Lecture 1 : Energy super critical singularity formation.The description of singularity formation for non linear PDE’s is a classical problem with deep physical roots. Immense progress have been done in the last thirty on the understanding of “bubbling” phenomenon for focusing problems. But recently, new mecanisms have been discovered for “defocusing” problems with a deep connection to fluid mechanics. This series of 4 lectures is intended as a graduate class and will propose an introduction to the key results and open problems in the field, as well as a self contained description of the essential steps of the proofs.

Machine learning and Data science algorithms involve in their last stage the need for optimization of complex random functions in very high dimensions. Simple algorithms like Stochastic Gradient Descent (with small batches) are used very effectively. I will concentrate on trying to understand why these simple tools can still work in these complex and very over-parametrized regimes. I will first introduce the whole framework for non-experts, from the structure of the typical tasks to the natural structures of neural nets used in standard contexts. l will then cover briefly the classical and usual context of SGD in finite dimensions. I will then survey recent work with Reza Gheissari (Northwestern), Aukosh Jagannath (Waterloo) giving a general view for the existence of projected “effective dynamics” for “summary statistics” in much smaller dimensions, which still rule the performance of very high dimensional systems, as well . These effective dynamics (as their so-called “critical regime”) define a dynamical system in finite dimensions which may be quite complex, and rules the performance of the learning algorithm.The next step will be to understand how the system finds these “summary statistics”. This is done in the next work with the same authors and with Jiaoyang Huang (Wharton, U-Penn). This is based on a dynamical spectral transition of Random Matrix Theory: along the trajectory of the optimization path, the Gram matix or the Hessian matrix develop outliers which carry these effective dynamics. I will naturally first come back to the Random Matrix Tools needed here (the behavior of the edge of the spectrum and the BBP transition) in a much broader context. And then illustrate the use of this point of view on a few central examples of ML: multilayer neural nets for classification (of Gaussian mixtures), and the XOR examples, for instance.

Machine learning and Data science algorithms involve in their last stage the need for optimization of complex random functions in very high dimensions. Simple algorithms like Stochastic Gradient Descent (with small batches) are used very effectively. I will concentrate on trying to understand why these simple tools can still work in these complex and very over-parametrized regimes. I will first introduce the whole framework for non-experts, from the structure of the typical tasks to the natural structures of neural nets used in standard contexts. l will then cover briefly the classical and usual context of SGD in finite dimensions. I will then survey recent work with Reza Gheissari (Northwestern), Aukosh Jagannath (Waterloo) giving a general view for the existence of projected “effective dynamics” for “summary statistics” in much smaller dimensions, which still rule the performance of very high dimensional systems, as well . These effective dynamics (as their so-called “critical regime”) define a dynamical system in finite dimensions which may be quite complex, and rules the performance of the learning algorithm.The next step will be to understand how the system finds these “summary statistics”. This is done in the next work with the same authors and with Jiaoyang Huang (Wharton, U-Penn). This is based on a dynamical spectral transition of Random Matrix Theory: along the trajectory of the optimization path, the Gram matix or the Hessian matrix develop outliers which carry these effective dynamics. I will naturally first come back to the Random Matrix Tools needed here (the behavior of the edge of the spectrum and the BBP transition) in a much broader context. And then illustrate the use of this point of view on a few central examples of ML: multilayer neural nets for classification (of Gaussian mixtures), and the XOR examples, for instance.

Machine learning and Data science algorithms involve in their last stage the need for optimization of complex random functions in very high dimensions. Simple algorithms like Stochastic Gradient Descent (with small batches) are used very effectively. I will concentrate on trying to understand why these simple tools can still work in these complex and very over-parametrized regimes. I will first introduce the whole framework for non-experts, from the structure of the typical tasks to the natural structures of neural nets used in standard contexts. l will then cover briefly the classical and usual context of SGD in finite dimensions. I will then survey recent work with Reza Gheissari (Northwestern), Aukosh Jagannath (Waterloo) giving a general view for the existence of projected “effective dynamics” for “summary statistics” in much smaller dimensions, which still rule the performance of very high dimensional systems, as well . These effective dynamics (as their so-called “critical regime”) define a dynamical system in finite dimensions which may be quite complex, and rules the performance of the learning algorithm.The next step will be to understand how the system finds these “summary statistics”. This is done in the next work with the same authors and with Jiaoyang Huang (Wharton, U-Penn). This is based on a dynamical spectral transition of Random Matrix Theory: along the trajectory of the optimization path, the Gram matix or the Hessian matrix develop outliers which carry these effective dynamics. I will naturally first come back to the Random Matrix Tools needed here (the behavior of the edge of the spectrum and the BBP transition) in a much broader context. And then illustrate the use of this point of view on a few central examples of ML: multilayer neural nets for classification (of Gaussian mixtures), and the XOR examples, for instance.

Machine learning and Data science algorithms involve in their last stage the need for optimization of complex random functions in very high dimensions. Simple algorithms like Stochastic Gradient Descent (with small batches) are used very effectively. I will concentrate on trying to understand why these simple tools can still work in these complex and very over-parametrized regimes. I will first introduce the whole framework for non-experts, from the structure of the typical tasks to the natural structures of neural nets used in standard contexts. l will then cover briefly the classical and usual context of SGD in finite dimensions. I will then survey recent work with Reza Gheissari (Northwestern), Aukosh Jagannath (Waterloo) giving a general view for the existence of projected “effective dynamics” for “summary statistics” in much smaller dimensions, which still rule the performance of very high dimensional systems, as well . These effective dynamics (as their so-called “critical regime”) define a dynamical system in finite dimensions which may be quite complex, and rules the performance of the learning algorithm.The next step will be to understand how the system finds these “summary statistics”. This is done in the next work with the same authors and with Jiaoyang Huang (Wharton, U-Penn). This is based on a dynamical spectral transition of Random Matrix Theory: along the trajectory of the optimization path, the Gram matix or the Hessian matrix develop outliers which carry these effective dynamics. I will naturally first come back to the Random Matrix Tools needed here (the behavior of the edge of the spectrum and the BBP transition) in a much broader context. And then illustrate the use of this point of view on a few central examples of ML: multilayer neural nets for classification (of Gaussian mixtures), and the XOR examples, for instance.

The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

The Zilber-Pink conjecture is a diophantine finiteness conjecture. It unifies and gives a far-reaching generalization of the classical Mordell-Lang and Andre-Oort conjectures, and is wide open in general.Point-counting results for definable sets in o-minimal structures provide a strategy for proving suitable cases which has had some success, in particular in its use in proving the Andre-Oort conjecture.The course will describe the Zilber-Pink conjecture and the point-counting approach to proving cases of it, eventually concentrating on the case of a curve in a power of the modular curve.We will describe the model-theoretic contexts of the conjectures and techniques, and the essential arithmetic ingredients.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.

Atttention : La première Leçon aura lieu à l’Ecole polytechnique, Amphithéâtre Becquerel, le 30 janvier à 15hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://www.fondation-hadamard.fr/en/events/advanced-courses/Abstract:Given a vector field in the euclidean space, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth. The theorem looses its validity as soon as the vector field is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields.The course presents a modern view, new results and open problems in the context of flows of irregular vector fields.We develop, in this framework, recent ideas and techniques such as quantitative regularity estimates on the flow of Sobolev vector fields, nonuniqueness of solutions via convex integration, similarity constructions, mixing, enhanced and anomalous dissipation.Such ideas have been proved useful to study nonlinear PDEs as well and we apply these results and techniques in the context of the mathematical understanding of phenomena in fluid dynamics, in particular for the Euler and Navier-Stokes equations and in relation to the Kolmogorov theory of turbulence.