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

• Hideo Kubo (Hokkaido University)

Room: Aula Dini 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.

Groups and Cayley graphs (part I )

• Motoko Kato (University of the Ryukyus)

Room: Aula Dini 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.

Mathematical aspects of cryptology and blockchain technology (part I)

• Massimiliano Sala (Università di Trento)

Room: Aula Dini 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.

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

• Alice Cortinovis (Università di Pisa)

Room: Aula Dini 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.

Groups and Cayley graphs (part II)

• Motoko Kato (University of the Ryukyus)

Room: Aula Dini 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.

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

• Hideo Kubo (Hokkaido University)

Room: Aula Dini 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.

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

• Alice Cortinovis (Università di Pisa)

Room: Aula Dini 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.

Mathematical aspects of cryptology and blockchain technology (part II)

• Massimiliano Sala (Università di Trento)

Room: Aula Dini 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.

Mathematical aspects of cryptology and blockchain technology (part III)

• Massimiliano Sala (Università di Trento)

Room: Aula Dini 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.

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

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

Room: Aula Dini 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.

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

• Hideo Kubo (Hokkaido University)

Room: Aula Dini 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.

Groups and Cayley graphs (part III)

• Motoko Kato (University of the Ryukyus)

Room: Aula Dini 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.

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

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

Room: Aula Dini 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.

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

• Hideo Kubo (Hokkaido University)

Room: Aula Dini 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.

Groups and Cayley graphs (part IV)

• Motoko Kato (University of the Ryukyus)

Room: Aula Dini 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.

Mathematical aspects of cryptology and blockchain technology (part IV)

• Massimiliano Sala (Università di Trento)

Room: Aula Dini 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.

Modelling Multimorbidity: From Disease Trajectories to Causal Mechanisms

• Fulvio Ricceri (Università di Torino)

Room: Aula Dini 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.

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

• Goro Akagi (Tohoku University)

Room: Aula Dini

Borsuk-Ulam theorem (part I)

• Daisuke Kishimoto (Kyushu University)

Room: Aula Dini 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.

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

• Norisuke Ioku (Tohoku University)

Room: Aula Dini 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.

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

• Alice Cortinovis (Università di Pisa)

Room: Aula Dini 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.

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

• Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Room: Aula Dini 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.

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

• Norisuke Ioku (Tohoku University)

Room: Aula Dini 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.

Borsuk-Ulam theorem (part II)

• Daisuke Kishimoto (Kyushu University)

Room: Aula Dini 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.

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

• Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Room: Aula Dini 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.

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

• Alice Cortinovis (Università di Pisa)

Room: Aula Dini 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.

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

• Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Room: Aula Dini 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.

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

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

Room: Aula Dini 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.

Borsuk-Ulam theorem (part III)

• Daisuke Kishimoto (Kyushu University)

Room: Aula Dini 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.

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

• Norisuke Ioku (Tohoku University)

Room: Aula Dini 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.

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

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

Room: Aula Dini 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.

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

• Mariarosa Mazza (Università degli studi di Roma Tor Vergata)

Room: Aula Dini 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.

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

• Norisuke Ioku (Tohoku University)

Room: Aula Dini 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.

Borsuk-Ulam theorem (part IV)

• Daisuke Kishimoto (Kyushu University)

Room: Aula Dini 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.