Programa XVII ENAMA

 
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Minicursos

Partial Differential Equations of Mixed Type
Gui-Qiang Chen, Mathematical Institute, University of Oxford

Three of the fundamental types of partial differential equations (PDEs) are elliptic, hyperbolic, and parabolic, following the standard classification for linear PDEs. While linear theories of PDEs of these types have been considerably better developed, many nonlinear PDEs encountered in Mathematics and Science are of mixed type naturally. Solving longstanding fundamental problems often hinges on a deep understanding of such nonlinear PDEs, particularly those of mixed elliptic-hyperbolic type. In this mincourse, we present the underlying connections between nonlinear PDEs of mixed type and fundamental problems ranging from the challenging Riemann problem to the intriguing isometric embedding problem. We analyse such examples including the Morawetz problem for transonic flow (esp. airfoil problems), the multidimensional Riemann problem (formulated first by Riemann in 1860 for the one-dimensional case) and related shock reflection/diffraction problems in fluid dynamics (governed by the compressible Euler equations), and the isometric embedding problem in differential geometry (governed by the Gauss-Codazzi-Ricci equations).  We also discuss both historical perspectives and recent developments in the analysis of these nonlinear PDEs, emphasising the development and identification of unified approaches, ideas, and techniques for addressing mixed-type problems. Some further developments and open problems in this direction will also be addressed.

Files: Chen-1 / Chen-2

Control and Machine Learning
Enrique Zuazua, Friedrich-Alexander Universität Erlangen-Nürnberg

Control theory and Machine Learning share common objectives, as evident in Norbert Wiener’s definition of “Cybernetics” as “The science of control and communication in animals and machines.”

The synergy between these fields is reciprocal. Control theory tools enhance our comprehension of the efficacy of certain Machine Learning algorithms and offer insights for their enhancement. However, this often bounces intricate queries back.

The interplay between Control and Machine Learning opens up a new captivating scientific lanscaüpe tp be explored but this can be a labyrinthine task. And this is part of the overall ambitious program of developing Digital Twins technologies.

In this talk, we will present some of the contributions from our team at the interface between Control and Machine Learning that can contribute to this ambitious complex task.

We will discuss some neural network architectures, whose success for Supervised Learning can be understood from a control perspective and explain how their dimension and complexity can be minimized. The attention mechanism of transformers will also be analyzed.

We will also present some challenging open problems.

File: Zuazua

Geometric Measure Theory and Plateau's problem
Giovanni Alberti, University of Pisa.
In its simplest form, Plateau’s problem concerns first of all the existence of a surface of minimal area among all surfaces spanned by a prescribed boundary (in general, the size of the surface and that of the ambient space are arbitrary). Proving the existence of such a surface has been one of the key problems of Geometric Measure Theory for almost a century, and has led to thedevelopment of several approaches,  all based on suitable compactifications of the space of surfaces. Since then, these results have been applied to many other problems in the Calculus of Variations and in Analysis. In these lectures I will briefly illustrate the fundamental concepts of Geometric Measure Theory that  lie behind these approaches, starting from the notion of rectifiable set, and give a summary description of some of these compactification results.

File: Alberti

 

Conferências

Teoremas do tipo Banach-Stone para álgebras de germes holomorfos em espaços de Banach
Daniela Vieira – Universidade de São Paulo (USP-SP)

O teorema clássico de Banach-Stone (1932) afirma que, se K e L são espaços topologicos com- pactos de Hausdorff, então K e L são homeomorfos se, e somente se, os espaços de funções contínuas C(K) e C(L) são isométricos. A versão algébrica deste teorema foi provada por Gelfand & Kolmogo- roff em 1939, quando eles mostram que K e L são homeomorfos se, e somente se, C(K) e C(L) são isomorfas como álgebras.

Nesta palestra, apresentaremos resultados desta natureza para álgebras de germes holomorfos em espaços de Banach. Tanto no caso do teorema clássico, bem como no contexto em questão, tais resultados estão relacionados com o estudo do espectro de tais álgebras, no sentido de que uma boa descrição do espectro pode ajudar a obter resultados do tipo Banach-Stone. No âmbito da nossa pesquisa, ambientes favoráveis a uma boa descrição do espectro são espaços de Banach com algum tipo de propriedade de aproximação, e conjuntos compactos com alguma propriedade geométrica.

Como exemplo, um dos resultados que serão apresentados afirma que se E e F são espaços de Banach separáveis com propriedade de aproximação, e K e L são subconjuntos compactos de E e F , respectivamente, de tal forma que K e L são polinomialmente convexos, equilibrados e determinantes, então as álgebras de germes holomorfos H(K) e H(L) são topologicamente isomorfas se, e somente se, K e L são biholomorficamente equivalentes, Isto é, existe uma bijeção holomorfa φ : K → L tal que sua inversa também é uma aplicação holomorfa.

Classification of radial solutions }{for fully nonlinear equations and systems
Liliane Maia – Universidade de Brasília (UnB)

We will present some recent results obtained on the existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations and system of equations involving Pucci extremal operators. Our study is entirely based on the analysis of the dynamics induced by an au- tonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. This is a work in collaboration with Gabrielle Nornberg (Universidad de Chile, Chile) and Filomena Pacella (Sapienza Uni- versità di Roma, Italy).

References

  1. [1]  Liliane Maia, Gabrielle Nornberg and Filomena Pacella, A dynamical sys- tem approach to a class of radial weighted fully nonlinear equations, Communications in Partial Differential Equations, 46(4):573?610, 2021. https://doi.org/10.1080/03605302.2020.1849281

  2. [2]  Liliane Maia, Gabrielle Nornberg and Filomena Pacella, Existence, nonexis- tence and uniqueness for Lane-Emden type fully nonlinear systems, Nonlinearity, 36(3): 1510, 2023. https://dx.doi.org/10.1088/1361-6544/acb399

  3. [3]  Liliane Maia, Gabrielle Nornberg and Filomena Pacella, Classification of radial solutions for fully nonlinear equations with Hardy potential, DCDS Series A 2024 (online). doi:10.3934/dcds.2024025