7th Italian-Japanese summer school in Mathematics

Contribution List

5. An Introduction to Isolated Singularities of Elliptic PDEs

Norisuke Ioku (Tohoku University)

This lecture begins with the observation that isolated singularities of harmonic functions naturally arise from the law of universal gravitation. In the first part, we review the classical theory, including the fundamental solution of the Laplace equation, the mean value property, and Harnack’s inequality, and apply these tools to classify isolated singularities of harmonic functions. In the...

8. Dimension reduction and low-rank approximation with randomization

Alice Cortinovis (Università di Pisa)

Randomized techniques have recently emerged as powerful tools for designing fast and scalable algorithms for performing linear algebra computations on very large matrices. This mini-course introduces some of the fundamental ideas of the field of randomized numerical linear algebra, focusing on dimension reduction and low-rank approximation. We will discuss randomized subspace embeddings for...

4. Fractional diffusion equations: a numerical linear algebra perspective

Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

This mini-course focuses on Fractional Diffusion Equations (FDEs), which extend classical diffusion equations by replacing standard derivatives with fractional ones. These models naturally capture non-local interactions, allowing for a more accurate description of anomalous diffusion phenomena arising in several applications, such as plasma physics and network dynamics.
The intrinsic...

1. Introduction to Physics-Informed Neural Networks and Their Applications to Partial Differential Equations

Hideo Kubo (Hokkaido University)

In this lecture, I introduce the basic ideas behind Physics-Informed Neural Networks (PINNs), a neural-network-based framework for computing approximate solutions to partial differential equations (PDEs). PINNs can be interpreted as a residual minimization framework based on a neural-network ansatz.
A key advantage of PINNs is their mesh-free formulation, which eliminates the need for...

3. Persistent Homology: A roadmap in Topological Data Analysis

Claudia Landi (Università degli studi di Modena e Reggio Emilia)

Persistent Homology provides a powerful framework for extracting robust topological features from data. In this course, we will introduce the foundational concepts underlying the theory, beginning with simplicial complexes constructed from point clouds, such as Čech and Vietoris–Rips complexes, and their associated filtrations. We will then review simplicial homology and show how it enables...

6. TBA

Massimiliano Sala (Università di Trento)

7. TBA

Fulvio Ricceri (Università di Torino)

2. Topological Tverberg theorem

Daisuke Kishimoto (Kyushu University)

Tverberg’s theorem states that any configuration of (d+1)(r-1)+1 points in d-dimensional Euclidean space admits a partition into r subsets whose convex hulls have a point in common. The topological Tverberg’s theorem is a topological generalization of Tverberg’s theorem, in which convex hulls are replaced by “flabby hulls”. In this lecture, I will introduce the basic tools in algebraic...