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: An o-minimal structure is a structure with a dense linear order in which there are as few definable subsets of the line as possible (namely just finite unions of points and intervals). This condition ensures rather nice topological properties of the definable sets in an o-minimal structure, the archetypical example here being the semialgebraic sets.

In order to understand the definable sets in an o-minimal field R, it is often helpful to understand the convex valuations on R in terms of the usually simpler residue field and value group. We shall discuss some related results, focusing mainly on the residue field.

Abstract: What is the maximum number of pieces of a cheese (or of a pizza) you can cut with $n$ cuts? A study of these kinds of problems goes back to the work of Jacob Steiner in the early 19th century. Over the years mathematicians have studied various aspects and generalizations of this problem. Questions of this type are collectively known as "topological dissection problems".

The aim of my talk is to introduce a unified way to solve a class of dissection problems. This new approach has helped in solving the topological dissection problems in a vast generality. The solution of this seemingly simple problem involves the use of a fundamental dimensionless invariant and some measure theory on posets.

I will try to explain this new approach with the help of simple diagrams and intuitive ideas avoiding technicalities. This is a part of an ongoing research project called "arrangements of submanifolds".