A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To attend this seminar, you must register in advance, by clicking HERE.

Abstract: Strongly F-regular singularities are one of the fundamental classes of singularities defined by the properties of Frobenius endomorphism. This class of mild singularities can be detected using F-signature, an invariant of a local ring with many good properties. Through this connection we obtain a powerful tool for studying strongly F-regular singularities, for example, several results on "mildness" of F-regular can be quantified using F-signature.  

Another fundamental class of singularities in positive characteristic are F-rational singularities. While generally more severe, this class of singularities is in many aspects analogous to strongly F-regular singularities and can be even understood by "moving" the definition of F-regularity to take place in the dualizing module. Naturally, there has been interest in adapting the definition of F-signature to work with F-rational singularities. 

While there is no complete solution yet, I am convinced that such a theory should exist. As my evidence, I will present results of a joint work with Kevin Tucker, and prior works of Hochster and Yao, and Sannai.

My talk will be self-contained. I will discuss all necessary background, such as definitions, properties, and relations between these notions, in the first half and then proceed to more technical results in the second half. 

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To attend this seminar, you must register in advance, by clicking HERE.

Abstract: The choice of a term order allows us to associate to any ideal I a monomial ideal, called the initial ideal of I. The initial ideal of I depends not only on the choice of a term order, but also on the system of coordinates. Nevertheless, many properties of I can be inferred from those of its initial ideal(s). For a given term order and in a generic coordinate system, however, the initial ideal of I is always the same and it is then called the generic initial ideal of I. In my talk, I will introduce a family of ideals whose generic initial ideal is independent of the choice of both the term order and of the system of coordinates. These are exactly the multigraded homogeneous ideals which have a radical generic initial ideal. Multigraded ideals which have a radical generic initial ideal show interesting rigidity properties, which e.g. allow us to deduce information on their universal Groebner bases. In the talk, I will present examples of ideals which belong to this class and of what we can deduce about them using this machinery. The original work in the talk is joint with Aldo Conca and Emanuela De Negri.

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no prerequisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To attend this seminar, you must register in advance, by clicking HERE.

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

To participate in this seminar, please register here: https://www.msri.org/seminars/25657