Monday, July 7th, 2025
Workshop Mini-course: Luis Silvestre
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Analysis of some kinetic equations Part 1
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Title: Analysis of some kinetic equations Part 1
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Workshop Mini-course: Jared Speck
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Shocks in multi-dimensional compressible fluids (Part 1)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Title: Shocks in multi-dimensional compressible fluids (Part 1)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Workshop Mini-course: Sung-Jin Oh
Time: 2:00 PM - 3:30 PM
Location: SCGP 102
Title: Long-term behavior of linear and nonlinear waves (Part 1)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Title: Long-term behavior of linear and nonlinear waves (Part 1)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Tuesday, July 8th, 2025
Workshop Mini-course: Sung-Jin Oh
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Long-term behavior of linear and nonlinear waves (Part 2)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Title: Long-term behavior of linear and nonlinear waves (Part 2)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Workshop Mini-course: Jared Speck
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Shocks in multi-dimensional compressible fluids (Part 2)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Title: Shocks in multi-dimensional compressible fluids (Part 2)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Workshop Mini-course: Luis Silvestre
Time: 2:00 PM - 3:30 PM
Location: SCGP 102
Title: Analysis of some kinetic equations Part 2
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Title: Analysis of some kinetic equations Part 2
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Wednesday, July 9th, 2025
Workshop Mini-course: Luis Silvestre
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Analysis of some kinetic equations Part 3
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Title: Analysis of some kinetic equations Part 3
Speaker: Luis Silvestre
Abstract: Kinetic equations govern the evolution of particle densities in gases and plasma. They focus on an intermediate scale between the macroscopic scale of fluid equations, and the microscopic description of many particles moving around and bouncing against each other. In these lectures we will present some mathematical problems in the study of these equations, their well posedness and their regularity estimates. We review some recent results for the Boltzmann and Landau equations. In particular, we will discuss the role of the Fisher information to show that there is no finite time blow-up in the space homogeneous setting.
Workshop Mini-course: Wilhelm Schlag
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 1)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 1)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.
Thursday, July 10th, 2025
Workshop Mini-course: Jared Speck
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Shocks in multi-dimensional compressible fluids (Part 3)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Title: Shocks in multi-dimensional compressible fluids (Part 3)
Speaker: Jared Speck
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Workshop Mini-course: Wilhelm Schlag
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 2)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 2)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.
Friday, July 11th, 2025
Workshop Mini-course: Sung-Jin Oh
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Long-term behavior of linear and nonlinear waves (Part 3)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Title: Long-term behavior of linear and nonlinear waves (Part 3)
Speaker: Sung-Jin Oh
Abstract: Recently, there has been tremendous progress in understanding the global stability of non-trivial stationary solutions for highly nonlinear wave equations and the precise long-time asymptotics of perturbed solutions. Motivated by such developments, we will discuss techniques for analyzing the long-term behavior (upper bound and asymptotics) of linear and nonlinear waves on curved (asymptotically flat) backgrounds, with an emphasis on physical space approaches.
Workshop Mini-course: Wilhelm Schlag
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 3)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.
Title: Long-term dynamics of Hamiltonian PDEs, scattering and asymptotic completeness (Part 3)
Speaker: Wilhelm Schlag
Abstract: This course will begin with a discussion of trapping, nontrapping, and scattering in Newtonian mechanics. Outgoing trajectories, and positive commutators will be presented, together with a version of Mourre’s theory in this context. We will then touch upon KAM theory and move on to dynamics in infinite dimensions described by dispersive PDEs. We will develop some basic ideas in the simple setting of the cubic nonlinear Klein-Gordon equation in three dimensions. The focusing equation exhibits surprisingly subtle dynamics the analysis of which hinges on results on elliptic PDEs and the calculus of variations. By means of this elementary equation we will study the method of concentration-compactness and how it leads to a proof of scattering for large data for this PDE. Other equations such as wave maps will appear as well.