The Hardy's inequality is a classical result in analysis, providing a relationship between the weighted norms of functions and their gradient norm, with several applications in Sobolev spaces, partial differential equations, and mathematical physics. In recent decades, interest has grown in fractional versions of this inequality, where the classical Laplacian is replaced by a non-local operator, such as the fractional Laplacian. In this talk, we introduce a weighted version of the classical Hardy inequality on bounded domains and extend it to the case of the fractional Laplacian. We also present some recent results related to this topic.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. We now explain the statement of the equivariant BSD conjecture.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. We now explain the statement of the equivariant BSD conjecture.

In four-dimensional topology, the smooth and topological categories have significant differences, called exotica. For instance, there are many smooth manifolds that are homeomorphic but not diffeomorphic. Moreover, there are many smoothly embedded surfaces in a 4-manifold that are isotopic topologically, but not smoothly. In this talk, we explore how exotic phenomena can be constructed and detected via knots.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the next two--three talks, I will explain the backgrounds on K_1 and relative K_0 of group rings for finite groups over local/global fields of characteristic 0 and their orders.