the notion of a classical automorphic form. I will discuss the definition and I will review some recent results.

Abstract: The idea of (left) regular representation has its origins in group theory and a well known theorem of Arthur Cayley from 19th century which is part of any introduction to group theory. The goal of this talk is to highlight a few of its ramifications, extensions, and applications in different areas of mathematics, including analysis and algebra. After a quick discussion of the general idea of representation, I shall try to point out - by way of examples drawn from different fields - how universal, as well as simple and useful, the idea of regular representation is. I shall make every effort to make this talk as self contained as possible for an undergraduate talk.