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.

In this talk, we introduce a two-dimensional semiflexible membrane model whose Hamiltonian is given by $H[\phi]=\sum_x |\nabla \phi_x|^2 +N^\lambda |\Delta \phi_x|^2$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its infinite-volume behavior and identify distinct regimes depending on the parameter $\lambda$. If $\lambda<0$, the covariance behaves similarly to that of the DGFF, while for $\lambda>2$, it resembles a rescaled MM. In the intermediate regime $0\le \lambda\le 2$, we observe a qualitatively different behavior: a nontrivial dependence on the distance between points. We further determine the leading order constant of the covariance. These results provide a unified description of the crossover from gradient-dominated to curvature-dominated behavior in this class of models.

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.

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 first three talks, I will explain the backgrounds on Selmer groups and flat cohomology.

Diffusion is a macroscopic phenomenon arising from the random movement of particles at the microscopic level. Fick’s law predicts uniform spreading of particles over time, while fractionation is often observed in heterogeneous environments, as in the Soret effect and Darken’s experiment. In this talk, we show that such heterogeneous diffusion can be described by a two-coefficient diffusion equation derived from particle dynamics. In particular, for persistent random walks, fractionation occurs only when both heterogeneity and anisotropy are present. We formally derive the limiting diffusion equation and present a methodology to rigorously establish convergence from a persistent discrete kinetic equation to the macroscopic diffusion equation.

his 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 first two or three talks, I will explain the backgrounds on Selmer groups and flat cohomology.