7th Italian-Japanese summer school in Mathematics

15 Sept 2026, 08:00 → 25 Sept 2026, 18:00 Europe/Rome

Aula Dini (Palazzo del Castelletto)

Goro Akagi (Tohoku University), Chiara Boccato (Università di Pisa), Luigi Caputi (University of Bologna), Lorenzo Cavallina (Tohoku University), Marco Gipo Ghimenti (Università di Pisa), Daisuke Kishimoto (Kyushu University), Kosuke Kita (Hokkaido University), Gerardo Morsella (Università Tor Vergata), Barbara Pacchiarotti (Università Tor Vergata), Aikaterini Papagiannouli (Università di Pisa), Lea Terracini (Università di Torino), Dario Trevisan (Università di Pisa)

The aim of the School is to provide intensive mini-courses in several areas of current mathematical interest, as well as to present methods for concretely applying the theoretical tools. This would be the seventh event in a series of Schools and Workshops organized both in Japan (Hokkaido) and in Italy (Pisa, Turin), starting in 2015. Here you can find the 2023 and the 2024 editions. The very first editions of the school have been hosted in Pisa, at Centro di Ricerca Matematica Ennio De Giorgi (see the 2016, 2018 and 2021 editions), where we plan on having the seventh edition.

At its seventh edition, the summer school is now well established and recognized for giving substantial visibility and support to young students and researchers; the organisers are committed to have contributed talks and/or a poster session. This way, the school aims to strengthen the ties between Italian and Japanese universities and their departments, giving continuity to a success ful format.

The school is intended mainly for master students (laurea magistrale) and Ph.D. students.

Learning goals and objectives: The aim of the lectures is to provide a practical approach to subjects such as algebra, analysis, geometry and probability. This also leads to a deeper understanding of the topics. To this end, we shall focus on various fields and stress their connections. Students who follow the lectures will be able to understand the practical aspects of the abstract the they learned in their undergraduate courses.

Structure of the summer school: We plan on having 8 courses. The courses shall be divided in 4 subjects, with 2 lecturers for subject. One of the lecturer per subject shall be affiliated to an Italian university and one to a Japanese university, so to engage both equally. In total, we plan on having 8 speakers.

Plan of proposed subjects:

• (Functional) analysis and PDE

• Algebra and geometry 

• Cryptography and applications

• Applications to biology/medicine/epidemiology

• Numerical analysis

 

LIST OF CONFIRMED SPEAKERS

Goro Akagi (Tohoku University)

Alice Cortinovis (Università di Pisa) 

Norisuke Ioku (Tohoku University)

Motoko Kato (University of the Ryukyus)

Daisuke Kishimoto (Kyushu University) 

Hideo Kubo (Hokkaido University)

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

Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Fulvio Ricceri (Università di Torino)

Massimiliano Sala (Università di Trento)

 

The deadline to apply for funding is the 30th of April 2026.

Please note that organizers will conduct a selection among the registered people. All details concerning the support will be given later on

More information at the REGISTRATION PAGE.

The number of participants attending the Workshop is limited, therefore the registration will be confirmed after its expiration.

 

Funded by:

• the MUR Excellence Department Project MatMod@TOV awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C23000330006.
• MEXT Grant-in-Aid for the WISE Program (Doctoral Program for World-leading Innovative & Smart Education).
• MEXT Grant - Top Global University Project (Type A), Tohoku University Global Initiative

book-of-abstracts.pdf

Tuesday 15 September

08:20 Registration

1 Introduction to Physics-Informed Neural Networks and Their Applications to Partial Differential Equations (part I)

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 explicit spatial discretization and the careful selection of mesh sizes required in classical finite difference or finite element methods. This feature makes PINNs particularly attractive for problems involving complex geometries or high-dimensional domains.
After presenting the fundamental principles of PINNs, I will explain how the method is constructed by incorporating governing equations and boundary conditions into the loss function of a neural network.
The second part of the lecture focuses on applications, illustrating how PINNs can be used to approximate solutions of the Eikonal equation and how this approach naturally connects to path planning problems. Practical considerations and representative examples will also be discussed to highlight both the strengths and limitations of the method.

Speaker: Hideo Kubo (Hokkaido University)

10:30 Coffee break

2 Groups and Cayley graphs (part I )

In this lecture, I will give a brief introduction to geometric group theory. After reviewing the basic terminology and concepts of groups, graphs and trees, I will explain the construction of Cayley graphs. I will then focus on the case of free groups. I describe how Cayley graphs work when you prove group theoretic results on free groups such as Nielsen-Schreier Theorem.

Speaker: Motoko Kato (University of the Ryukyus)

3 Mathematical aspects of cryptology and blockchain technology (part I)

Digital signatures and hash functions are the building blocks of blockchain technology.
Blockchains have been the most significant innovation in distributed applications, especially concerning finance with cryptocurrencies (Bitcoin...) and smart contracts (Ethereum).
We will explain their essential features and how they are used. We will also provide the relevant mathematical theory, such as elliptic curves and security reductions, assuming the students have elementary notions of group theory and number theory.

Speaker: Massimiliano Sala (Università di Trento)

16:00 Coffee break

4 Dimension reduction and low-rank approximation with randomization (part I)

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 reducing the dimensionality of data while approximately preserving its geometric structure, with applications to the fast solution of least-squares problems. We will also talk about randomized algorithms for low-rank matrix approximation, including the randomized rangefinder, the Nyström method, and ideas related to column subset selection. The course will highlight the interplay between (numerical) linear algebra and probability.

Speaker: Alice Cortinovis (Università di Pisa)

Wednesday 16 September

08:55 Welcome Address (Prof. Malchiodi)

5 Groups and Cayley graphs (part II)

In this lecture, I will give a brief introduction to geometric group theory. After reviewing the basic terminology and concepts of groups, graphs and trees, I will explain the construction of Cayley graphs. I will then focus on the case of free groups. I describe how Cayley graphs work when you prove group theoretic results on free groups such as Nielsen-Schreier Theorem.

Speaker: Motoko Kato (University of the Ryukyus)

10:30 coffee break

6 Introduction to Physics-Informed Neural Networks and Their Applications to Partial Differential Equations (part II)

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 explicit spatial discretization and the careful selection of mesh sizes required in classical finite difference or finite element methods. This feature makes PINNs particularly attractive for problems involving complex geometries or high-dimensional domains.
After presenting the fundamental principles of PINNs, I will explain how the method is constructed by incorporating governing equations and boundary conditions into the loss function of a neural network.
The second part of the lecture focuses on applications, illustrating how PINNs can be used to approximate solutions of the Eikonal equation and how this approach naturally connects to path planning problems. Practical considerations and representative examples will also be discussed to highlight both the strengths and limitations of the method.

Speaker: Hideo Kubo (Hokkaido University)

7 Dimension reduction and low-rank approximation with randomization (part II)

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 reducing the dimensionality of data while approximately preserving its geometric structure, with applications to the fast solution of least-squares problems. We will also talk about randomized algorithms for low-rank matrix approximation, including the randomized rangefinder, the Nyström method, and ideas related to column subset selection. The course will highlight the interplay between (numerical) linear algebra and probability.

Speaker: Alice Cortinovis (Università di Pisa)

16:00 Coffee break

8 Mathematical aspects of cryptology and blockchain technology (part II)

Digital signatures and hash functions are the building blocks of blockchain technology.
Blockchains have been the most significant innovation in distributed applications, especially concerning finance with cryptocurrencies (Bitcoin...) and smart contracts (Ethereum).
We will explain their essential features and how they are used. We will also provide the relevant mathematical theory, such as elliptic curves and security reductions, assuming the students have elementary notions of group theory and number theory.

Speaker: Massimiliano Sala (Università di Trento)

Thursday 17 September

9 Mathematical aspects of cryptology and blockchain technology (part III)

Digital signatures and hash functions are the building blocks of blockchain technology.
Blockchains have been the most significant innovation in distributed applications, especially concerning finance with cryptocurrencies (Bitcoin...) and smart contracts (Ethereum).
We will explain their essential features and how they are used. We will also provide the relevant mathematical theory, such as elliptic curves and security reductions, assuming the students have elementary notions of group theory and number theory.

Speaker: Massimiliano Sala (Università di Trento)

10:30 Coffee break

10 Persistent Homology: A roadmap in Topological Data Analysis (part I)

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 the passage from filtered simplicial complexes to persistent homology modules. Next, we will discuss how these modules decompose to yield persistence barcodes and in what sense these barcodes encode the underlying geometric and topological structure of the initial data. Time permitting, we will conclude with a proof of the stability of the full pipeline, from data to barcodes, establishing its 1-Lipschitz continuity.

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

11 Introduction to Physics-Informed Neural Networks and Their Applications to Partial Differential Equations (part III)

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 explicit spatial discretization and the careful selection of mesh sizes required in classical finite difference or finite element methods. This feature makes PINNs particularly attractive for problems involving complex geometries or high-dimensional domains.
After presenting the fundamental principles of PINNs, I will explain how the method is constructed by incorporating governing equations and boundary conditions into the loss function of a neural network.
The second part of the lecture focuses on applications, illustrating how PINNs can be used to approximate solutions of the Eikonal equation and how this approach naturally connects to path planning problems. Practical considerations and representative examples will also be discussed to highlight both the strengths and limitations of the method.

Speaker: Hideo Kubo (Hokkaido University)

16:00 Coffee break

12 Groups and Cayley graphs (part III)

In this lecture, I will give a brief introduction to geometric group theory. After reviewing the basic terminology and concepts of groups, graphs and trees, I will explain the construction of Cayley graphs. I will then focus on the case of free groups. I describe how Cayley graphs work when you prove group theoretic results on free groups such as Nielsen-Schreier Theorem.

Speaker: Motoko Kato (University of the Ryukyus)

Friday 18 September

13 Persistent Homology: A roadmap in Topological Data Analysis (part II)

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 the passage from filtered simplicial complexes to persistent homology modules. Next, we will discuss how these modules decompose to yield persistence barcodes and in what sense these barcodes encode the underlying geometric and topological structure of the initial data. Time permitting, we will conclude with a proof of the stability of the full pipeline, from data to barcodes, establishing its 1-Lipschitz continuity.

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

10:30 Coffee break

14 Introduction to Physics-Informed Neural Networks and Their Applications to Partial Differential Equations (part IV)

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 explicit spatial discretization and the careful selection of mesh sizes required in classical finite difference or finite element methods. This feature makes PINNs particularly attractive for problems involving complex geometries or high-dimensional domains.
After presenting the fundamental principles of PINNs, I will explain how the method is constructed by incorporating governing equations and boundary conditions into the loss function of a neural network.
The second part of the lecture focuses on applications, illustrating how PINNs can be used to approximate solutions of the Eikonal equation and how this approach naturally connects to path planning problems. Practical considerations and representative examples will also be discussed to highlight both the strengths and limitations of the method.

Speaker: Hideo Kubo (Hokkaido University)

15 Groups and Cayley graphs (part IV)

In this lecture, I will give a brief introduction to geometric group theory. After reviewing the basic terminology and concepts of groups, graphs and trees, I will explain the construction of Cayley graphs. I will then focus on the case of free groups. I describe how Cayley graphs work when you prove group theoretic results on free groups such as Nielsen-Schreier Theorem.

Speaker: Motoko Kato (University of the Ryukyus)

16:00 Coffee break

16 Mathematical aspects of cryptology and blockchain technology (part IV)

Digital signatures and hash functions are the building blocks of blockchain technology.
Blockchains have been the most significant innovation in distributed applications, especially concerning finance with cryptocurrencies (Bitcoin...) and smart contracts (Ethereum).
We will explain their essential features and how they are used. We will also provide the relevant mathematical theory, such as elliptic curves and security reductions, assuming the students have elementary notions of group theory and number theory.

Speaker: Massimiliano Sala (Università di Trento)

Monday 21 September

08:30 Registration

17 Modelling Multimorbidity: From Disease Trajectories to Causal Mechanisms

Multimorbidity—the coexistence of multiple chronic diseases in the same individual—is one of the major challenges for modern public health, considering the ageing of population. Understanding how chronic conditions develop, accumulate and interact over the life course requires moving beyond traditional epidemiological analyses based on single outcomes.
This lecture will illustrate how increasingly complex epidemiological questions naturally lead to increasingly sophisticated statistical models. Starting from population-based cohort studies, we will discuss the challenges posed by multiple correlated outcomes, competing events and disease trajectories over time. These problems motivate the use of methods such as competing-risk models and multistate models, which provide a more realistic description of disease progression.
The second part of the lecture will focus on causal inference. In particular, we will discuss how causal mediation analysis can be used to investigate the mechanisms linking socioeconomic disadvantage to multimorbidity, helping to distinguish association from causation and to identify potential intervention targets.
Rather than providing a technical review of statistical methods, the lecture aims to show how mathematical and statistical modelling can address fundamental questions in epidemiology and contribute to understanding the complex pathways leading to chronic disease.

Speaker: Fulvio Ricceri (Università di Torino)

10:30 Coffee break

18 Optimal rate of convergence to asymptotic profiles for fast diffusion in bounded domains

Speaker: Goro Akagi (Tohoku University)

Akagi_abstract.pdf

Poster session

16:00 Coffee break

Poster session

Tuesday 22 September

19 Borsuk-Ulam theorem (part I)

The Borsuk--Ulam theorem is one of the most widely used and influential results in algebraic topology, with applications in mathematics, physics, computer science, robotics, economics, and many other fields. In dimension two, it states that if one continuously flattens a beach ball without tearing or puncturing it, then there exists a point on the surface that lies directly above its antipodal point.
In these lectures, we introduce two of the most fundamental invariants in algebraic topology, homology and homotopy groups. Using these invariants, we establish a generalization of the Borsuk--Ulam theorem. We then apply this generalized theorem to prove the topological Tverberg theorem, a topological extension of Tverberg's theorem on the combinatorics of points in Euclidean space.

Speaker: Daisuke Kishimoto (Kyushu University)

10:30 Coffee break

20 An Introduction to Isolated Singularities of Elliptic PDEs (part I)

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 second part, after a brief overview of distribution theory, we present an alternative approach to isolated singularities due to Brezis and Lions, which is based on the support of distributions.

Speaker: Norisuke Ioku (Tohoku University)

21 Dimension reduction and low-rank approximation with randomization (part III)

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 reducing the dimensionality of data while approximately preserving its geometric structure, with applications to the fast solution of least-squares problems. We will also talk about randomized algorithms for low-rank matrix approximation, including the randomized rangefinder, the Nyström method, and ideas related to column subset selection. The course will highlight the interplay between (numerical) linear algebra and probability.

Speaker: Alice Cortinovis (Università di Pisa)

16:00 coffee break

22 Fractional diffusion equations: a numerical linear algebra perspective (part I)

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 non-locality of fractional operators improves the physical modeling of the underlying processes but also leads to important computational challenges. In particular, when FDEs are discretized, the resulting coefficient matrices typically lose the sparsity structure that characterizes classical discretizations of partial differential equations, making the associated linear systems significantly more demanding from a computational perspective.
After introducing the main modeling ideas behind FDEs, we focus on the structural and spectral properties of the matrices arising from standard discretizations and on how these can be exploited to design efficient iterative solvers for the resulting linear systems. In particular, we will highlight strategies based on preconditioning techniques and multigrid methods, showing how suitable matrix analysis can lead to fast and scalable numerical algorithms.

Speaker: Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Wednesday 23 September

23 An Introduction to Isolated Singularities of Elliptic PDEs (part II)

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 second part, after a brief overview of distribution theory, we present an alternative approach to isolated singularities due to Brezis and Lions, which is based on the support of distributions.

Speaker: Norisuke Ioku (Tohoku University)

10:30 Coffee break

24 Borsuk-Ulam theorem (part II)

The Borsuk--Ulam theorem is one of the most widely used and influential results in algebraic topology, with applications in mathematics, physics, computer science, robotics, economics, and many other fields. In dimension two, it states that if one continuously flattens a beach ball without tearing or puncturing it, then there exists a point on the surface that lies directly above its antipodal point.
In these lectures, we introduce two of the most fundamental invariants in algebraic topology, homology and homotopy groups. Using these invariants, we establish a generalization of the Borsuk--Ulam theorem. We then apply this generalized theorem to prove the topological Tverberg theorem, a topological extension of Tverberg's theorem on the combinatorics of points in Euclidean space.

Speaker: Daisuke Kishimoto (Kyushu University)

25 Fractional diffusion equations: a numerical linear algebra perspective (part II)

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 non-locality of fractional operators improves the physical modeling of the underlying processes but also leads to important computational challenges. In particular, when FDEs are discretized, the resulting coefficient matrices typically lose the sparsity structure that characterizes classical discretizations of partial differential equations, making the associated linear systems significantly more demanding from a computational perspective.
After introducing the main modeling ideas behind FDEs, we focus on the structural and spectral properties of the matrices arising from standard discretizations and on how these can be exploited to design efficient iterative solvers for the resulting linear systems. In particular, we will highlight strategies based on preconditioning techniques and multigrid methods, showing how suitable matrix analysis can lead to fast and scalable numerical algorithms.

Speaker: Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

16:00 Coffee break

26 Dimension reduction and low-rank approximation with randomization (part IV)

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 reducing the dimensionality of data while approximately preserving its geometric structure, with applications to the fast solution of least-squares problems. We will also talk about randomized algorithms for low-rank matrix approximation, including the randomized rangefinder, the Nyström method, and ideas related to column subset selection. The course will highlight the interplay between (numerical) linear algebra and probability.

Speaker: Alice Cortinovis (Università di Pisa)

Thursday 24 September

27 Fractional diffusion equations: a numerical linear algebra perspective (part III)

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 non-locality of fractional operators improves the physical modeling of the underlying processes but also leads to important computational challenges. In particular, when FDEs are discretized, the resulting coefficient matrices typically lose the sparsity structure that characterizes classical discretizations of partial differential equations, making the associated linear systems significantly more demanding from a computational perspective.
After introducing the main modeling ideas behind FDEs, we focus on the structural and spectral properties of the matrices arising from standard discretizations and on how these can be exploited to design efficient iterative solvers for the resulting linear systems. In particular, we will highlight strategies based on preconditioning techniques and multigrid methods, showing how suitable matrix analysis can lead to fast and scalable numerical algorithms.

Speaker: Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

10:30 Coffee break

28 Persistent Homology: A roadmap in Topological Data Analysis (part III)

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 the passage from filtered simplicial complexes to persistent homology modules. Next, we will discuss how these modules decompose to yield persistence barcodes and in what sense these barcodes encode the underlying geometric and topological structure of the initial data. Time permitting, we will conclude with a proof of the stability of the full pipeline, from data to barcodes, establishing its 1-Lipschitz continuity.

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

29 Borsuk-Ulam theorem (part III)

The Borsuk--Ulam theorem is one of the most widely used and influential results in algebraic topology, with applications in mathematics, physics, computer science, robotics, economics, and many other fields. In dimension two, it states that if one continuously flattens a beach ball without tearing or puncturing it, then there exists a point on the surface that lies directly above its antipodal point.
In these lectures, we introduce two of the most fundamental invariants in algebraic topology, homology and homotopy groups. Using these invariants, we establish a generalization of the Borsuk--Ulam theorem. We then apply this generalized theorem to prove the topological Tverberg theorem, a topological extension of Tverberg's theorem on the combinatorics of points in Euclidean space.

Speaker: Daisuke Kishimoto (Kyushu University)

16:00 Coffee break

30 An Introduction to Isolated Singularities of Elliptic PDEs (part III)

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 second part, after a brief overview of distribution theory, we present an alternative approach to isolated singularities due to Brezis and Lions, which is based on the support of distributions.

Speaker: Norisuke Ioku (Tohoku University)

Friday 25 September

31 Persistent Homology: A roadmap in Topological Data Analysis (part IV)

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 the passage from filtered simplicial complexes to persistent homology modules. Next, we will discuss how these modules decompose to yield persistence barcodes and in what sense these barcodes encode the underlying geometric and topological structure of the initial data. Time permitting, we will conclude with a proof of the stability of the full pipeline, from data to barcodes, establishing its 1-Lipschitz continuity.

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

10:30 Coffee break

32 Fractional diffusion equations: a numerical linear algebra perspective (part IV)

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 non-locality of fractional operators improves the physical modeling of the underlying processes but also leads to important computational challenges. In particular, when FDEs are discretized, the resulting coefficient matrices typically lose the sparsity structure that characterizes classical discretizations of partial differential equations, making the associated linear systems significantly more demanding from a computational perspective.
After introducing the main modeling ideas behind FDEs, we focus on the structural and spectral properties of the matrices arising from standard discretizations and on how these can be exploited to design efficient iterative solvers for the resulting linear systems. In particular, we will highlight strategies based on preconditioning techniques and multigrid methods, showing how suitable matrix analysis can lead to fast and scalable numerical algorithms.

Speaker: Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

33 An Introduction to Isolated Singularities of Elliptic PDEs (part IV)

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 second part, after a brief overview of distribution theory, we present an alternative approach to isolated singularities due to Brezis and Lions, which is based on the support of distributions.

Speaker: Norisuke Ioku (Tohoku University)

16:00 Coffee break

34 Borsuk-Ulam theorem (part IV)

The Borsuk--Ulam theorem is one of the most widely used and influential results in algebraic topology, with applications in mathematics, physics, computer science, robotics, economics, and many other fields. In dimension two, it states that if one continuously flattens a beach ball without tearing or puncturing it, then there exists a point on the surface that lies directly above its antipodal point.
In these lectures, we introduce two of the most fundamental invariants in algebraic topology, homology and homotopy groups. Using these invariants, we establish a generalization of the Borsuk--Ulam theorem. We then apply this generalized theorem to prove the topological Tverberg theorem, a topological extension of Tverberg's theorem on the combinatorics of points in Euclidean space.

Speaker: Daisuke Kishimoto (Kyushu University)